finsler-lagrange geometry. applications to dynamical systems ioan...

FINSLER-LAGRANGE GEOMETRY.Applications to dynamical systems

Ioan Bucataru, Radu Miron

July 6, 2007

In memory of three great Finslerists:Shiing-Shen Chern, Mendel Haimovici and Makoto Matsumoto.

This work has been supported by grants CEEX ET 3174/2005-2007 andCEEX M III 12595/2007.

Contents

Part I Differential Geometry of Tangent Bundles

1 Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Tangent and cotangent bundles of a manifold . . . . . . . . . . 31.2 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Vertical subbundle . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Vertical and complete lifts . . . . . . . . . . . . . . . . . . . . . 131.5 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Nonlinear Connections . . . . . . . . . . . . . . . . . . . . . . 192.1 Nonlinear connections on a manifold . . . . . . . . . . . . . . . 202.2 Local representations of a connection . . . . . . . . . . . . . . . 232.3 Nonlinear connections on the tangent bundle . . . . . . . . . . 242.4 Characterizations of nonlinear connections . . . . . . . . . . . . 272.5 d-tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Curvature and torsion of a nonlinear connection . . . . . . . . . 322.7 Dynamical covariant derivative . . . . . . . . . . . . . . . . . . 332.8 Autoparallel curves . . . . . . . . . . . . . . . . . . . . . . . . . 362.9 Symmetries of a nonlinear connection . . . . . . . . . . . . . . 372.10 Homogeneous connections and linear connections . . . . . . . . 40

3 N-Linear Connections . . . . . . . . . . . . . . . . . . . . . . . 433.1 N -linear connections . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Berwald connection . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Horizontal and vertical covariant derivatives . . . . . . . . . . . 473.4 Torsion of an N -linear connection . . . . . . . . . . . . . . . . . 483.5 Curvature of an N -linear connection . . . . . . . . . . . . . . . 503.6 N -linear connections induced by a complete parallelism . . . . 523.7 Structure equations of an N -linear connection . . . . . . . . . . 543.8 Geodesics of an N -linear connection . . . . . . . . . . . . . . . 573.9 Homogeneous Berwald connection . . . . . . . . . . . . . . . . 59

viii Contents

4 Second Order Differential Equations . . . . . . . . . . . . . . 614.1 Second order differential vector field . . . . . . . . . . . . . . . 614.2 Nonlinear connections and semisprays . . . . . . . . . . . . . . 634.3 Berwald connection of a semispray . . . . . . . . . . . . . . . . 664.4 Jacobi equations of a semispray . . . . . . . . . . . . . . . . . . 684.5 Symmetries of a semispray . . . . . . . . . . . . . . . . . . . . . 704.6 Geometric invariants of an SODE . . . . . . . . . . . . . . . . . 724.7 Homogeneous SODE . . . . . . . . . . . . . . . . . . . . . . . . 73

Part II Finsler-Lagrange geometry

5 Finsler Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.1 Finsler metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Geometric objects of a Finsler space . . . . . . . . . . . . . . . 825.3 Geodesics of a Finsler space . . . . . . . . . . . . . . . . . . . . 865.4 Geodesic spray and symmetries . . . . . . . . . . . . . . . . . . 905.5 Cartan nonlinear connection . . . . . . . . . . . . . . . . . . . . 945.6 Finsler linear connections . . . . . . . . . . . . . . . . . . . . . 995.7 Geodesic deviation and symmetries . . . . . . . . . . . . . . . . 1055.8 Two dimensional Finsler space . . . . . . . . . . . . . . . . . . 1085.9 Three dimensional Finsler space . . . . . . . . . . . . . . . . . . 1105.10 Randers spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.11 Ingarden spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.12 Anisotropic inhomogeneous media . . . . . . . . . . . . . . . . 122

6 Lagrange Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.1 Lagrange metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.2 Geometric objects of a Lagrange space . . . . . . . . . . . . . . 1336.3 Variational problem . . . . . . . . . . . . . . . . . . . . . . . . 1356.4 Canonical semispray . . . . . . . . . . . . . . . . . . . . . . . . 1376.5 Symmetries and Noether type theorems . . . . . . . . . . . . . 1406.6 Canonical nonlinear connection . . . . . . . . . . . . . . . . . . 1436.7 Almost Kahlerian model of a Lagrange space . . . . . . . . . . 1486.8 Metric N -linear connections . . . . . . . . . . . . . . . . . . . . 1516.9 Almost Finslerian Lagrange spaces . . . . . . . . . . . . . . . . 1556.10 Geometry of ϕ-Lagrangians . . . . . . . . . . . . . . . . . . . . 1596.11 Gravitational and electromagnetic fields . . . . . . . . . . . . . 1636.12 Einstein equations of Lagrange spaces . . . . . . . . . . . . . . 166

Contents ix

7 Generalized Lagrange spaces . . . . . . . . . . . . . . . . . . . 1737.1 Metric classes on TM . . . . . . . . . . . . . . . . . . . . . . . 1747.2 Metric nonlinear connections and semisprays . . . . . . . . . . 1797.3 Metric N -linear connections . . . . . . . . . . . . . . . . . . . . 1827.4 Regular metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.5 Variational problem for regular GL-metrics . . . . . . . . . . . 1897.6 Deformations of Finsler metrics . . . . . . . . . . . . . . . . . . 1907.7 Connections for a deformed Finsler metric . . . . . . . . . . . . 1957.8 New metric classes . . . . . . . . . . . . . . . . . . . . . . . . . 1987.9 Nonholonomic Finsler frames . . . . . . . . . . . . . . . . . . . 200

Part III Dynamical systems

8 Dynamical Systems. Lagrangian Geometries . . . . . . . . . 2078.1 Riemannian mechanical systems . . . . . . . . . . . . . . . . . . 2098.2 Finslerian mechanical systems . . . . . . . . . . . . . . . . . . . 2118.3 Nonlinear connection of a Finslerian mechanical system . . . . 2148.4 Metric N -linear connection of a Finslerian mechanical system . 2158.5 Electromagnetic tensors of a Finslerian mechanical system . . . 2178.6 Almost Hermitian model of a Finslerian mechanical system . . 2198.7 Lagrangian mechanical systems . . . . . . . . . . . . . . . . . . 2208.8 Almost Hermitian model of a Lagrangian mechanical system . 2268.9 Generalized Lagrangian mechanical systems . . . . . . . . . . . 227

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Preface

The present book is devoted to Finsler-Lagrange geometry and its applica-tions to dynamical systems. The geometries of Finsler, Lagrange and Hamil-ton spaces were studied by different schools lead by M. Matsumoto (Japan),S.S. Chern (USA), P.L. Antonelli (Canada), L. Tamassy (Hungary), R. Miron(Romania) and others. Remarkable applications of these geometries are dueto G.S. Asanov (Russia), R.G. Beil and R.M. Santilli (USA), R.S. Ingarden(Poland), K. Kondo and S. Ikeda (Japan).

Among the very few books regarding Finsler geometry, this book presents,from our point of view, the geometry of Finsler spaces as a subgeometry ofLagrange spaces, which can be viewed as a subgeometry of the differentialgeometry of tangent bundle. Hence, the geometry of the tangent bundle isa natural framework for the Finsler-Lagrange geometry we develop in thisbook, while Finsler-Lagrange geometry is presented as a natural frameworkfor applications.

This monograph is a natural and necessary continuation of the authors’work on the theory of Lagrange spaces published by Kluwer, in the FTPHseries, in the volumes [21], [132], [124], [130] or by Hadronic Press in the vol-ume [125]. It contains some new important chapters as: a geometrical theoryof connections, a geometrical theory of systems of differential equations anddynamical systems. Following some R.M. Santilli’s ideas from his treatise ofAnalytical Mechanics, [166], [167], we define the most general concept of La-grangian (and in particular Finslerian) mechanical system, where the externalforces depend also of velocity coordinates. Thus, the corresponding dynamicalsystem can be introduced only on the phase space as a canonical semispray.The geometry of this semispray is the geometry of the considered Lagrangianmechanical system.

Differential geometry of the total space of the tangent bundle of a mani-fold has its roots in various problems from Differential Equations, Calculusof Variations, Mechanics, Theoretical Physics and Biology. Nowadays, it is adistinct domain of differential geometry and has important applications in thetheory of physical fields and special problems from mathematical Biology.

This book is devoted to the geometry of Finsler and Lagrange spaces and

xii Preface

their applications to the geometrical theory of Finslerian and Lagrangian dy-namical systems, as being an intrinsic part of the geometry of the total spaceTM of a differentiable manifold M .

The differential manifold TM has a special geometric structure: it is ori-entable, admits a globally defined vector field, the Liouville vector field C,possesses an integrable distribution, the vertical distribution V TM and anintegrable tensor field of (1,1)-type, the tangent structure J . We refer to theseas to the natural geometric structures on TM . Due to these aspects of thegeometry of the manifold TM we expect that some other geometric structures,like connections, systems of second order differential equations, metric struc-tures and symplectic structures, cannot be studied on TM without requiringsome compatibility conditions with the natural geometric structures on TM .However, it is not possible to study these geometric structures on TM by us-ing methods borrowed from the geometry of the base manifold M . This is thereason we have to introduce new concepts that are specific to TM : semispraysand nonlinear connections and to investigate the geometrical properties of thephase space TM , determined by the configuration space M , [166].

Indeed, the geometry of a system of second order differential equationsis the geometry of a semispray. A semispray S is a globally defined vectorfield on TM such that JS = C. If a semispray S is given, then one canassociate to it different geometric objects like nonlinear connections and N-linear connections. Based on these entities we can develop the differentialgeometry of the pair (TM, S). Such a geometry is imposed by a geometricstudy of systems of second order differential equations, SODE, that appear inthe theory of dynamical systems or the theory of mechanical systems.

Different metric structures on TM are induced, in physical examples, eitherby regular Lagrangians, by Finsler metrics, or by generalized Lagrange metricswhich at their turn can be induced by Ehlers-Pirani-Schield axiomatic systemor by different metric structures from Relativistic Optics. Metric geometry ofTM that corresponds to these metric structures determines the backgroundof Lagrangian geometries we discuss in this book. Within these geometries,there are some particular aspects we want to emphasize:

1) One can build the geometry of a Lagrange space from the principles ofAnalytical Mechanics.

2) The geometry of a Finsler space is a particular form of the Lagrangegeometry with specific properties due to the homogeneity condition. Hence,one can build this geometry from the principles of Theoretical Mechanics.

3) One cannot study the geometry of generalized Lagrange spaces usingmethods from Riemannian geometry, one has to approach it as metric geom-etry on TM .

4) Geometric properties from Calculus of Variations can be obtained bymeans of an associated semispray and its differential geometry.

Preface xiii

5) A geometric theory for Lagrangian dynamical systems can be obtainedif we use differential geometry of the manifold TM . This problem is studiedand published for the first time in this monograph.

6) It is easy to extend this theory to the differential manifold T kM , thespace of accelerations of order k > 1.

These considerations prove that the present book is a new introduction inFinsler-Lagrange geometry, having new openings for applications.

Historically speaking, a systematic study of the differential geometry oftangent bundles started in 1960’s and 1970’s with the work of P. Dombrowski[76], S. Kobayashi and K. Nomizu [101], K. Yano and S. Ishihara [194]. An im-portant contribution to the geometry of tangent bundle is due to M. Crampin[66] and J. Grifone [80] who associated a nonlinear connection on the tangentbundle to a system of second order differential equations on a manifold. Also,we refer to work of V. Oproiu [154] for the geometry of the tangent bundle.Since then, theories of vertical and complete lifts and of nonlinear connectionshave been studied using the modern apparatus of differential geometry. Forthe theory of different lifts from a manifold to associated vector bundles werefer to the work of V. Cruceanu [71] and K. Yano and S. Ishihara [194]. Forthe theory of nonlinear connections and compatible linear connection we referto the work of R. Miron and M. Anastasiei [130] and [131]. A rigorous investi-gations of the differential geometry of the total space of a vector bundle can befound in R. Miron and M. Anastasiei [130] and [131]. Also the geometry of thetotal space of a covector bundle appears in the book of R. Miron, D. Hrimiuc,H. Shimada and S. Sabau [138]. The notion of generalized Lagrange spaceswas introduced and studied by R. Miron in [119]. Systematic studies regard-ing geometric objects and covariant derivatives one can associate to a systemof second order differential equations have been done by M. Crampin et al.[67], O. Krupkova [105], B. Lackey [107], W. Sarlet [168], P.L. Antonelli andI. Bucataru [21]. Geometric theories for systems of higher order differentialequations were proposed by G.B. Byrnes [58], I. Bucataru [53], M. Crampin,W. Sarlet and F. Cantrijn [68], M. de Leon and P.R. Rodrigues [109]. R. Mironin [124] introduced and studied the notion of higher order Lagrange spaces.

In the first part of this book, we introduce the geometry of the first ordertangent bundle of a manifold. Within this context, we investigate — usinga modern apparatus of differential geometry — geometric objects such aslifts, nonlinear connections, semisprays and N -linear connections. We needthese concepts for the second part, where a metric compatibility of all thesestructures is studied.

Since most of the geometric objects we study in this book live on the totalspace of the tangent bundle of a manifold, the first chapter is dedicated tothe geometry of the tangent bundle. In this chapter, we introduce and pay aspecial attention to natural geometric objects that live on the total space of the

xiv Preface

tangent bundle, such as vertical distribution, Liouville vector field and tangentstructure. Natural relations between the base manifold and the tangent spaceare studied using vertical and complete lifts and the natural projection of thetangent bundle.

In the geometry of the tangent bundle, and particularly in Finsler andLagrange geometries, an important role is played by the notion of horizontaldistribution and its associated concept of nonlinear connection. Despite thefact that a nonlinear connection lives on the total space of the tangent bundleof a manifold, it can be introduced in the same manner as a linear connection,using the concept of parallel transport. We shall see in the second chapter thata parallel transport always defines a connection, which in general is nonlin-ear. If additional conditions are required for the parallel transport we obtainhomogeneous connection and linear connections.

The geometry of horizontal distributions and the associated nonlinear con-nections are studied in chapter two. Using geometric structures like almostproduct structure, almost complex structure, adjoint structure, connectionmap, horizontal lift, we provide characterization for the existence of a nonlin-ear connection. The relation between the integrability of these structures andthe integrability of the nonlinear connection is studied. Then we prove thateach connection, generated by a parallelism on the base manifold, induces anonlinear connection on the tangent space. We also prove that each nonlinearconnection on the tangent space is the lift of such a connection from the basemanifold.

The disadvantage of “nonlinearity” for nonlinear connections induced bya parallel transport can be removed by considering its lift to an N -linear con-nection on the tangent bundle. The price we pay for this is that sometimes itis not easy to go back to the base manifold and study its geometry. However,all geometric objects we derive from a nonlinear connection on the tangentbundle are linear. An important object we derive from a nonlinear connectionis a linear connection on the tangent bundle that preserves the vertical andhorizontal distributions. This is the so called Berwald connection studied forthe first time by L. Berwald in [44]. In chapter three we pay attention to theclass of linear connections on tangent bundles that preserve both horizontaland vertical distributions. For such linear connections we determine the struc-ture equations and study their geodesics. We determine the relation betweenthe geodesics of a nonlinear connection and the geodesics of an associatedN -linear connection.

The geometry of systems of second order differential equations, which westudy in chapter four, is closely related to the geometry of nonlinear con-nections. We shall see that all geometric invariants, which characterize thesolutions of a system of second order differential equations, can be determinedfrom an associated nonlinear connection. This theory is called KCC-theory

Preface xv

by P.L. Antonelli in [23], based on work of its initiators, D.D. Kosambi [103],E. Cartan [61] and S.S. Chern [64]. A recent treatment of KCC-theory canbe found in the work of P.L. Antonelli and I. Bucataru [20] and [21]. In thistheory, integral curves of a system of SODE are viewed as geodesics for anassociated linear connection on the tangent bundle. Hence we can use thetheory developed in chapter three for the geometry of a system of SODE.

In the second part of the book we study the geometry of the total spaceof the tangent bundle if a metric d-tensor, which is a metric structure onthe vertical subbundle, is given. If such a metric tensor is derived from thefundamental function of a Finsler space or Lagrange space, then its geometryis the geometry of the corresponding Finsler or Lagrange space, studied inchapters five and six. For the geometry of Finsler and Lagrange spaces asa subgeometry of the geometry of the tangent bundle, we refer to work ofR. Miron [119], [121] and [130] and J. Kern [96]. Important contributions tothe geometry of Finsler spaces were obtained by M. Abate and G. Patrizio[1], D. Bao, S.S. Chern and Z. Shen [32], A. Bejancu [39], L. Berwald [43],H. Busemann [57], E. Cartan [62], M. Haimovici [81], M. Matsumoto [114],R. Miron and M. Anastasiei [130], H. Rund [162]. The complex Finsler andLagrange geometry by Gh. Munteanu in [149].

For Finsler and Lagrange spaces we give conditions that uniquely de-termine the geometric objects as semispray, nonlinear connection and N-linear connections. All linear connections that are usually associated with aFinsler space, Berwald, Cartan, Chern-Rund, and Hashiguchi connections areuniquely determined using a system of axioms for each of them. The canoni-cal nonlinear connection of a Finsler or Lagrange space is uniquely determinedby two compatibility conditions: one with the metric structure and one withthe symplectic structure of the space. For both Finsler and Lagrange spacesif the canonical semispray is given, we derive the whole family of nonlinearconnections that are compatible with the metric structure. This is part of thesymplectic geometry we develop for Finsler and Lagrange spaces.

For a Finsler space, its geodesics with the arclength parameterization co-incide with the integral curves of the geodesic spray, with the autoparallelcurves of the canonical nonlinear connection; even more, they coincide withgeodesic curves of Berwald, Cartan, Chern-Rund, or Hashiguchi connection.Consequently, we can use the theory developed in chapters two, three andfour to investigate these geodesics, their variation and symmetries. A specialattention is paid to the Noether-type theorems of Finsler geometry. If theFinsler space is two or three-dimensional, we study also the stability of thegeodesics as it has been done in [22].

If the metric structure, which we refer to as a generalized Lagrange metric,is not reducible to a Finsler or Lagrange space, in general, there are no canon-ical semisprays and nonlinear connection one can associate to such a space.

xvi Preface

However, a compatibility condition between the metric structure, semispraysand induced nonlinear connection is studied. We focus our attention to theparticular cases when a generalized Lagrange metric has a canonical semispray.These are the regular generalized Lagrange metrics studied by R. Miron, [119],S. Watanabe and F. Ikeda, [192], and J. Szilasi [179].

Important applications of the geometry of Finsler and Lagrange spaces aredue to G. Randers [159], P.L. Antonelli, R.S. Ingarden and M. Matsumoto[23], R.G. Beil [37], R.M. Santilli [166], S. Ikeda [175], G.S. Asanov [28], A.K.Aringazin [30], S. Rutz [164].

The theory we developed in the first two parts of the book has good appli-cations for a geometric study of dynamical systems determined by mechanicalLagrangian systems. In the last part of the book we investigate Riemannian,Finslerian and Lagrangian mechanical systems, whose evolution curves aregiven, on the phase space TM , by Lagrange equations of the form:

ddt

(∂L

∂yi

)− ∂L

∂xi= Fi(x, y), yi =

dxi

dt= xi,

where Fi(x, x) are the external forces of the system. If Fi are the componentsof a globally defined d-covector field on TM , then one can associate to themechanical system Σ = (M,L, Fi) a globally defined vector field S on TM ,which will be called the canonical semispray, or the dynamical system asso-ciated to Σ. The geometry of the Lagrangian mechanical system Σ is thegeometry of the phase space manifold TM endowed with the semispray S. Allgeometric objects one can derive from S, such as a nonlinear connection, anN-linear metric connection, will be used to study the system Σ. The stabilityof Σ is investigated as the stability of the integral curves of S. The dissipativecase is studied and we provide examples of dissipative Lagrangian MechanicalSystems.

If one lift the above mentioned theory to TM , we obtain an almost Her-mitian model (TM,G,F) of the considered mechanical system Σ. The geo-metric theory of Σ can be deduced from that of the almost Hermitian space(TM,G,F). The theory presented in this chapter is based on the papers [55],[140].

In our opinion, the book is useful to a large class of readers: graduatestudents, mathematicians, physicists and to everybody else interested in thesubject of Finsler-Lagrange geometry and its applications.

We are aware that we could have not reached this form of the book withoutthe enormous work we refer to in the Bibliography. We want to address ourthanks to all authors mentioned there and to everybody else we forgot tomention, without any intention, in our Bibliography.

We would like to acknowledge the direct improvement of this book that has

Preface xvii

resulted from numerous suggestions and comments from M. Anastasiei, P.L.Antonelli, A. Balmus, D. Bao, R.G. Beil, M. Brailovschi, M. Crasmareanu, S.V. Sabau, R.M. Santilli and H. Shimada.

Iasi, Summer 2007

Ioan BucataruFaculty of Mathematics“Al.I. Cuza University” Iasiemail: [email protected]://www.math.uaic.ro/∼bucataru

Radu MironFaculty of Mathematics“Al.I. Cuza University” Iasiemail: [email protected]

Part I

Differential Geometry ofTangent Bundles

Chapter 1

Tangent Bundles

We start by studying the geometry of the tangent bundle (TM, π,M) over asmooth, real, n-dimensional manifold M . This geometry is one of the mostimportant field of modern differential geometry. The tangent bundle TMcarries some natural geometric object fields like: the Liouville vector fieldC, the tangent structure J , the vertical distribution V TM . They allow usto introduce the notion of a semispray and a nonlinear connection in thenext chapters. By studying the compatibility of semisprays and nonlinearconnections with a metric structure, we shall develop later a metric geometryof the tangent bundle. Particular cases of this geometry are given by thegeometry of Lagrange and Finsler spaces.

In this book all geometric objects and mappings are considered to be ofC∞-class, and we shall express this by using the words “differentiable” or“smooth”.

We shall see that a rich geometry of the manifold TM can be developedfrom the notion of a nonlinear connection. It is more natural for a nonlinearconnection to be defined on the tangent space of a manifold rather than on thebase manifold. However, in chapter two we shall define a nonlinear connectionas usually one does for a linear connection, starting from a parallel transport.Then, we shall lift this to what is usually called a nonlinear connection onthe total space of the tangent bundle of a manifold. There, we shall studyrelations between a nonlinear connection and some natural geometric objectsone can define on the tangent bundle of a manifold.

1.1 Tangent and cotangent bundles of a manifold

In this section, the tangent and cotangent bundles over a real, finite dimen-sional manifold are presented.

We consider M a real, n-dimensional manifold, with A = (Uα, φα)α∈I

4 Chapter 1. Tangent Bundles

an atlas of C∞-class on M . For every local chart (U, φ) at p ∈ U ⊂ Mwe denote by (xi)i=1,n the local coordinates induced by φ, which means thatφ(p) = (xi(p)) ∈ Rn. We shall denote this by φ = (xi) or (U, φ = (xi)). If wehave two local charts (U, φ = (xi)) and (V, ψ = (xi)), then their compatibilitymeans that ψφ−1 : (xi) ∈ φ(U∩V ) 7→ xi(xj) ∈ ψ(U∩V ) is a diffeomorphism.This will imply that rank(∂xi/∂xj) = n.

For each point p ∈ M , we introduce now the tangent space TpM at p tothe manifold M . Let Cp(M) = σ : I ⊂ R −→ M, σ is smooth and σ(0) = p.Two curves ρ and σ ∈ Cp(M) have a contact of order 1 or the same tangent lineat the point p if there is a local chart (U, φ = (xi)) at p such that d0(ϕ ρ) =d0(ϕ σ). The relation “contact of order 1” does not depend on the localchart we choose and it is an equivalence on Cp(M). An equivalence class willbe denoted by [σ]p and it will be called a tangent vector at the point p ∈ M .The set of all tangent vectors at the point p ∈ M will be denoted by TpMand it is called the tangent space to the manifold M at p ∈ M . We considerthe union of tangent spaces at all points of M , TM = ∪p∈MTpM. Considerthe canonical projection π : TM 7→ M defined by π([σ]p) = p. Clearly, π is asurjection and π−1(p) = TpM , ∀p ∈ M .

The set TM carries a natural differentiable structure, induced by that ofthe base manifold M , such that the natural projection π is a differentiablesubmersion. This differentiable structure will be described bellow. First, wepresent the structure of locally trivial vector bundle one can introduce on TM .

For a local chart (U, φ = (xi)) on M , we define τU : π−1(U) −→ U × Rn

through τU ([σ]p) = (σ(0), d0(ϕσ)). The n-dimensional Euclidean space Rn iscalled the typical fibre. The mapping τU is a bijection and satisfies pr1τU = π,which means that the following diagram is commutative:

π−1(U)

π%%LLLLLLLLLLL

τU // U × Rn

pr1

²²U

The pair (U, τU ) is called a trivialization chart. As pr1 τU = π, the mapτU preserves the fibres. Consequently, we obtain that the restriction of thebijection τU to the fibre at p, π−1(p) = TpM , τU,p : TpM −→ Rn is also abijection. If we consider the composition

τU,p τ−1V,p = dψ(p)(ϕ ψ−1) : Rn −→ Rn,

we obtain a linear isomorphism of the typical fibre Rn which can be identifiedwith an element of the Lie group GL(n,R). The following functions

gUV : p ∈ U ∩ V 7→ gUV (p) = τU,p τ−1V,p ∈ Gl(n,Rn)

1.1. Tangent and cotangent bundles of a manifold 5

are called structural functions. It is a straightforward calculation to checkthat the following properties are true for the structural functions:

1) gUV (p) gV W (p) = gUW (p), ∀p ∈ U ∩ V ∩W ;

2) gUU (p) = IdRn , ∀p ∈ U ;

3) g−1UV (p) = gV U (p), ∀p ∈ U ∩ V .

Here U, V and W are domains of local charts.The pair (π−1(U), Φ), where Φ = (ϕ × IdRn) τU , is a local chart on

TM to which we refer to as an induced local chart. Next, on TM , weshall consider only induced local charts. Therefore, a differentiable atlasAM = (Uα, ϕα)α∈I of the differentiable manifold M determines a differen-tiable atlas ATM = (π−1(Uα), Φα = (ϕα×IdRn)τUα)α∈I on TM . ThereforeTM is a differentiable manifold of dimension 2n and the canonical projection πis a differentiable submersion. This implies also that (TM, π,M,Rn, GL(n,R))is a differentiable vector bundle, with typical fibre Rn and structural groupGL(n,R). We call this fibre bundle the tangent bundle of the manifold Mand we refer to it sometimes by (TM, π,M).

Let us fix now a local chart (U, φ = (xi)) at a point p ∈ M . Any curveσ ∈ Cp(M) is represented in the given local chart by xi = xi(t), t ∈ I,φ(p) = xi(0). Then the tangent vector [σ]p is determined by the coefficients

xi = xi(0), yi =dxi

dt

∣∣∣∣t=0

.

Then the coordinates of [σ]p induced by the local chart (π−1(U),Φ) are givenby Φ([σ]p) = (xi, yi) ∈ R2n, [σ]p ∈ π−1(U). Such an induced local chartwill be denoted by (π−1(U),Φ = (xi, yi)). With respect to the induced localcoordinates, the canonical submersion has the expression π : (xi, yi) 7→ (xi).At each point p ∈ M , the fibre of this vector bundle is TpM , which is a linearn-dimensional space isomorph with the typical fibre Rn. This isomorphism isinduced by a local chart and it is explained bellow.

Every local chart (U, φ) at p ∈ M induces an isomorphism τU,p : TpM →Rn, such that (τU,pτ−1

V,p)([σ]p) = dψ(p)(φψ−1)([σ]p), where (V, ψ) it is anotherlocal chart at p ∈ M . In local coordinates, if φ = (xi) and ψ = (xi), then (τU,pτ−1V,p)(y

i) = yj(∂xi/∂xj). We remark here that there is no canonical way ofdefining isomorphisms between tangent spaces TpM and Rn, and consequentlywe cannot define natural isomorphisms between two tangent spaces TpM andTqM .

Let (U, φ = (xi)) be a local chart at p ∈ M and τU,p : TpM → Rn theinduced isomorphism. If eii=1,n is the natural basis of Rn, we denote by


∂/∂xi|p = τ−1U,p(ei). Then ∂/∂xi|pi=1,n is called the natural basis of TpM .

Consider a vector [σ]p ∈ TpM , with σ(t) = (xi(t)). Then, one can expressthe vector as [σ]p = yi∂/∂xi|p, where yi = dxi/dt|t=0. This is equivalent toτU,p([σ]p) = yiei.

For every p ∈ M , one can define the cotangent space T ∗p M at p to Mas the dual space of the tangent space TpM . Hence, T ∗p M = ωp : TpM −→R, ωp is linear. We consider the union T ∗M = ∪p∈MT ∗p M of cotangent spacesto M .

The space T ∗M carries a differentiable structure of C∞-class and dimen-sion 2n. The set (T ∗M, π∗, M,Rn, GL(n,R)) is a vector bundle, called thecotangent bundle. The canonical submersion π∗ : T ∗M → M is definedby π∗(ω) = q if and only if ω ∈ T ∗q M . Every local chart (U, φ = (xi)) atq ∈ M induces a natural basis dxi|q of the cotangent space T ∗q M such thatdxi|q(∂/∂xj |q) = δi

j .The local coordinates on the cotangent space T ∗M are denoted by (xi, pi),

which means that a covector ωq ∈ T ∗p M can be expressed as ωq = pidxi|q. Thecanonical submersion π∗ : T ∗M → M has the local expression π∗ : (xi, pi) 7→(xi).

If (U, φ = (xi)) and (V, ψ = (xi)) are local charts around p ∈ M , the localcoordinates (xi) and (xi) are related by xi = xi(xj), with rank(∂xi/∂xj) = n.The corresponding change of coordinates on TM , induced by (π−1(U), Φ =(xi, yi)) and (π−1(V ), Ψ = (xi, yi)) is given by:

xi = xi(xj), rank(

∂xi

∂xj

)= n,

yi =∂xi

∂xjyj .

(1.1)

We call (1.1) the change of induced local coordinates formula on TM . Since∂yi/∂yj = ∂xi/∂xj we have that the Jacobian of ΨΦ−1 is always positive (itis equal to det(∂xi/∂xj)2), so TM is an orientable manifold.

The change of local coordinates formula on T ∗M (the corresponding for-mula of (1.1) on T ∗M) is:

xi = xi(xj), rank(

∂xi

∂xj

)= n,

pi =∂xj

∂xipj .

(1.2)

A geometric object defined on the tangent bundle or cotangent bundle of amanifold has to be invariant under the change of coordinates (1.1) or (1.2).

1.2. Tensor fields 7

1.2 Tensor fields

Since at each point p on a manifold M we have defined a linear space TpMone can use these linear spaces to define tensors. Then, if p varies on themanifold, we can define tensor fields. Consider F(M) the set of C∞ realfunctions defined on the manifold M .

If Xp = [σ]p ∈ TpM is a vector on M it acts on functions nearby p ∈ Maccording to the following formula:

Xp(f) =ddt

(f σ)|t=0 =dxi

dt

∣∣∣∣t=0

∂f

∂xi,

where (U,ϕ = (xi)) is a local chart at p. A vector field on M is a smooth mapX : M −→ TM such that πX = IdM , which means that ∀p ∈ M , Xp ∈ TpM .The set of all vector fields over the manifold M is denoted by χ(M) and it isan F(M)-module. We recall here that a vector field X ∈ χ(M) can be viewedalso as a derivation on the ring F(M) of real functions on M . This meansthat if X ∈ χ(M) is a vector field, then one can define X : F(M) −→ F(M),through X(f)(p) = Xp(f). Vector field X has the following properties:

1) X(af + bg) = aX(f) + bX(g), ∀f, g ∈ F(M), ∀a, b ∈ R;

2) X(fg) = X(f)g + fX(g), ∀f, g ∈ F(M).

The above two properties can be used as defining axioms for a vector field ona manifold. If we consider also the Lie bracket of two vector fields [X,Y ](f) =X(Y (f)) − Y (X(f)), then χ(M) is an infinite dimensional real algebra. Avector field X ∈ χ(M) can be expressed locally as X = Xi(∂/∂xi), where Xi

are functions defined on the domain of a local chart (U,ϕ = (xi)). Conse-quently, the Lie bracket of two vector fields X, Y ∈ χ(M) has the followinglocal expression:

[X, Y ] =[Xi ∂

∂xi, Y j ∂

∂xj

]=

(Xi ∂Y j

∂xi− Y i ∂Xj

∂xi

)∂

∂xj.

For a vector field X ∈ χ(M) and a fixed point p ∈ M , there is an open subsetU in M , an open interval I in R that contains 0 and a map ϕ : I ×U −→ M ,such that dϕ(t, q)/dt|t=0 = Xq, ∀q ∈ U . For each fixed t ∈ I, map ϕt =ϕ(t, ·) : U ⊂ M −→ ϕt(U) ⊂ M is a local diffeomorphism on M . It has theproperties that ϕ0(q) = q, ∀q ∈ U and ϕt ϕs = ϕs+t, ∀s, t ∈ I, such thats + t ∈ I. The family of diffeomorphisms ϕt is called the one-parameter groupof transformations (or the flow) induced by X.

An 1-form on the manifold M can be defined as a smooth map ω : M −→T ∗M such that π∗ ω = IdM . The set of 1-forms on M is denoted by Λ1(M)


and it is an F(M)-module. One can define the set of 1-forms also as Λ1(M) =ω : χ(M) −→ F(M), ω is F(M)-linear. The set of q-forms is denoted byΛq(M), where a q-form is a q-F(M)-linear and skew-symmetric map:

ω : χ(M)× · · · × χ(M)︸︷︷︸q-times

−→ F(M).

A tensor field of (r,s)-type is an F(M)-linear map

T : Λ1(M)× · · · × Λ1(M)︸︷︷︸r-times

×χ(M)× · · · × χ(M)︸︷︷︸s-times

−→ F(M)

If one fix a local chart (U,ϕ = (xi)), then a tensor field T determines nr+s

functions defined over U ,

T i1i2···irj1j2···js

:= T

(dxi1 , . . . , dxir ,

∂

∂xj1, . . . ,

∂

∂xjs

).

The nr+s functions T i1i2···irjij2···js

are called the components of the tensor field T

with respect to the local chart (U,ϕ = (xi)). Under a change of coordinatesxi 7→ xj(xi) on M the components of a tensor field T transform as follows:

T i1i2···irj1j2···js

=∂xi1

∂xk1· · · ∂xir

∂xkrT k1k2···kr

l1l2···ls∂xl1

∂xj1· · · ∂xls

∂xjs.(1.3)

If u ∈ TM , we denote by TuTM the tangent space at u to TM . Thisis a 2n-dimensional vector space and the natural basis induced by a localchart (π−1(U), Φ = (xi, yi)) at u is ∂/∂xi|u, ∂/∂yi|ui=1,n. After a change ofcoordinates (1.1) on TM , the natural basis changes as follows:

∂

∂xi

∣∣∣∣u

=∂xj

∂xi(u)

∂

∂xj

∣∣∣∣u

+∂yj

∂xi(u)

∂

∂yj

∣∣∣∣u

, rank(

∂xi

∂xj

)= n,

∂

∂yi

∣∣∣∣u

=∂xj

∂xi(u)

∂

∂yj

∣∣∣∣u

.

(1.4)

A vector Xu ∈ TuTM has the form Xu = Xi(u)(∂/∂xi)|u + Y i(u)(∂/∂yi)|uwith respect to the natural basis. Under a change of coordinates (1.1) on TM ,the coordinates of a vector Xu ∈ TuTM change as follows:

Xi =∂xi

∂xjXj ,

Y i =∂xi

∂xjY j +

∂yi

∂xjXj .

(1.5)

1.3. Vertical subbundle 9

We denote by χ(TM) and F(TM) the set of all vector fields on TM and theset of all real differentiable functions on TM , respectively. Then χ(TM) isan F(TM)-module finite generated and it is an infinite dimensional, real Liealgebra with respect to the Lie bracket.

We remark here that the tangent manifold TTM carries two natural pro-jections. One is the natural projection τ of the tangent bundle (TTM, τ, TM)and the second one is the linear map π∗ induced by π. In local coordinatesthe two projections are expressed as follows:

τ : (x, y, X, Y ) ∈ TTM 7→ (x, y) ∈ TM, and

π∗ : (x, y, X, Y ) ∈ TTM 7→ (x,X) ∈ TM.

If a section for the first projection τ defines a vector field on TM , we shallsee later that a section for both projections defines an important second ordervector field that will be called a semispray.

If T is a (1,1)-type tensor field and X is a vector field then the Frolicker-Nijenhuis bracket of T and X is a (1, 1)-type tensor field [T, X], defined asfollows:

[T, X](Y ) = [T (Y ), X]− T [Y, X].

The Frolicker-Nijenhuis bracket of two (1, 1)-type tensors K and L is a vectorvalued 2-form [K, L] and is defined as follows:

[K, L](X, Y ) = [K(X), L(Y )] + [L(X), K(Y )] + (K L)[X, Y ]

+(L K)[X, Y ]−K[X, L(Y )]−K[L(X), Y ]− L[X,K(Y )]− L[K(X), Y ].

In particular,

12[K,K](X, Y ) = [K(X),K(Y )] + K2[X, Y ]−K[X,K(Y )]−K[K(X), Y ].

The vector valued 2-form NK = (1/2)[K, K] is called the Nijenhuis tensor ofK. It is a (1, 2)-type tensor field, skew symmetric with respect to its vectorarguments. Its vanishing is equivalent to the integrability of the tensor (1, 1)-type tensor K.

1.3 Vertical subbundle

For a natural study of the geometry of the tangent bundle of a manifold thereare two bundles one usually can associate, the vertical subbundle and the pull-back bundle of the tangent bundle through its projection. Even though thereis a natural isomorphism between these two bundles, there are authors thatprefer one or another. The vertical subbundle appears in work of R. Miron and


M. Anastasiei, [130] and A. Bejancu [39]. The pull-back bundle is preferredby D. Bao, S.S. Chern and Z. Shen, [32] and M. Crampin, E. Martinez andW. Sarlet, [67]. In our work we shall use the vertical subbundle of the tangentbundle.

The natural submersion π : TM −→ M determines a simple foliation ofdimension n (and codimension n) on the manifold TM . The leafs of this foli-ation are the tangent spaces TpM = π−1(p), they are embedded submanifoldsof dimension n in TM . If (xi, yi) are local coordinates on TM , then yi arecoordinates for the leafs of the foliation, while xi are transverse coordinatesof the foliation. One can see this from the change of coordinates on TM ,given by formula (1.1). The relation between the geometry of this foliationand the geometry of Finsler spaces was recently studied by A. Bejancu andH.R. Farran in [42].

This foliation induces a regular, n-dimensional and integrable distributionV : u ∈ TM 7→ VuTM ⊂ TuTM , where VuTM are tangent spaces to the leafsof the foliation. We call this the vertical distribution of the tangent bundle.As π : TM → M is a submersion it follows that π∗,u : TuTM → Tπ(u)M isan epimorphism of linear spaces, for ∀u ∈ TM , where π∗,u is the linear mapinduced by π at u ∈ TM . The kernel of this linear map, π∗,u, is exactly thevertical subspace, which means that VuTM = Ker(π∗,u), ∀u ∈ TM . If wedenote by V TM = ∪u∈TMVuTM , then V TM is a subbundle of the tangentbundle (TTM, τ, TM), which we call the vertical subbundle. This subbundlehas the fibre, at each point u ∈ TM , an n-dimensional vector space VuTM ,generated by ∂/∂yi|u. Consequently, most of the geometric objects that liveon this bundle transform, under a change of induced coordinates, in a similarway with corresponding geometric objects from the base manifold M . Suchgeometric objects will be called “distinguished” or d-geometric objects, forshort.

The set of all vertical vector fields on TM is denoted by χv(TM). A verticalvector field has the form X = Xi(x, y)∂/∂yi. As for any two vertical vectorfields X, Y , their Lie bracket is a vertical vector field, [X,Y ] ∈ χv(TM), wehave that χv(TM) is a real Lie subalgebra of χ(TM). An important verticalvector field is C = yi(∂/∂yi). One can check using (1.1) and (1.4) that thevector field C is globally defined on TM . We call this vector field, the Liouvillevector field.

We consider now the pull-back bundle (π∗(TM) = TM×πTM, π∗(π), TM)of the tangent bundle (TM, π, M) through its projection π. For each u ∈ TM ,its fibre at u is π∗(TM)u = Tπ(u)M , which is an n-dimensional linear space,generated by ∂/∂xi|π(u). This fibre Tπ(u)M is isomorph with VuTM , thefibre of the vertical subbundle at point u ∈ TM .

Consider now T ∗uTM the cotangent space of TM at u ∈ TM and denoteby dxi|u, dyi|u the natural cobasis. In other words, dxi|u, dyi|u is the dual

1.3. Vertical subbundle 11

basis of ∂/∂xi|u, ∂/∂yi|u. After a change of local coordinates (1.1) on TM ,the dual basis changes as follows:

dxi =∂xi

∂xjdxj , rank

(∂xi

∂xj

)= n,

dyi =∂xi

∂xjdyj +

∂yi

∂xjdxj .

(1.6)

For each point u ∈ TM , we have from the above formula (1.6) that dxi|u spanan n-dimensional subspace V ∗

u TM of T ∗uTM . This way we determine a regular,n-dimensional, integrable distribution V ∗ : u ∈ TM 7→ V ∗

u TM ⊂ T ∗uTM .

Definition 1.3.1 A vector field Y ∈ χ(TM) is a symmetry of the verticaldistribution V TM if [X,Y ] ∈ χv(TM), ∀X ∈ χv(TM).

Proposition 1.3.1 A vector field Y = Y i(x, y)(∂/∂xi)+ Y i(x, y)(∂/∂yi) is asymmetry of the vertical distribution if and only if the coefficient functions Y i

depend on position only, which means that Y i = Y i(x).

One can reformulate the above proposition by saying that a vector fieldY ∈ χ(TM) is a symmetry of the vertical distribution if and only if the vectorfield Y is projectable, which means that π∗Y = Y i(x)(∂/∂xi) is a vector fieldon the base manifold M .

The total space of the vertical subbundle V TM is an embedded sub-manifold in TTM . The canonical inclusion ι : V TM −→ TTM preservesthe linear structure of the fibres. We look now for a canonical submersionJ : TTM −→ V TM that preserves also the linear structures of the fibres andmakes the following sequence exact:

0 −→ V TMι−→ TTM

J−→ V TM −→ 0.(1.7)

The mapping J is therefore a morphism of vector bundles from TTM to V TMsuch that Ker J = Im ι = V TM . This mapping is called the tangent structureof the tangent bundle (or the vertical endomorphism ) and it is defined asfollows:

J =∂

∂yi⊗ dxi, or

J

(∂

∂xi

)=

∂

∂yiand

J

(∂

∂yi

)= 0.

(1.8)

Using the transformation rules (1.4) and (1.6) one can check that the (1, 1)-type tensor field J is globally defined on TM . For the tangent structure J wehave the following properties:


1) J2 = 0;

2) Ker J = Im J = V TM.

The Nijenhuis tensor field of the tangent structure J is given by the followingformula:

NJ(X,Y ) = [JX, JY ]− J [X, JY ]− J [JX, Y ], ∀X, Y ∈ χ(TM).(1.9)

A direct calculation shows that NJ = 0, which means that the tangent struc-ture J is integrable. From (1.8) one can also see that the tangent structureJ has constant coefficients with respect to the natural basis and cobasis andtherefore it is integrable.

The tangent structure J acts also linearly on vector fields that live on thetangent bundle TM . We can also consider J∗, the cotangent structure thatacts on 1-forms that live on the tangent space TM . The cotangent structureJ∗ is defined by

J∗ = dxi ⊗ ∂

∂yi, or

J∗(dyi) = dxi and

J∗(dxi) = 0.(1.10)

It is globally defined on TM and it has similar properties with the tangentstructure. Similarly as for the tangent structure, we have that NJ∗ = 0,and the cotangent structure J∗ is integrable. From (1.10) one can see thatcotangent structure J∗ has constant coefficients with respect to the naturalbasis and cobasis and therefore it is integrable. The two structures J and J∗

are related through the following formula:

J∗(df)(X) = J(X)(f),(1.11)

where f is an arbitrary function on TM . More generally, one can prove thatfor a vector field X ∈ χ(TM), the Frolicker-Nijenhuis brackets of X and thetangent and cotangent structures are related by

[X,J∗] = [X,J ]∗.

One can extend the action of the cotangent structure J∗ to arbitrary k-forms.If ω is a k-form on TM and X1, ..., Xk are vector fields on TM we define J∗ωthrough

(J∗(ω))(X1, ..., Xk) = ω(J(X1), ..., J(Xk)).(1.12)

With this extension, we have now that J∗ is an F(TM)-linear map on Λk(M).

1.4. Vertical and complete lifts 13

1.4 Vertical and complete lifts

In the geometry of the tangent bundle it is important to extend some geometricobjects that live on the base manifold M to the tangent space TM . This taskcan be done using the lifting process. There are two natural important typesof lifts from M to TM , the vertical and the complete lifts, [71] [194].

For a function f ∈ F(M), we define fv = f π, the vertical lift of f . Afunction f ∈ F(TM) is said to be a basic function if it is the vertical lift of afunction f ∈ F(M), which means that f = fv. A basic function f ∈ F(TM)is constant along the leafs of the vertical foliation.

For every u ∈ TM , we define the linear map lv,u : Tπ(u)M → TuTM as

lv,u

(Xi(π(u))

∂

∂xi

∣∣∣∣π(u)

)= Xi(π(u))

∂

∂yi

∣∣∣∣u

.

We can see that lv,u : Tπ(u)M → VuTM is a linear isomorphism. It is calledthe vertical lift of vectors of the tangent bundle. We may also think to thevertical lift lv as an F(M)-linear map between χ(M) and χ(TM). In thiscase lv is defined as follows: for every vector field X = Xi(∂/∂xi) ∈ χ(M),(lvX)(u) = lv,u(Xπ(u)). The vertical lift of a vector field X ∈ χ(M) will bedenoted also by Xv ∈ χ(TM). It can be written as Xv = (Xi)v(∂/∂yi) ∈χv(TM) for X = Xi(∂/∂xi) ∈ χ(M).

Consider X a vector field on the base manifold M and ϕt its one-parametergroup. Then, the vertical lift Xv has as one-parameter group, the vertical liftϕv

t = ϕt π.For a function f ∈ F(M) we define f c = yi∂f/∂xi its complete lift. The

set of all complete lifts of functions from the base manifold M is a subring ofF(TM). A vector field on TM is perfectly determined if one knows its actionon complete lifts of functions. This means that if X, Y ∈ χ(TM) such thatX(f c) = Y (f c), ∀f ∈ χ(M), then X = Y .

The complete lift Xc of a vector field X = Xi∂/∂xi ∈ χ(M) is defined asfollows:

Xc = (Xi)v ∂

∂xi+ (Xi)c ∂

∂yi.(1.13)

Consider X a vector field on the base manifold M and ϕt its one-parametergroup. Then, the complete lift Xc has as one-parameter group the completelift ϕc

t = yi(∂ϕt/∂xi).The vertical lift ωv of an 1-form ω = ωidxi ∈ Λ1(M) is defined by ωv =

(ωi)vdxi. The complete lift ωc of an 1-form ω is defined as ωc = (ωi)cdxi +(ωi)vdyi.

Next results show that vertical and complete lifts of functions and vectorscan be characterized using tangent and cotangent structures.


Proposition 1.4.11) A function f ∈ F(TM) is the vertical lift of some function from the

base manifold M if and only if J∗(df) = 0.2) A function f ∈ F(TM) is the sum of a complete lift and a vertical lift

of some functions from the base manifold M if and only if dJ∗(df) = 0.

Proof.1) As J∗(df) = (∂f/∂yi)dxi we have that J∗(df) = 0 if and only if

∂f/∂yi = 0, which means that f is constant along the leafs of the verticaldistribution. Consequently f is the vertical lift of some function from the basemanifold M .

2) We have the following expression for dJ∗(df):

dJ∗(df) =∂2f

∂yi∂yjdyj ∧ dxi +

12

(∂2f

∂yi∂xj− ∂2f

∂xi∂yj

)dxj ∧ dxi.

Using this expression we have that dJ∗(df) = 0 if and only if

∂2f

∂yi∂yj= 0 and

∂2f

∂yi∂xj=

∂2f

∂xi∂yj.(1.14)

First equation from (1.14) implies that f has the form f(x, y) = Ai(x)yi+ψ(x).If we ask for this function to satisfy second equations (1.14), then we obtain∂Ai/∂xj = ∂Aj/∂xi. Last equation is equivalent to the fact that Ai is thegradient of some function ϕ ∈ F(M), which means that Ai = ∂ϕ/∂xi. Weconclude now that dJ∗(df) = 0 is equivalent to the following form of f ,f = ϕc + ψv, where ϕ,ψ ∈ F(M). q.e.d.

Proposition 1.4.21) A vector field X ∈ χ(TM) is the vertical lift of some vector field from

the base manifold M if and only if J(X) = 0 and LXJ = 0.2) A vector field X ∈ χ(TM) is the sum of a complete and a vertical lift

of some vector fields from the base manifold M if and only if LXJ = 0.

Proof. With respect to the natural basis we have:

(LXJ)(

∂

∂xi

)=

[X,

∂

∂yi

]− J

[X,

∂

∂xi

]

= −∂Xj

∂yi

∂

∂xj+

(∂Xj

∂xi− ∂Y j

∂yi

)∂

∂yj,

which is zero if and only if

∂Xj

∂yi= 0, which is equivalent to Xj = Xj(x) and

∂Xj

∂xi=

∂Y j

∂yi, which is equivalent to Y j =

∂Xj

∂xiyi + V i(x).

1.5. Homogeneity 15

Consequently, we have that

(LXJ)(

∂

∂xi

)= 0 if and only if X = Xi(x)

∂

∂xi+

∂Xi

∂xj(x)yj ∂

∂yi+ V i(x)

∂

∂yi.

We remark here that condition (LXJ)(∂/∂yi) = 0 is satisfied if X is the sumof a complete and a vertical lift. q.e.d.

For the vertical and the complete lifts we have the following properties:1) (f · g)c = fv · gc + f c · gv, ∀f, g ∈ F(M);2) (fX)v = fvXv, (fX)c = fvXc + f cXv, ∀X ∈ χ(M), ∀f ∈ F(M);3) J(Xc) = Xv, [Xv, Y v] = 0, [Xv, Y c] = [X,Y ]v, [Xc, Y c] = [X, Y ]c;4) (fω)v = fvωv, (fω)c = fvωc + f cωv, ∀ω ∈ Λ1(M), ∀f ∈ F(M);5) J∗(ωc) = ωv, (df)v = d(fv), (df)c = d(f c), ∀ω ∈ Λ1(M), ∀f ∈ F(M).A vector field X ∈ χ(TM) is called basic if there is a vector field X ∈ χ(M)

such that π∗(X) = X. Sometimes we say that vector fields X and X are π-related. In local coordinates we have that a vector field X = Xi(∂/∂xi) +Y i(∂/∂yi) is basic if the coefficients Xi are basic functions. As an example wehave that the complete lift Xc of a vector field X ∈ χ(M) is basic.

According to Proposition 1.3.1, we have that a vector field X ∈ χ(TM) isa symmetry of the vertical distribution if and only if X is a basic vector field.We have also that the complete lift Xc of any vector field X from the basemanifold is a symmetry of the vertical distribution.

1.5 Homogeneity

When studying the geometry of a manifold and its tangent bundle, there aregeometric objects defined along curves. First question one has to answer is ifthese objects depend on the parameterization of such curves. The indepen-dence of parameterization is equivalent to certain homogeneity of the discussedgeometric objects.

For a curve c : t ∈ I ⊂ R 7→ c(t) ∈ M , if we change the parameterizationt 7→ t(s), then the tangent vector dc/dt|t=0 changes according to dc/dt =dc/ds · ds/dt. An affine transformation of parameter t(s) = as + b, a ∈ R∗ fora curve c implies the following change of the coordinates of the tangent vectordc/dt:

xi = xi, yi = ayi.(1.15)

Therefore, the change of coordinates (1.1) on the total space TM preservesthe above transformation for the coordinates of a vector.

Since it is important to decide if geometric objects that can be definedalong curves on M depend or not on their parameterization, we shall studytheir behavior under transformation (1.15).


Denote by TM = TM \ 0 the tangent space with zero section removed.If λ ∈ (0, +∞), we define hλ : TM → TM by hλ(x, y) = (x, λy) and wecall hλ the dilatation of ratio λ. The set of all dilatations hλ, λ ∈ (0, +∞)constitutes a one-parameter group. The vector field that has this group as aone-parameter group is the Liouville vector field and in local coordinates ithas the expression C = yi(∂/∂yi).

Definition 1.5.1 A function f : TM −→ R that is differentiable on TM andcontinuous only on the null section 0 : M −→ TM is called homogeneous oforder r (r ∈ Z) on the fibres of TM (or r-homogeneous with respect to yi) if:

f ha = arf, ∀a ∈ R+.

The following Euler theorem holds true:

Theorem 1.5.1 A function f ∈ F(TM) differentiable on TM and continu-ous only on the null sections is homogeneous of order r if and only if

LCf = yi ∂f

∂yi= rf.(1.16)

The following properties hold true:1) If f1, f2 are r-homogeneous functions, then the function λ1f1 + λ2f2,

λ1, λ2 ∈ R is r -homogeneous, too.2) If f1 is r-homogeneous and f2 is s-homogeneous, then the function f1 ·f2

is (r + s)-homogeneous.

Definition 1.5.2 A vector field X ∈ χ(TM) is r-homogeneous if

X ha = ar−1h∗,a X, ∀a ∈ R+.

An equivalent Euler-type theorem holds true for vector fields:

Theorem 1.5.2 A vector field X ∈ χ(TM) is r-homogeneous if and only if

LCX = [C, X] = (r − 1)X.(1.17)

The following properties hold true:1) The vector fields ∂/∂xi, ∂/∂yi are 1 and 0-homogeneous, respectively.2) If f ∈ F(TM) is s-homogeneous and X ∈ χ(TM) is r-homogeneous

then fX is (s + r)-homogeneous.3) A vector field on TM , X = Xi(∂/∂xi) + Y i(∂/∂yi) is r-homogeneous

if and only if Xi are homogeneous functions of order (r − 1) and Y i arehomogeneous functions of order r.

1.5. Homogeneity 17

4) If X ∈ χ(TM) is an r-homogeneous vector field and f ∈ F(TM) is ans-homogeneous function, then Xf ∈ F(TM) is an (r + s − 1)-homogeneousfunction.

5) The Liouville vector field C is 1-homogeneous.6) If f ∈ F(TM) is an arbitrary s-homogeneous function, then ∂f/∂yi are

(s− 1)-homogeneous functions.7) If X ∈ χ(TM) is an r-homogeneous vector field and Y ∈ χ(TM) is an

s-homogeneous vector field then [X,Y ] is an (r + s − 1)-homogeneous vectorfield. Consequently, we have that the set of 1-homogeneous vector fields is aLie subalgebra of χ(TM).

In the case of q-forms the definition of homogeneity can be stated as follows:

Definition 1.5.3 A q-form ω ∈ Λq(TM) is s-homogeneous if

ω h∗a = asω, ∀a ∈ R+.(1.18)

We have also an Euler type theorem for q-forms:

Theorem 1.5.3 A q-form ω ∈ Λq(TM) is s-homogeneous if and only if

LCω = sω.(1.19)

The following properties hold true:1) If ω ∈ Λq(TM) is s-homogeneous and ω′ ∈ Λq′(TM) is s′-homogeneous,

then ω ∧ ω′ is (s + s′)-homogeneous.2) If ω ∈ Λq(TM) is s-homogeneous and X1, ..., Xq are r-homogeneous

vector fields then ω(X1, ..., Xq) is an (s + r − 1)-homogeneous function.3) dxi (i = 1, ..., n) are 0-homogeneous 1-forms, dyi (i = 1, ..., n) are 1-

homogeneous 1-forms.4) An 1-form on TM , ω = ωidxi + θidyi is r-homogeneous if and only if

ωi are homogeneous functions of order r and θi are homogeneous functions oforder (r − 1).

More generally, a vector field T of (1, s)-type is homogeneous of order r ifand only if LCT = (r−1)T . As an example we have that the tangent structureJ is a (1, 1)-type tensor field homogeneous of order 0. In order to prove this wehave to show that LCJ = −J , which is equivalent to [C, JX]−J [C, X] = −JX,∀X ∈ χ(TM). This can be done by taking X ∈ ∂/∂xi, ∂/∂yi.

Now, we give some examples of homogeneous functions, vectors and 1-forms.

1) The vertical lift fv and f c of a function f ∈ χ(M) are homogeneousfunctions, the first one is homogeneous of order zero, while the second one ishomogeneous of order one.

2) For a vector field X ∈ χ(M), its vertical lift Xv is homogeneous of orderzero and the complete lift Xc is homogeneous of order one.


3) For an 1-form ω ∈ Λ1(M), its vertical lift ωv is homogeneous of orderzero and the complete lift ωc is homogeneous of order one.

Let us explain now why do we have to consider that a homogeneous objectis differentiable only on TM , the tangent bundle with zero section removed,and not on the entire tangent bundle TM .

If a function f ∈ F(TM) is differentiable on TM and it is homogeneousof order r, then there exist the functions (φi1i2···is(x)), such that f(x, y) =φi1i2···is(x)(yi1)λ1 · · · (yis)λs , λ1 + λ2 + · · · + λs = r, so f is a polynomial oforder r with respect to y. The homogeneous functions, vectors and 1-forms wepresented in the example above are differentiable on whole TM , consequentlywe have for example that f c is homogeneous of order one and then f c = φiy

i,where φi = ∂f/∂xi. If we want to avoid this particular cases, we have toassume that the function f is of C∞-class on TM and only continuous on thenull section. A similar remark works also for tensor fields. Next, if we arereferring to a homogeneous object this will be supposed to be of C∞-class onTM and only continuous on the null section.

Chapter 2

Nonlinear Connections

In the geometry of the tangent bundle, and particularly in Finsler and La-grange geometry, an important role is played by the notion of horizontal dis-tribution and its associated concept of nonlinear connection. Despite the factthat a nonlinear connection lives on the total space of the tangent bundle ofa manifold, it can be introduced in the same manner as a linear connection,using the concept of parallel transport. We shall see in this chapter that aparallel transport defines always a connection, which in general is nonlinear.If additional conditions are required for the parallel transport, we obtain ho-mogeneous connections and linear connections.

By studying geodesics of such connections, we obtain that the geometry ofconnections is the same with the geometry of systems of second order differ-ential equations.

If we lift such connections to the tangent bundle of a manifold, we obtainwhat is usually called a nonlinear connection on the tangent bundle. As wehave seen in the first chapter, the vertical distribution V TM is a regular, n-dimensional, integrable distribution on the tangent space. Then it is naturallyto look for complementary distributions of the vertical one in TM . Such dis-tributions that will be called horizontal distributions are induced by nonlinearconnections. In this chapter we introduce the notion of a nonlinear connec-tion on the manifold TM and some geometric structures whose existence isequivalent to the existence of a nonlinear connection. We also study the in-tegrability of a nonlinear connection. Then, we determine the necessary andsufficient conditions for the integrability of a nonlinear connection in terms ofthe integrability of some induced geometric structures.

The existence of a horizontal distribution together with the vertical onedetermine a decomposition of the tangent bundle TTM into a Whitney sumTTM = HTM ⊕ V TM . If we express geometric objects using an adaptedbasis to this decomposition, their components are easier to handle. These

20 Chapter 2. Nonlinear Connections

components behave in a similar manner with corresponding objects from thebase manifold.

In this chapter we shall see that a nonlinear connection can be defineddirectly on the total space of the tangent bundle of a manifold and it definesa parallel transport on the base manifold. A nonlinear connection is an im-portant tool in the geometry of systems of second order differential equations.Also, the presence of a nonlinear connection on TM will allow us to extendsome results and geometric objects from the vertical subbundle V TM to thewhole tangent bundle TTM .

Equivalent definitions for a nonlinear connection on TM are given anda study of the main geometric objects induced by it is presented. As weintend to apply this theory to dynamical systems, we pay a special attentionto the autoparallel curves of a nonlinear connection and their symmetries. Theparticular cases when the connection is either homogeneous or linear are alsostudied.

2.1 Nonlinear connections on a manifold

When one defines a linear connection using the covariant derivative inducedby a parallel transport, one usually makes the assumption that the map thatassigns to each direction the total derivative in that direction is linear, [65],[111], [113]. If we remove such an assumption the connection we obtain is calleda nonlinear connection. In this chapter we study such nonlinear connectioninduced by a parallel transport. For the introduction of parallel transport andabsolute derivative we follow the book of M. Crampin and F.A.E. Pirani, [65],but we do not make the assumption that the linear transport is linear withrespect to the direction.

Consider M a real, n-dimensional manifold of C∞-class. As we have seen inthe previous chapter, there is no canonical isomorphism between two tangentspaces TpM and TqM at two points p and q to M . Consequently, there is nocanonical way of deciding if vectors that live on different tangent spaces areparallel or not.

In order to define a notion of parallelism on a manifold one should beable to identify the tangent spaces at any two points if a curve joining thetwo points is given. This identification has to preserve the linear structureof tangent spaces. Consequently, a parallel transport on a manifold can bedefined as a collection of linear isomorphisms γc

q,p : TpM −→ TqM , wherep, q ∈ M and c is a smooth curve on M joining p and q such that

γc1r,q γc2

q,p = γc1∪c2r,p .

From this we have that γcp,p = IdTpM for any closed curve c at p ∈ M and

2.1. Nonlinear connections on a manifold 21

(γcp,q)

−1 = γc−1

q,p , where c−1 is the reverse curve from q to p.A vector field X along a curve c is called a parallel vector field along the

curve if Xc(t) = γcc(t),p0

Xp0 , ∀t, where Xp0 is a vector at some point p0 = c(t0)on the curve. We also say that the vector field X is parallel-transported alongthe curve. The definition does not depend on the particular Xp0 we choose.This means that if Xp1 is any other vector on the curve c at p1 = c(t1) andYc(t) = γc

c(t),p1Xp1 is the vector field constructed by parallel transporting Xp1

along the curve, then X and Y coincide.The parallel transport is a collection of maps that preserve the linear struc-

ture of tangent spaces. Consider now X and Y two parallel vector fields alonga curve c obtained by parallel transport of Xp and Yp. Then, for any two realnumbers a and b, the vector field aX + bY is a parallel vector field obtainedby parallel transport of aXp + bYp.

We shall use parallel transport to define an absolute derivative such thatif a vector is parallel along a curve, then its absolute derivative vanishes.Let X(t) = Xc(t) be a vector field along a curve c. For ε > 0, we considerX(t+ε)|| = γc

c(t),c(t+ε)X(t+ε), the vector at c(t) obtained by parallel transportof X(t + ε). Then, the absolute derivative, along the curve c with parametert, of X at c(t) is defined as

DX

dt= lim

ε→0

1ε(X(t + ε)|| −X(t)) =

ddλγc

c(t),c(t+λ)X(t + λ)|λ=0.(2.1)

From this definition, we have that a vector field X is parallel along a curve cwith parameter t if and only if DX/dt = 0. We remark here that at this level,the concepts of parallelism and absolute derivative depend on the parameter-ization of the curve. The following is an assumption we shall use sometimes:

(A1) The parallel transport is independent on the parameterization of thecurve. Under this assumption we have that the absolute derivative satisfies

DX

dt=

DX

ds

ds

dt,(2.2)

where s and t are two parameters along curve c. In his case we say that theabsolute derivative is homogeneous.

Using the definition (2.1) of the absolute derivative, we obtain the followingimmediate properties:

Ddt

(X1 + X2) =DX1

dt+

DX2

dt,

Ddt

(fX) = fDX

dt+

df

dtX,

(2.3)

where X1, X2 and X are vector fields and f is a function along a curve c.


The absolute derivative at a point of a vector field on a curve should dependon the direction of the curve at that point and not on the particular curve withthat direction. From now we assume that the absolute derivative along a curvedepends at a point only on the tangent vector to the curve and not on theparticular curve that defines the vector. According to this assumption, we canassociate to each vector Y ∈ TpM and each vector field near p an element∇Y X ∈ TpM such that

∇Y X =DX

dt

∣∣∣∣t=0

,(2.4)

where the absolute derivative is taken along a curve c such that c(0) = pand (dc/dt)|t=0 = Y . We call ∇Y X the covariant derivative of X in thedirection Y . Using the covariant derivative, we can construct for two vectorfields X,Y near a point p another local vector field ∇Y X whose value at p is(∇Y X)p = ∇YpX. We require also that the rule of association (Y, X) 7→ ∇Y Xis smooth.

The map ∇ : (X,Y ) 7→ ∇XY is called a connection and it has, accordingto (2.3), the following properties:

C1) ∇X(Y + Z) = ∇XY +∇XZ,

C2) ∇X(fY ) = f∇XY + X(f)Y , where X,Y and Z are local vector fieldsand f is a local function.

If we do not require any other properties of linearity or homogeneity for ∇with respect to the first argument, then we refer to ∇ as a nonlinear connec-tion.

If we require for the absolute derivative to be homogeneous, which meansthat assumption (A1) is satisfied, then the covariant derivative (2.4) is calledhomogeneous and the induced connection ∇ is called a homogeneous connec-tion. In such a case, connection ∇ satisfies also the property:

C3) ∇fXY = f∇XY .In order to obtain what is usually called a linear connection we have to

make an additional assumption for the covariant derivative:(A2) For a fixed vector field X, the map Y ∈ TpM 7→ ∇Y X ∈ TpM is a

linear map. If assumption (A2) is satisfied, then∇ is called a linear connection.A linear connection satisfies C1), C2), C3) and

C4) ∇X+Y Z = ∇XZ +∇Y Z.A linear connection is also a homogeneous connection.One can extend the connection ∇ to act on 1-forms through

(∇Xθ)(Y ) + θ(∇XY ) = X(θ(Y )).(2.5)

Then, for this, properties that are similar to C1) and C2) hold true.

2.2. Local representations of a connection 23

2.2 Local representations of a connection

In this section we give local expressions for a connection ∇. Using its local co-efficients we obtain characterizations for homogeneous and linear connections.Consider (U,ϕ = (xi)) a coordinate chart on M , and X, Y local vector fieldsover U , such that Y = Y i(∂/∂xi). According to C1) and C2), we have that

∇X

(Y i ∂

∂xi

)= X(Y i)

∂

∂xi+ Y i∇X

(∂

∂xi

).

If we denote

∇X

(∂

∂xi

)= N j

i (x,X)∂

∂xj,(2.6)

then we have

∇X

(Y i ∂

∂xi

)=

X(Y j) + N j

i (x,X)Y i ∂

∂xj= Y i

|X∂

∂xi.(2.7)

We call Y i|X = X(Y i)+N i

j(x, X)Y j the covariant derivative of Y i in directionX. Functions N i

j(x,X), which depend on both point and direction are calledthe local coefficients of the connection. If we consider another local chart(V, (xi)), then, on the intersection U ∩ V , the coefficients N i

j and N ij are

related by

Nki

∂xj

∂xk=

∂xk

∂xiN j

k + X

(∂xj

∂xi

).(2.8)

By examining the homogeneity condition C3 and expression (2.6) we have thefollowing result:

Proposition 2.2.1 Connection ∇ is a homogeneous connection if and onlyif its local coefficients N i

j(x,X) are homogeneous functions of order one withrespect to second argument, which means that N i

j(x, fX) = fN ij(x,X), where

f is a local function on M .

Using conditions C3 and C4 and expression (2.6) we have the followingresult:

Proposition 2.2.2 Connection ∇ is a linear connection if and only if its localcoefficients N i

j(x,X) are linear with respect to second argument, which meansthat N i

j(x,X) = γijk(x)Xk.

For a linear connection ∇ its local coefficients γijk(x) are defined over do-

mains of local chart and they correspond to formula (2.6) for a linear connec-tion, which means

∇ ∂

∂xj

∂

∂xi= γk

ij(x)∂

∂xk.(2.9)


If we consider two coordinate charts, then according to expression (2.8), thecorresponding coefficients of a linear connection are related by

γijk

∂xj

∂xp

∂xk

∂xl=

∂xi

∂xmγm

pl +∂2xi

∂xp∂xl.(2.10)

For a covector field θ = θidxi, we have according to expression (2.5) that∇Xθ = X(θi)dxi + θi∇Xdxi. If we use again expression (2.5), we have

(∇Xdxi)(

∂

∂xj

)= −dxi

(∇X

∂

∂xj

)= −N i

j(x,X).(2.11)

From the above expression (2.11) we have that

∇Xθ = X(θj)−N ij(x,X)θidxj = θj|Xdxj .

A curve c : t ∈ I ⊂ R 7→ c(t) = (xi(t)) ∈ M is said to be a geodesic for aconnection ∇ if the tangent field dc/dt = (dxi/dt) is parallel along the curvec. This implies the following equations:

Ddt

(dxi

dt

)=

d2xi

dt2+ N i

j

(x,

dx

dt

)dxj

dt= 0.(2.12)

Next, we shall see that by lifting the connection ∇ to the tangent bundle,geodesics (2.12) are autoparallel curves given by equations (2.39) for the non-linear connection.

2.3 Nonlinear connections on the tangent bundle

As we have seen in the second section of the first chapter, the vertical distribu-tion V TM is a regular, n-dimensional, integrable distribution on the tangentbundle of a manifold M . It is naturally then to look for a supplementary dis-tribution of the vertical one in TTM . Such a distribution that will be calleda horizontal distribution is induced by a nonlinear connection, [130]. In thissection we introduce the notion of a nonlinear connection on the manifold TMand some geometric structures whose existence is equivalent to the existenceof a nonlinear connection. We shall study also the integrability of a nonlinearconnection. Then we determine necessary and sufficient conditions for theintegrability of a nonlinear connection in terms of the integrability of someinduced geometric structures.

The vertical distribution V TM and the tangent structure J determine thefollowing exact sequence:

0 −→ V TMι−→ TTM

J−→ V TM −→ 0.(2.13)

Using this exact sequence we can define supplementary distributions for thevertical distribution.

2.3. Nonlinear connections on the tangent bundle 25

Definition 2.3.1 A nonlinear connection on the tangent bundle TM of amanifold M is a left splitting of the exact sequence (2.13).

Therefore, a nonlinear connection on TM is a vector bundle morphismv : TTM → V TM , with the property that v ι = IdV TM .

The kernel of the morphism v is a vector subbundle of the tangent bundle(TTM, π∗, TM), denoted by HTM and called the horizontal subbundle. Itsfibres HuTM determine a regular n-dimensional distribution u ∈ TM →HuTM ⊂ TuTM , which is supplementary to the vertical distribution u ∈TM → VuTM ⊂ TuTM . Therefore, a nonlinear connection on TM inducesthe following decomposition for the tangent space TuTM , ∀u ∈ TM :

TuTM = HuTM ⊕ VuTM.(2.14)

The reciprocal of the above stated property holds true. So, we can formulate:

Theorem 2.3.1 A nonlinear connection on TM is characterized by the exis-tence of a subbundle HTM of the tangent bundle TTM such that the decom-position (2.14) holds true.

Next, we refer to a nonlinear connection by N , while we use the notationHTM for the associated horizontal subbundle or for the horizontal distribu-tion.

The restriction of the morphism J : TTM → V TM to HTM is an iso-morphism of vector bundles. Using the inverse of this isomorphism J |HTM

we can define a morphism of vector bundles Θ : V TM → TTM, such thatJ Θ = Id|V TM . In other words, Θ is a right splitting of the exact sequence(2.13). One can easily see that the bundle ImΘ coincides with the horizontalsubbundle HTM . The tangent bundle TTM will decompose then as a Whit-ney sum of the horizontal and the vertical subbundle. We can define now themorphism v : TTM → V TM on fibres as being the identity on vertical vectorsand zero on the horizontal vectors. It follows that v is a left splitting of theexact sequence (2.13). Moreover, the mappings v and Θ satisfy the relation

ι v + Θ J = IdTTM .

Theorem 2.3.2 A nonlinear connection on the tangent bundle (TM, π,M)is characterized by a right splitting of the exact sequence (2.13), Θ : V TM →TTM , such that J Θ = Id|V TM .

If a nonlinear connection is given on the tangent bundle TM , then thesequence (2.13) can be represented as follows:

0 −→ V TMι−→←−v

TTMJ−→←−Θ

V TM −→ 0.(2.15)


For a nonlinear connection N we denote by h and v the horizontal and thevertical projectors that correspond to decomposition (2.14), respectively.A vector field X ∈ χ(TM) is called horizontal if h(X) = X and vertical ifv(X) = X. We denote by χh(TM) the F(TM)-module of horizontal vectorfields.

Since π∗,u : TuTM → Tπ(u)M is an epimorphism, from (2.14) we can seethat the restriction of π∗,u to HuTM from HuTM to Tπ(u)M is an isomorphismof linear spaces. We denote by lh,u : Tπ(u)M → HuTM the inverse map ofthe above mentioned isomorphism. We call lh,u the horizontal lift induced bythe given nonlinear connection. The horizontal lift lh can be viewed also asan F(M)-linear map between χ(M) and χ(TM) and it is defined as follows:if X = Xi(∂/∂xi) ∈ χ(M) we define

lh(X)(u) = lh,u(Xπ(u)) = Xi(π(u))lh,u

(∂

∂xi

∣∣∣∣π(u)

).

The horizontal lift of a vector field X ∈ χ(M) is also denoted by Xh ∈ χ(TM).The horizontal lift lh induced by a nonlinear connection N and the vertical

lift lv are related by:J lh = lv,(2.16)

that is the following diagram is commutative:

TuTM

π∗,u %%KKKKKKKKKJu // TuTM

Tπ(u)M

lv,u

OO

One can prove that if lh : χ(M) → χ(TM) is an F(M)-linear map suchthat (2.16) holds, then H : u ∈ TM 7→ HuTM = lh,u(Tπ(u)M) is a horizontaldistribution on TM .

Denote by δ/δxi|u = lh,u(∂/∂xi|π(u)). We have that δ/δxi|ui=1,n is abasis of HuTM , ∀u ∈ TM and under a change of coordinates (1.1) on TM wehave that

δ

δxi=

∂xj

∂xi

δ

δxj.(2.17)

As π∗,u(δ/δxi|u) = ∂/∂xi|π(u), ∀u ∈ TM , then with respect to the naturalbasis ∂/∂xi|u, ∂/∂yi|u of TuTM , δ/δxi|u has the following expression:

δ

δxi

∣∣∣∣u

=∂

∂xi

∣∣∣∣u

−N ji (u)

∂

∂yj

∣∣∣∣u

.(2.18)

The set of functions (N ij) are defined on domains of induced local charts and

they are called the local coefficients of the nonlinear connection.

2.4. Characterizations of nonlinear connections 27

Proposition 2.3.1 To give a nonlinear connection N on the tangent bundleTM it is equivalent to give a set of functions N i

j(x, y) on every domain ofinduced local chart such that on intersections of such domains, they are relatedby

∂xj

∂xkNk

i = N jk

∂xk

∂xi+

∂yj

∂xi.(2.19)

Proof. The “if” part is a consequence of (2.17) and the action of the pseudo-group of coordinate transformations (1.1).

For the “only if” part we suppose that on every domain of induced localchart we have a set of functions N i

j such that on the intersection of any twodomains the corresponding functions N i

j and N jk are related by (2.19). Then

we may define δ/δxi|u as in (2.18). It is a straight forward calculation to checkthat (2.17) is true and then δ/δxi|u span a n-dimensional subspace HuTMof TuTM . As δ/δxi|u, ∂/∂yi|u are linearly independent, then HuTM andVuTM satisfy (2.14). q.e.d.

Formula (2.19) is the lift to the tangent bundle of formula (2.8). Thereforea nonlinear connection presented in this section as a horizontal subbundle isthe lift of a connection induced by a parallel transport.Examples Let γi

jk(x) be the local coefficients of a symmetric linear connection∇ on the base manifold M . Under a change of local coordinates on M we havethat

γijk =

∂xi

∂xlγl

pq

∂xp

∂xj

∂xq

∂xk− ∂2xi

∂xp∂xq

∂xp

∂xj

∂xq

∂xk.

If we denote by N ij(x, y) = γi

jk(x)yk and take into account the above lawof transformation, we find that N i

j(x, y) satisfy (2.19), so they are the localcoefficients of a nonlinear connection.

2.4 Characterizations of nonlinear connections

Due to its importance in the geometry of the tangent bundle we study neces-sary and sufficient conditions for the existence of a nonlinear connection usingassociated almost product structures, almost complex structure, etc. Thesestructures were studied also in [76], [80], [130].

For a given nonlinear connection N , we have a basis δ/δxi|u, ∂/∂yi|uof TuTM adapted to the decomposition (2.14). We call it the Berwald basisof the nonlinear connection. The adapted dual basis of this basis, or theadapted cobasis, is given by dxi, δyi = dyi +N i

j(x, y)dxj. Consequently, thehorizontal space HuTM at each point u ∈ TM is given by

HuTM = Xu ∈ TuTM, δyi(Xu) = 0,∀i ∈ 1, ..., n.


A nonlinear connection N on TM induces also a regular, n-dimensional dis-tribution (the horizontal codistribution) H∗ : u ∈ TM −→ H∗

uTM ⊂ T ∗uTMsuch that

T ∗uTM = H∗uTM ⊕ V ∗

u TM.(2.20)

Here H∗uTM = ωu : TuTM −→ R, ωu is R-linear and ωu(δ/δxi) = 0. A

horizontal 1-form can be expressed locally as ω = ωiδyi, while a vertical 1-

form has the expression ω = ωidxi.The horizontal and the vertical projectors of the nonlinear connection can

be expressed with respect to the Berwald basis as follows:

h =δ

δxi⊗ dxi and v =

∂

∂yi⊗ δyi.(2.21)

The horizontal and vertical projectors that correspond to decomposition (2.20)are

h∗ = δyi ⊗ ∂

∂yiand v∗ = dxi ⊗ δ

δxi.(2.22)

Next we shall refer to both projectors h and h∗ by h and we shall see fromthe context to which one we refer to. A similar remark applies to projectorsv and v∗.

From expression (1.8) we can see that the tangent structure J acts on theBerwald basis as follows:

J

(δ

δxi

)=

∂

∂yiand J

(∂

∂yi

)= 0, that is J =

∂

∂yi⊗ dxi.

Then, for ∀u ∈ TM , the restriction of Ju to HuTM , Ju : HuTM → VuTM ,is an isomorphism. The inverse map of this isomorphism is denoted by Θu :VuTM → HuTM . We can extend the structure Θu to the whole TuTM bytaking Θu := Θu vu. This is equivalent to

Θ =δ

δxi⊗ δyi or

Θ(

∂

∂yi

)=

δ

δxi,

Θ(

δ

δxi

)= 0.

(2.23)

We call the morphism Θ the adjoint structure. It has the following properties:

1) Θ2 = 0, ImΘ = KerΘ = HTM ;

2) Θ J = h, J Θ = v and consequently Id = Θ J + J Θ.

These two global properties uniquely determine a nonlinear connection withrespect to which Θ is locally given by expression (2.23). This can be seen fromthe following proposition:

2.4. Characterizations of nonlinear connections 29

Proposition 2.4.1 An F(TM)-linear morphism Θ : χ(TM) → χ(TM) suchthat Θ2 = 0 and Id = Θ J + J Θ determines a nonlinear connectionHTM = KerΘ.

Proof. With respect to the natural basis of the tangent space TuTM , the linearmap Θu can be expressed as follows: Θ(∂/∂xi) = Aj

i (∂/∂xj)+Bji (∂/∂yj) and

Θ(∂/∂yi) = Cji (∂/∂xj)+Dj

i (∂/∂yj). Using the second property ΘJ(∂/∂xi)+J Θ(∂/∂xi) = ∂/∂xi we have that Cj

i (∂/∂xj) + Dji (∂/∂yj) + Aj

i (∂/∂yj) =∂/∂xi, so Cj

i = δji and Aj

i = −Dji . From the second property 0 = Θ2(∂/∂xi)

we get that Bki = −Aj

iAkj . Denote by N j

i = Aji . Under a change of coordinates

(1.1) on TM the set of functions N ji obey the transformation rule (2.19), so

they are the local coefficients of a nonlinear connection N . We have also thatΘ(∂/∂xi − N j

i ∂/∂yj) = 0 and Θ(∂/∂yi) = ∂/∂xi − N ji (∂/∂yj), which mean

that Θ is locally given by expression (2.23) and the statement is proved. q.e.d.

Proposition 2.4.2 To give a nonlinear connection N on the tangent bundleTM it is equivalent to give for every u ∈ TM a linear map Ku : TuTM →Tπ(u)M such that Ku Ju = π∗,u.

Proof. If we have a nonlinear connection N , then we consider the structureΘ. For each u ∈ TM we define the linear map Ku : TuTM → Tπ(u)M , byKu = π∗,u Θu. Using the fact that Θu Ju = hu and π∗,u hu = π∗,u, weobtain that the linear Ku satisfies Ku Ju = π∗,u and therefore we proveddirect part of the proposition.

Conversely, let Ku : TuTM → Tπ(u)M be a linear map such that Ku Ju =π∗,u. Since π∗,u is an epimorphism then Ku is also an epimorphism, ∀u ∈ TM .If we denote by HuTM = KerKu we have an n-dimensional distribution onTM . The vertical distribution VuTM = KerJu is n-dimensional, too and fromKuJu = π∗,u we have that HuTM ∩VuTM = 0 and then (2.14) is satisfied.q.e.d.

The map K we used in the above proposition is called the connection map.This map was introduced first by P. Dombrowski [76] for the particular caseof a linear connection.

All structures Ju, Θu, π∗,u, lh,u, lv,u,Ku discussed above are related throughthe following diagram:

HuTMJu //

π∗,u

##HHHHHHHHHHHHHHHHHHH VuTMΘu

oo

Ku

²²Tπ(u)M

lv,u

OO

lh,u

ccHHHHHHHHHHHHHHHHHHH


Next, we present two other geometric structures, the almost product struc-ture and the almost complex structure whose existence is equivalent to theexistence of a nonlinear connection.

Proposition 2.4.3 To give a nonlinear connection N on the tangent bundleTM it is equivalent to give an F(TM)-linear morphism P : χ(TM) → χ(TM)such that

J P = J, P J = −J.(2.24)

Proof. If a nonlinear connection N is given, we define the F(TM)-linearmorphism P : χ(TM) → χ(TM) as

P(

Xi δ

δxi+ Y i ∂

∂yi

)= Xi δ

δxi− Y i ∂

∂yi.

Then, J P = J and P J = −J , which means that P satisfies the formulae(2.24).

Conversely, let P : χ(TM) → χ(TM) be an F(TM)-linear morphism suchthat (2.24) is true. Therefore, in the natural basis, the morphism P has theform

P(

∂

∂xi

)=

∂

∂xi− 2N j

i

∂

∂yjand P

(∂

∂yi

)= − ∂

∂yi.

It can be shown that under a change of induced local coordinates (1.1) onTM , the functions N i

j satisfy the formula (2.19) and, consequently, they arethe local coefficients of a nonlinear connection N . q.e.d.

The morphism P defined on the “if” part of the above proof satisfies alsoP2 = Id, and consequently it is called the almost product structure of the non-linear connection. It has the property that the distribution of eigenspacescorresponding to +1 is the horizontal distribution and the distribution ofeigenspaces corresponding to −1 is the vertical distribution. With respect tothe Berwald basis of the nonlinear connection, the almost product structureP has the expression

P =δ

δxi⊗ dxi − ∂

∂yi⊗ δyi = h− v.(2.25)

Proposition 2.4.4 To give a nonlinear connection N on the tangent bundleTM it is equivalent to give an F(TM)-morphism F : χ(TM) → χ(TM), suchthat

F2 = −Id, and F J + J F = Id.(2.26)

Proof. If we have a nonlinear connection N , we consider the adjoint structureΘ and define F = Θ−J . Then F2 = Θ2−ΘJ−J Θ+J2 = −(h+v) = −Id.Also, we have that F J + J F = Θ J + J Θ = h + v = Id.

2.5. d-tensor fields 31

Conversely, consider an F(TM)-linear morphism F : χ(TM) → χ(TM)such that (2.26) are true. If we define Θ = J + F, we have that Θ2 = 0and Θ J + J Θ = Id. According to Proposition 2.4.1, HTM = KerΘ is anonlinear connection on TM . q.e.d.

Structure F is called the almost complex structure of the nonlinear connec-tion and it has the following expression with respect to the Berwald basis:

F =δ

δxi⊗ δyi − ∂

∂yi⊗ dxi.(2.27)

From the above formula, we see that the almost complex structure has constantcoefficients with respect to Berwald basis and cobasis. The integrability of thisstructure is then equivalent to the fact that δ/δxi is a holonomic frame, whichis the same to the fact that δyi are exact 1-forms and we shall see next sectionthat this is equivalent to the integrability of the nonlinear connection.

2.5 d-tensor fields

A tensor field on the total space of the tangent bundle of a manifold is a quitecomplicated structure, while tensor fields defined over the vertical subbundleare very similar to tensor fields over the base manifold. These are the so-called d-tensor fields. The algebra of d-tensor fields on the tangent bundle ofa manifold is studied in [130].

A tensor field T of (r, s)-type on TM is said to be a distinguished tensorfield (or a d-tensor field for short) if under a change of local coordinates (1.1)on TM , its local components change as the local components of a (r, s)-typetensor field on the base manifold, which means that they satisfy a formulathat is similar to (1.3).

More precisely, we have the following characterization for a d-tensor field.Let T be a tensor field of (r, s)-type on TM , so T is an F(TM)-linear morphism

T : Λ1(TM)× · · · × Λ1(TM)︸︷︷︸r−times

×χ(TM)× · · · × χ(TM)︸︷︷︸s−times

→ F(TM).

Every 1-form ω ∈ Λ1(TM) and every vector field X ∈ χ(TM) can be de-composed into a horizontal and a vertical component ω = hω + vω andX = hX + vX. If T is a tensor field of (r, s)-type on TM , then T (hω1 +vω1, ..., hωr + vωr, hX1 + vX1, ..., hXs + vXs) is a sum of 2r+s terms. Each ofthese 2r+s terms is a d-tensor field on TM . Then we have that T is a d-tensorfield of (r,s)-type if and only if it reduces to only one term from those 2r+s

possible terms. This means that T is a d-tensor field if and only if

T (ω1, ..., ωr, X1, ..., Xs) = T (ε1ω1, ..., εrω

r, ε1X1, ..., εsXs),


for some choice of ε1, ..., εr, ε1, ..., εs ∈ h, v.

For example a (0, 2)-type tensor field g = gij(x, y)dxi ⊗ dxj is a d-tensorfield since under a change of coordinates (1.1), if we make use of (1.6), thecomponents gij transform as the components of a (0, 2)-type tensor field onthe base manifold, i.e.:

gij =∂xk

∂xi

∂xl

∂xjgkl.

If such a d-tensor field is symmetric and rank(gij) = n, we shall refer to it asa metric d-tensor field (or generalized Lagrange metric). If a metric d-tensorfield g = gijdxi ⊗ dxj is given, then one can consider also gv = gijδy

i ⊗ δyj

and gd = gijdxi ⊗ δyj . Both gv and gd are d-tensor fields on TM . Accordingto the characterization we gave above, this is true also because

g(hX + vX, hY + vY ) = g(hX, hY ),

gv(hX + vX, hY + vY ) = g(vX, vY ),

gd(hX + vX, hY + vY ) = g(hX, vY ).

Some other examples of d-tensor fields that we studied so far are given bellow.The tangent structure

J = δij

∂

∂yi⊗ dxj

is a (1,1)-type d-tensor field.A vertical vector field X = Xi(x, y)(∂/∂yi) is a (1,0)-type d-tensor field.

We call it a d-vector field. A vertical 1-form ω = ωi(x, y)dxi is a (0,1)-typed-tensor field. We call it a d-covector field.

2.6 Curvature and torsion of a nonlinear connection

So far we proved that a nonlinear connection is perfectly characterized bysome geometric structures on TM . Next, we study the relation between theintegrability of a nonlinear connection and the integrability of these geometricstructures. If N is a nonlinear connection, according to Frobenius theorem, wehave that N is integrable if and only if the corresponding horizontal distribu-tion is involutive, that is χh(TM) is a Lie subalgebra of χ(TM). As δ/δxiare generators for χh(TM) we have that N is integrable if and only if their Liebracket is horizontal, which means that [δ/δxi, δ/δxj ] ∈ χh(TM). We havethat [

δ

δxi,

δ

δxj

]= Rk

ij

∂

∂yk, where Rk

ij =δNk

i

δxj− δNk

j

δxi.(2.28)

Therefore, a nonlinear connection is integrable if and only if its curvature Rkji,

which is a d-tensor field of (1,2)-type, vanishes.

2.7. Dynamical covariant derivative 33

The curvature tensor of a nonlinear connection is defined as

R = −Nh = −12[h, h],(2.29)

where h is the horizontal projector and Nh is the Nijenhuis tensor of h. Wehave that:

R =12Rk

ijdxj ∧ dxi ⊗ ∂

∂yk.

For a nonlinear connection N on TM , we call the weak torsion of the nonlinearconnection the vertical valued 2-form:

t(X, Y ) = J [hX, hY ]− v[hX, JY ]− v[JX, hY ].(2.30)

With respect to the Berwald basis we have that the weak torsion has the form

t =12

(∂N i

j

∂yk− ∂N i

k

∂yj

)dxk ∧ dxj ⊗ ∂

∂yi=:

12tijkdxk ∧ dxj ⊗ ∂

∂yi.(2.31)

We have immediately that J t = 0 and t(JX, Y ) = t(X, JY ) = t(JX, JY ) =0, ∀X,Y ∈ χ(TM).

A nonlinear connection is said to be symmetric if its weak torsion t van-ishes, which meant that ∂N i

j/∂yk = ∂N ik/∂yj .

The Nijenhuis tensors of the adjoint structure Θ, the almost complex struc-ture F, and the almost product structure P are given by:

NΘ = NF =12tijkδy

k ∧ δyj ⊗ δ

δxi+

12Ri

jkδyk ∧ δyj ⊗ ∂

∂yi;

NP = 4Rkjidxj ∧ dxi ⊗ ∂

∂yk.

(2.32)

Using the above formulae (2.32) we can state the following theorem:

Theorem 2.6.1 a) A nonlinear connection N is integrable if and only if thecorresponding almost product structure P is integrable, which means that theNijenhuis tensor NP vanishes.

b) A symmetric nonlinear connection is integrable if and only if the almostcomplex structure F is integrable, which is equivalent to say that the adjointstructure Θ is integrable.

2.7 Dynamical covariant derivative

In this section, we study the autoparallel curves of a nonlinear connectionand the corresponding Jacobi vector fields. In order to do this we define first


the dynamical covariant derivative induced by a nonlinear connection. Thedynamical covariant derivative has been associated to a connection map in[130]. It has been studied also as a covariant derivative along the projectionmap π of the tangent bundle in [67].

Definition 2.7.1 The dynamical covariant derivative induced by a nonlinearconnection N is defined by

∇ : χv(TM) → χv(TM),

∇(

Xi ∂

∂yi

)=

(∂Xi

∂xjyj − ∂Xi

∂yjN j

kyk + N ijX

j

)∂

∂yi.

(2.33)

Denote nowS := yi δ

δxi= yi ∂

∂xi−N j

i yi ∂

∂yj.

We have that S is a globally defined vector field on TM . With this notation,formula (2.33) can be written as follows:

∇(

Xi ∂

∂yi

)= (S(Xi) + N i

jXj)

∂

∂yi.(2.34)

The action of ∇ on functions f ∈ F(TM) and components Xi of a d-vectorfield are given by:

∇f = S(f) =: f|, ∇Xi = S(Xi) + N ijX

j =: Xi| .

Then ∇fv = f c and ∇Xv = v(Xc). Dynamical covariant derivative ∇ acts onnatural basis ∂/∂yi of vertical vector fields according to the following formula

∇(

∂

∂yi

)= N j

i

∂

∂yj.

The covariant derivative ∇ has the following properties:

1) ∇(X + Y ) = ∇X +∇Y,∀X, Y ∈ χv(TM),

2) ∇(fX) = S(f)X + f∇X, ∀X ∈ χv(TM),∀f ∈ F(TM).

We can extend the action of covariant derivative ∇ to the algebra of d-tensorfields, by asking it to satisfy the Leibniz rule for tensor product. For exampleif gv = gijδy

i ⊗ δyj is a (0,2)-type d-tensor field, then its dynamical covariantderivative is given by:

(∇g)(X,Y ) = S(g(X, Y ))− g(∇X, Y )− g(X,∇Y ), where(2.35)

X, Y are vertical vector fields. In local coordinates (2.35) can be written asfollows:

gij| := (∇g)(

∂

∂yi,

∂

∂yj

)= S(gij)− gmjN

mi − gimNm

j .(2.36)

2.7. Dynamical covariant derivative 35

Definition 2.7.2 We say that a nonlinear connection N is metric with respectto a d-tensor metric g if its dynamical covariant derivative vanishes, whichmeans that ∇g = 0.

In local coordinates we have that a nonlinear connection is metric if andonly if S(gij) = gmjN

mi + gimNm

j . If we denote Nij = gimNmj , we have that a

nonlinear connection is metric if and only if

N(ij) =12(Nij + Nji) =

12S(gij).

One can compose ∇ with the vertical lift

∇v = ∇ lv : χ(M) −→ χv(TM).

In such a case we have ∇vX = v(Xc). In local coordinates we have: ifX = Xi(∂/∂xi) ∈ χ(M), then ∇vX = ∇v(Xi)(∂/∂yi), where:

∇vXi =∂Xi

∂xjyj + N i

jXj .(2.37)

Dynamical covariant derivative ∇v has similar properties with ∇, the differ-ence is that ∇v acts on vector fields from the base manifold.

1) ∇v(X + Y ) = ∇vX +∇vY ; ∇v(fX) = f cXv + fv∇vX;

2) If X ∈ χ(M), then Xc = lh(X) +∇v(X).

The action of the dynamical covariant derivative ∇ can be extended nowto the whole tangent bundle TM by:

∇(hX + vX) = Θ∇J(hX) +∇(vX).(2.38)

According to this definition, ∇ preserves the horizontal and vertical distri-butions. One can check this also by verifying that horizontal and verticalprojectors h and v are parallel with respect to ∇, which can be written as∇h = ∇v = 0. It is a straightforward calculation to check that the otherstructures like tangent structure J , the almost product structure P, the almostcomplex structure F and the adjoint structure Θ are parallel with respect to∇. This means that:

∇J = ∇P = ∇F = ∇Θ = 0.

Let us check one of this, for example ∇J = 0. We have to prove that(∇J)(X) = ∇J(X) − J(∇X) = 0, ∀X ∈ χ(TM). Since every vector X ∈χ(TM) can be decomposed as X = hX + vX and because of linearity of∇ and J one can check this formula for X = hX and X = vX. We have(∇J)(hX) = ∇J(hX) − J(∇hX) = ∇J(X) − JΘ∇JX = 0 because Jθ = vand v∇J(X) = ∇J(X). For vertical vector fields we have (∇J)(vX) =∇J(vX)− J(∇vX) = 0 because Jv = 0 and J(∇vX) = 0.


2.8 Autoparallel curves

Having the dynamical covariant derivative induced by a nonlinear connectionone can define parallel vector fields along a curve. An autoparallel curve is thena curve whose tangent vectors are parallel along the curve. A first variationof such autoparallel curves is studied also and corresponding Jacobi equationsare derived. The variation of autoparallel curves of a nonlinear connection hasbeen studied in [21] inspired especially by the following two papers: [44] and[67].

Definition 2.8.1 A smooth curve c : t ∈ I ⊂ R 7→ c(t) = (xi(t)) ∈ M iscalled an autoparallel curve of the nonlinear connection N if its natural lift toTM , c : t ∈ I 7→ c(t) = (xi(t), (dxi/dt)(t)) ∈ TM is a horizontal curve, whichmeans that the tangent vector field to c(t) is horizontal.

In local coordinates, a smooth curve c(t) = (xi(t)) is an autoparallel curveif and only if:

d2xi

dt2+ N i

j

(x,

dx

dt

)dxj

dt= 0.(2.39)

Using the dynamical covariant derivative, the invariant equivalent form of(2.39) is

∇(

dxi

dt

)= 0.(2.40)

Consider now, c(t) = (xi(t)) a trajectory of (2.39) and let vary it into nearbyones, according to

xi(t) = xi(t) + εξi(t),(2.41)

where ε denotes a scalar parameter with small value |ε|, and ξi(t) are compo-nents of a contravariant vector field along the curve c. If we substitute (2.41)into (2.39) and let ε approach zero we get the so-called variational equationsof (2.39):

d2ξi

dt2+

(∂N i

k

∂yj

dxk

dt+ N i

j

)dξj

dt+

∂N ik

∂xj

dxk

dtξj = 0.(2.42)

Theorem 2.8.1 For the variational equations (2.42), the equivalent invariantform (the Jacobi equations) are given by:

∇2ξi +

(∂N i

j

∂yk

dxj

dt−N i

k

)∇ξk + Ri

jk

dxj

dtξk = 0.(2.43)

A vector field ξi(t) along a trajectory c(t) = (xi(t)) of (2.39) is called a Jacobivector field if it satisfies the equations (2.43).

2.9. Symmetries of a nonlinear connection 37

Proof. Along a trajectory of (2.39), we have that

ddt

= S =: yi ∂

∂xi−N j

i yi ∂

∂yj.

Consequently, equations (2.42) are equivalent to

S2(ξi) +(

∂N ik

∂yjyk + N i

j

)S(ξj) +

∂N ik

∂xjykξj = 0.(2.44)

Since ∇ξi = S(ξi)+ N ijξ

j and ∇2ξi = S2(ξi)+ S(N ij)ξ

j +2N ijS(ξj)+N i

kNkj ξj

it is a straightforward calculation to check that equations (2.43) and (2.44)are equivalent. q.e.d.

For every vector field on the base manifold X = Xi(∂/∂xi) ∈ χ(M), weconsider the (1,1)-type d-tensor field ai

j(x, y, X):

aij =

∂2Xi

∂xj∂xkyk − ∂Xi

∂xkNk

j +∂Xk

∂xjN i

k +∂N i

j

∂xkXk +

∂N ij

∂yk

∂Xk

∂xryr.(2.45)

The Lie derivative of N ij with respect to the complete lift Xc of a vector field

X = Xi(∂/∂xi) ∈ χ(M) is defined as follows:

LXc

(N i

j

)= Xc(N i

j) + N ik

∂Xk

∂xj−Nk

j

∂Xi

∂xk+

∂2Xi

∂xj∂xkyk.(2.46)

For every X ∈ χ(M) we have that aij(x, y,X) = LXc(N i

j) is a (1,1)-typed-tensor field on TM . Then, variational equations (2.42), or the equivalentinvariant forms (2.43) or (2.44) can be written as follows:

aij

(x(t),

dx

dt, ξ(t)

)dxj

dt= LξcN i

j

(x,

dx

dt

)dxj

dt= 0.(2.47)

In the next section we will extend all these formulae from vector fields alongautoparallel curves to vector fields that are globally defined on M or TM .

2.9 Symmetries of a nonlinear connection

We are looking now for globally defined vector fields whose restriction to anautoparallel curve is a Jacobi vector field. Such vectors are symmetries for theconsidered nonlinear connection. First integrals for a nonlinear connectionare studied also. Symmetries and first integrals for the general case of adistribution are presented in [105].


Definition 2.9.1a) A diffeomorphism Φ of TM is an invariant transformation of a non-

linear connection N if Φ∗ preserves the horizontal distribution that is for allu ∈ TM , Φ∗,u(HuTM) = HΦ(u)TM .

b) A vector field X ∈ χ(M) is a Lie symmetry of a nonlinear connectionN if LXcY = [Xc, Y ] ∈ χh(TM), ∀Y ∈ χh(TM).

c) A vector field X ∈ χ(TM) is a dynamical symmetry of a nonlinearconnection N if LXY = [X, Y ] ∈ χh(TM), ∀Y ∈ χh(TM).

d) A function f ∈ F(TM) is a first integral of a nonlinear connectionN if its differential df is a horizontal 1-form, which means that Y (f) = 0,∀Y ∈ χh(TM).

Remarks:1) A vector field X from the base manifold M is a Lie symmetry of a

nonlinear connection if and only if its complete lift Xc is a dynamical symmetryof the nonlinear connection.

2) Let X ∈ χ(TM) and Φt its one-parameter group. Then X is a symmetryof a nonlinear connection if and only if Φt are invariant transformations ∀t thatis

(Φt)∗,u(HuTM) = HΦt(u)TM, ∀u ∈ TM.

3) Let f ∈ F(TM) be a first integral of a nonlinear connection N . Thenf is constant along autoparallel curves of the nonlinear connection.

Theorem 2.9.1 A vector field X ∈ χ(TM) is a symmetry of a nonlinearconnection N if and only if one of the following equivalent conditions holdtrue:

1) (LXP)(Y ) = 0, ∀Y ∈ χh(TM), which is equivalent to Ker(LXP) ⊂χh(TM);

2) (LXΘ)(Y ) = 0, ∀Y ∈ χh(TM), which is equivalent to Ker(LXΘ) ⊂χh(TM).

Proof.1) For a horizontal vector field Y , we have that Y = hY and P(Y ) = Y .

Then we have that (LXP)(Y ) = [X,P(Y )] − P[X, Y ] = (Id − P)[X,Y ] =2v[X, Y ]. A vector field X is a symmetry if and only if v[X,Y ] = 0, ∀Y ∈χh(TM). Then X is a symmetry if and only if (LXP)(Y ) = 0, ∀Y ∈ χh(TM).

2) A vector field Y ∈ χ(TM) is horizontal if and only if ΘY = 0. For ahorizontal vector field Y , we have that (LXΘ)(Y ) = −Θ[X, Y ]. A vector fieldX is a symmetry if and only if Θ[X, Y ] = 0, ∀Y ∈ χh(TM). Then X is asymmetry if and only if (LXΘ(Y ) = 0, ∀Y ∈ χh(TM). q.e.d.

Next, we pay a special attention to Lie symmetries as their restrictionalong autoparallel curves will give Jacobi vector fields.

2.9. Symmetries of a nonlinear connection 39

Since δ/δxii=1,n is a local basis for a nonlinear connection N on TM wehave that a vector field X ∈ χ(M) is a Lie symmetry of N if and only if

0 = v

[Xc,

δ

δxi

]= −aj

i (x, y, X)∂

∂yj,

where aij(x, y, X) is defined by (2.45). Consequently, we have that a vector

field X ∈ χ(M) is a Lie symmetry of a nonlinear connection N if and only if:

LXcN ij = 0.(2.48)

Proposition 2.9.1 A vector field X ∈ χ(M) is a Lie symmetry of a nonlinearconnection N if and only if LXcP = 0, where P is the almost product structureof the given nonlinear connection.

Proof. We have to prove that X ∈ χ(M) is a Lie symmetry of N if and onlyif [Xc,P(Y )]− P[Xc, Y ] = 0, ∀Y ∈ χ(TM). As LXcP(Y ) is an F(TM)-linearmap we can check this only for Y = δ/δxi and Y = ∂/∂yi. Theorem 2.9.1assures us that Xc is a symmetry if and only if [Xc,P(Y )] = P[Xc, Y ] = 0,∀Y ∈ χh(TM).

Consider now Y = ∂/∂yi, then [Xc,−∂/∂yi] − P[Xc, ∂/∂yi] = −(Id +P)([Xc, ∂/∂yi]) = −2h([Xc, ∂/∂yi]) = 0 and the theorem is proved. q.e.d.

Proposition 2.9.2 A vector field X ∈ χ(M) is a Lie symmetry of a nonlinearconnection N if and only if LXcΘ = 0, Θ being the adjoint structure inducedby N .

Proof. One can use directly the Theorem 2.9.1. Another direct proof is pre-sented here. We have that:

LXcΘ(

δ

δxi

)=

[Xc, Θ

(δ

δxi

)]−Θ

[Xc,

δ

δxi

]= −Θ

(−aj

i

∂

∂yj

)= aj

i

δ

δxj;

LXcΘ(

∂

∂yi

)=

[Xc,

δ

δxi

]−Θ

[Xc,

∂

∂xi

]= −aj

i

∂

∂yj.

Then we obtain

[Xc, Θ] = LXcΘ = LXcN ij

δ

δxi⊗ dxj − LXcN i

j

∂

∂yi⊗ δyj .

From these we have that LXcΘ = 0 if and only if aij = LXcN i

j = 0 that is Xis a Lie symmetry of the given nonlinear connection. q.e.d.

Proposition 2.9.3 A vector field X ∈ χ(M) is a symmetry of a nonlinearconnection N if and only if LXcF = 0, F being the almost complex structureof the nonlinear connection N .


Proof. For the complete lift Xc, we have that LXcJ = 0. As F = Θ − J , wehave that LXcΘ = LXcF. Consequently, X is a Lie symmetry if and only ifLXcF = 0. q.e.d.

Theorem 2.9.2 Every Lie symmetry of a nonlinear connection N is a Jacobivector field along an autoparallel curve of N .

Proof. A vector field X ∈ χ(M) is a Lie symmetry of a nonlinear connec-tion N if and only if ai

j(x, y,X) = LXcN ij = 0. Consequently we have that

aij(x, y, X)yj = 0. The restriction of the last equation along an autoparallel

curve of N gives us the equations (2.48), which is an equivalent form of theJacobi equations (2.43). q.e.d.

Let X ∈ χ(M) be a vector field on the base manifold and (φt) be its localone-parameter group of transformations. The complete lift Xc has (Φt) =((φt)∗) as its local one parameter group of transformations. The vector fieldX is a Lie symmetry of a nonlinear connection N if and only if

(Φt)∗,u(HuTM) = HΦt(u)TM, ∀u ∈ TM.

Consider c an autoparallel curve of the nonlinear connection. Then the naturallift c of c to TM is a horizontal curve. This means that the tangent vectorfield dc/dt is a horizontal vector and (Φt)∗,c(t)(dc/dt) is horizontal too.

As (Φt)∗,c(t)(dc/dt) = d/dt(Φt c(t)) and Φt c(t) is the natural lift ofφt c(t) we have that the one-parameter group of transformations φt mapsautoparallel curves into autoparallel curves.

2.10 Homogeneous connections and linear connec-tions

In this section we study autoparallel curves, Jacobi equations and symmetriesfor the particular case when the connection is either homogeneous or linear,by specializing the results obtained in the previous section. For the latter casewe obtain standard results about variation of geodesics and their symmetries,[59].

If hλ : TM → TM , hλ(x, y) = (x, λy) is the dilatation of ratio λ, λ ∈(0,∞), then (hλ)∗,u : TuTM → Thλ(u)TM is an isomorphism of linear spaces,∀u ∈ TM . If these isomorphisms preserve the horizontal distribution, HTM ,which means that

(hλ)∗,u(HuTM) = Hhλ(u)TM, ∀u ∈ TM,(2.49)

the connection is said to be homogeneous. Accordingly, a nonlinear connec-tion is homogeneous if and only if hλ are invariant transformations for the

2.10. Homogeneous connections and linear connections 41

horizontal distribution. The family (hλ)λ∈R is the one-parameter group forthe Liouville vector field C. Consequently, a nonlinear connection is homoge-neous if and only if Liouville vector field C is a symmetry for the nonlinearconnection that is

[C, X] ∈ χh(TM), ∀X ∈ χh(TM).

Then, from the above condition we have that a nonlinear connection is homo-geneous if and only if C(N i

j) = N ij that is the local coefficients of the nonlinear

connection are homogeneous functions of order one. One can express this alsoby N i

j(x, λy) = λN ij(x, y), which comes directly from (2.49).

For homogeneous nonlinear connection we assume that the local coefficientsN i

j are of C∞-class on TM and only continuous on the null section. If the localcoefficients N i

j of a nonlinear connection are of C∞-class on whole TM , thenN i

j(x, y) = γijk(x)yk. In this particular case, the connection is called linear.

The functions γijk(x) are the local coefficients of a linear connection on the

base manifold M .

Proposition 2.10.1 A nonlinear connection N is homogeneous if and only ifLCP = 0, where C = yi(∂/∂yi) is the Liouville vector field and P is the almostproduct structure.

Proof. In local coordinates we have that LCP = (N ij−yk∂N i

j/∂yk)∂/∂yi⊗dxj .

Thus we have that LCP = 0 if and only if yk∂N ij/∂yk = N i

j that is thenonlinear connection is homogeneous. Condition LCP = 0 expresses also thatC is a symmetry of the nonlinear connection. The (1,1)-type d-tensor fieldLCP is called the tension of the nonlinear connection N . q.e.d.

Consider that a curve c, with parameter t, is an autoparallel curve of ahomogeneous nonlinear connection. If one changes the parameter t 7→ t(s),then

d2xi

dt2+ N i

j

(x,

dx

dt

)dxj

dt=

(ds

dt

)2 [d2xi

ds2+ N i

j

(x,

dx

ds

)dxj

ds

]+

dxi

dt

d2s

dt2= 0.

Curve c with parameter s is again an autoparallel curve of the nonlinear con-nection N if and only if the change of parameter is affine that is t = λs + µ,where λ, µ ∈ R. For nonlinear connections that are not homogeneous, wecannot change the parameter of an autoparallel curve in order to obtain againautoparallel curves.

For a vector field X = Xi(∂/∂xi) ∈ χ(M) and a homogeneous nonlinear


connection N , we define:

aijk(x, y,X) =

∂2Xi

∂xj∂xk− ∂Xi

∂xp

∂Npj

∂yk+

∂N ip

∂yk

∂Xp

∂xj

+∂N i

p

∂yk

∂Xp

∂xj+

∂2N ip

∂yk∂ypXp +

∂2N ij

∂yp∂yk

∂Xp

∂xsys.

(2.50)

The Lie derivative of the geometric object F ijk(x, y) = ∂N i

j/∂yk with respectto the complete lift Xc of a vector field X = Xi∂/∂xi ∈ χ(M) is given by:

LXcF ijk = Xc(F i

jk) + F ipk

∂Xp

∂xj+ F i

jp

∂Xp

∂xk− F p

jk

∂Xi

∂xp+

∂2Xi

∂xj∂xk.(2.51)

We have that for every vector field X ∈ χ(M), aijk(x, y, X) is a (1,2)-type

d-tensor field and aijk(x, y, X)yk = ai

j(x, y,X). Therefore, a vector field X ∈χ(M) is a Jacobi vector field for N if and only if

aijk

(x,

dx

dt,X

)dxj

dt

dxk

dt= 0.

A vector field X ∈ χ(M) is a Lie symmetry of N if and only if aijk(x, y, X)yk =

aij(x, y, X) = 0.

The Jacobi equations for a symmetric, homogeneous connection are now:

∇2ξi + Rijk

dxj

dtξk = 0.(2.52)

Here Rijk(x, y) is the curvature d-tensor of the nonlinear connection.

If the connection is linear then N ij(x, y) = γi

jk(x)yk, where γijk(x) are

the local coefficients of a linear connection on the base manifold M . Thenautoparallel curves of the connection are geodesics of the linear connection,and equations (2.39) can be written as

d2xi

dt2+ γi

jk(x)dxj

dt

dxk

dt= 0.

The Jacobi equations for geodesics of a linear connection have the usual form

∇2ξi + Riljk(x)

dxl

dt

dxj

dtξk = 0,

where Riljk(x) are the components of the curvature tensor of the linear con-

nection.

Chapter 3

N-Linear Connections

It is well known that for an arbitrary manifold M there is no canonical isomor-phism between two tangent spaces TpM and TqM at p, q ∈ M . The existenceof such isomorphism, which is be called a parallel transport, is equivalent tothe existence of a connection on the manifold. In general such a connection isnot linear. In the previous chapter we have seen that one can deal with thenonlinearity of such a connection by lifting it to the total space of the tan-gent bundle. There, we can associate to it a dynamical covariant derivativealong the autoparallel curves of a nonlinear connection. Moreover, one canassociate to a nonlinear connection a linear connection, which is called theBerwald connection. This connection has special features and contains mostof the information we need about the nonlinear connection. In this chapter westudy a class of linear connections on the tangent bundle that preserve bothhorizontal and vertical distributions. One of these linear connections is theBerwald connection we can associate to a nonlinear connection.

If the tangent bundle TM of a manifold is endowed with a nonlinear con-nection, then at each point u ∈ TM we have the decomposition TuTM =HuTM ⊕ VuTM . For two points u, v ∈ TM we are interested to define aparallel transport between TuTM and TvTM that preserves the above decom-position. The linear connection that corresponds to such a parallel transportis called an N-linear connection on TM . The compatibility between a para-llel transport on TM and a nonlinear connection is characterized using variousgeometric structures we studied in the previous chapter: almost product struc-ture, almost complex structure and adjoint structure. The Berwald connectionthat corresponds to a nonlinear connection is an N-linear connection. Conse-quently, an N -linear connection on TM is a special linear connection D onTM that preserves by parallelism the horizontal and vertical distributions. Inthis chapter we study such linear connections, we determine all components oftorsion and curvature and we give some examples of such linear connections.

44 Chapter 3. N -Linear Connections

The Cartan’s structure equations of an N -linear connection are determinedand the integrability conditions for these equations are studied. When we ap-ply these results to the Berwald connection we obtain important informationabout the nonlinear connection and its autoparallel curves.

3.1 N-linear connections

In this section we define linear connections that preserve the decomposition(2.14) and with respect to which all geometric structures that define a non-linear connection are parallel. Such linear connections were studied in [80],[130].

Throughout this chapter a nonlinear connection N with local coefficientsN i

j is fixed. Let h and v be the horizontal and vertical projectors induced byN . Consider also the almost product structure P, the adjoint structure Θ, andthe almost complex structure F induced by the nonlinear connection N .

Definition 3.1.1 A linear connection D on TM is called a d-connection if itpreserves by parallelism the horizontal distribution, which means that Dh = 0.

Proposition 3.1.1 A linear connection D on TM is a d-connection if andonly if one of the following conditions is true:

1) Dv = 0;2) DP = 0.

Proof. Since Id = h + v we have that the two conditions Dh = 0 and Dv = 0are equivalent. Also, from 2h = Id + P and 2v = Id− P we have that Dh = 0and DP = 0 are equivalent. q.e.d.

Consequently, we have that a linear connection D on TM is a d-connectionif and only if it preserves by parallelism both horizontal and vertical distribu-tions. This is equivalent with the fact the the Schouten connection one canassociate to D and the projectors h and v coincides with D, in other words:

DXY = hDXhY + vDXvY, ∀X,Y ∈ χ(TM).

Proposition 3.1.2 For a d-connection D on TM the following conditions areequivalent:

1) DJ = 0;2) DΘ = 0;3) DF = 0.

Proof. Since F = Θ − J , it is enough to prove that two of the above threeconditions are equivalent. Suppose that DJ = 0 and let us prove that DΘ = 0.As DXΘY −ΘDXY is horizontal, we have that DXΘY −ΘDXY = h(DXΘY −

3.2. Berwald connection 45

ΘDXY ) = ΘJ(DXΘY − ΘDXY ) = ΘDXJΘY − ΘJΘDXY = ΘDXvY −ΘDXY = ΘvDXY −ΘDXY = ΘDXY −ΘDXY = 0, and then DΘ = 0.

Conversely, suppose that DΘ = 0. As D preserves the vertical distri-bution we have that DXJY − JDXY is a vertical vector field. Consequently,DXJY −JDXY = v(DXJY −JDXY ) = JΘ(DXJY −JDXY ) = JDXΘJY −JΘJDXY = JDXhY − JDXY = JhDXY − JDXY = JDXY − JDXY = 0.So, we have proved that DJ = 0. q.e.d.

Definition 3.1.2 A d-connection is called an N -linear connection if one ofthe equivalent conditions of Proposition 3.1.2 holds good.

With respect to the Berwald basis of the nonlinear connection N , an N -linear connection has the form

D δ

δxi

δ

δxj= F k

ji(x, y)δ

δxk, D δ

δxi

∂

∂yj= F k

ji(x, y)∂

∂yk,

D ∂∂yi

δ

δxj= Ck

ji(x, y)δ

δxk, D ∂

∂yi

∂

∂yj= Ck

ji(x, y)∂

∂yk.

(3.1)

From expression (3.1) we can see that an N -linear connection D transports byparallelism horizontal vectors into horizontal ones and vertical vectors into ver-tical ones. Moreover, this parallelism acts on the same manner on horizontaland vertical vectors.

The set of functions F kij(x, y), Ck

ij(x, y) are called the local coefficients ofthe N -linear connection D. Sometimes we refer to an N -linear connection Dby the set DΓ = (N i

j(x, y), F kij(x, y), Ck

ij(x, y)) or DΓ(N) = (F kij , C

kij) if the

nonlinear connection is fixed. Under a change of coordinates (1.1) on TM , wehave:

F kij =

∂xk

∂xlF l

pq

∂xp

∂xi

∂xq

∂xj− ∂2xk

∂xp∂xq

∂xp

∂xi

∂xq

∂xj,

Ckij =

∂xk

∂xlC l

pq

∂xp

∂xi

∂xq

∂xj.

(3.2)

One can see from the above formula that horizontal coefficients F kij of an

N -linear connection D on TM have the same rule of transformation as thelocal coefficients of a linear connection on the base manifold M . The verticalcoefficients Ck

ij are the components of a (1,2)-type d-tensor field.

3.2 Berwald connection

In this section we show that every nonlinear connection induces an N -linearconnection on the tangent bundle. This connection appears for the first timein Berwald’s papers, [44]. It has been studied later on by [130], [67]. A globaldefinition as a linear map on the tangent bundle has been proposed in [53].


Next theorem gives a first example of an N -linear connection on TM .

Theorem 3.2.1 The map D : χ(TM)× χ(TM) → χ(TM), given by

DXY = v[hX, vY ] + h[vX, hY ] + J [vX,ΘY ] + Θ[hX, JY ](3.3)

is an N -linear connection on TM .

Proof. As all the operators involved in the right hand side of (3.3) are additivewe have that D is additive too, with respect to both arguments. To provethat DfXY = fDXY , ∀f ∈ F(TM), we have to use that vh = hv = Jv =Θh = 0. Now let us prove that DXfY = X(f)Y +fDXY . From (3.3) we havethat DXfY = fDXY + (hX)(f)v2(Y ) + (vX)(f)h2(Y ) + (vX)(f)JΘ(Y ) +(hX)(f)ΘJ(Y ).

As v2 = v, h2 = h, JΘ = v, and ΘJ = h we have that DXfY = X(f)Y +fDXY . At this moment we have proved that D is a linear connection on TM .As vJ = J and Θv = Θ we have that DXvY = v[hX, vY ] + J [vX, ΘY ] =v(DXY ) that is Dv = 0 and D preserves by parallelism the vertical distribu-tion. Consequently, D is a d-connection on TM .

Next, we have that DXΘY = v[hX, vΘY ] + h[vX, hΘY ] + J [vX, Θ2Y ] +Θ[hX, JΘY ] = h[vX,ΘY ]+Θ[hX, vY ] = Θ(DXY ), because vΘ = 0, hΘ = Θ,Θ2 = 0, and JΘ = v. So, DΘ = 0, and D is an N -linear connection. q.e.d.

The connection given by expression (3.3) is called the Berwald connectionof the nonlinear connection N . With respect to the Berwald basis, the Berwaldconnection has the expression

D δδxi

δ

δxj= Θ

[δ

δxi,

∂

∂yj

]= Θ

(∂Nk

i

∂yj

∂

∂yk

)=

∂Nki

∂yj

δ

δxk; and

D ∂

∂yi

δ

δxj= h

[∂

∂yi,

δ

δxj

]= 0.

Consequently, we have also

D δδxi

∂

∂yj=

∂Nki

∂yj

∂

∂ykand D ∂

∂yi

∂

∂yj= 0.

The local coefficients of the Berwald connection are given then by

F kij =

∂Nkj

∂yi, and Ck

ij = 0.

Sometimes we refer to the Berwald connection as BΓ = (N ij , ∂Nk

j /∂yi, 0).Let ∇ be a linear connection on the base manifold M , with local coeffi-

cients γijk(x). Then N i

j(x, y) = γijk(x)yk are local coefficients of a nonlinear

connection N on TM . Berwald connection that corresponds to this nonlinearconnection N has local coefficients F i

jk(x, y) = γijk(x), and Ci

jk(x, y) = 0.

3.3. Horizontal and vertical covariant derivatives 47

3.3 Horizontal and vertical covariant derivatives

Since N -linear connection preserves by parallelism both horizontal and verticaldistribution, we look now for two associated covariant derivatives, called h-and v-covariant derivatives that correspond to these two distributions. For anN -linear connection D on TM we associate two operators of h- and v-covariantderivation in the algebra of d-tensor fields, [130]. For each X ∈ χ(TM), weset:

DhXY = DhXY, Dh

Xf = (hX)(f), ∀Y ∈ χ(TM), ∀f ∈ F(TM).(3.4)

If ω ∈ Λ1(TM), we define

(DhXω)(Y ) = (hX)(ω(Y ))− ω(Dh

XY ), ∀Y ∈ χ(TM).(3.5)

We can extend the action of the operator DhX to any d-tensor field by asking

that DhX preserves the type of d-tensor fields, is R-linear, satisfies the Leibnitz

rule with respect to tensor product and commutes with all contractions. Weshall keep the notation Dh

X for this operator on the algebra of d-tensor fields.We call it the operator of h-covariant derivative. We remark here that Dh

is not a covariant derivative, it has the usual properties as the operator D,except Dh

Xf = hX(f) 6= X(f).In a similar way, for every vector field X ∈ χ(TM) we define

DvXY = DvXY, Dv

Xf = (vX)(f), ∀Y ∈ χ(TM), ∀f ∈ F(TM).(3.6)

If ω ∈ Λ1(TM), we define

(DvXω)(Y ) = (vX)(ω(Y ))− ω(Dv

XY ), ∀Y ∈ χ(TM).(3.7)

We extend the action of DvX to any d-tensor field in a similar way, as we did

for DhX . We obtain an operator on the algebra of d-tensor fields on TM , this

will be denoted also by DvX and will be called the v-covariant derivative. We

remark here that Dv is not a covariant derivative, it has the usual propertiesas the operator D, except Dv

Xf = vX(f) 6= X(f).If T is a d-tensor field of (r, s)-type with local components T i1···ir

j1···js(x, y),

then its h-covariant derivative is an (r, s + 1)-type d-tensor field DhXT given

by

DhXT = XkT i1···ir

j1···js|kδ

δxi1⊗ · · · ⊗ ∂

∂yir⊗ dxj1 ⊗ · · · ⊗ δyjs ,(3.8)

where

T i1···irj1···js|k =

δT i1···irj1···js

δxk+ F i1

pkTpi2···irj1···js

+ · · ·+ F irpkT

i1···ir−1pj1···js

−F pj1kT

i1···irpj2···js

− · · · − F pjskT

i1···irj1···js−1p .

(3.9)


The v-covariant derivative of a d-tensor field T of (r, s)-type is an (r, s+1)-typed-tensor field Dv

XT , given by

DvXT = XkT i1···ir

j1···js|k δ

δxi1⊗ · · · ⊗ ∂

∂yir⊗ dxj1 ⊗ · · · ⊗ δyjs ,(3.10)

where

T i1···irj1···js

|k =∂T i1···ir

j1···js

∂yk+ Ci1

pkTpi2···irj1···js

+ · · ·+ CirpkT

i1···ir−1pj1···js

−Cpj1kT

i1···irpj2···js

− · · · − CpjskT

i1···irj1···js−1p .

(3.11)

We shall use these formula for metric tensors, which are d-tensor fields of(0,2)-type.

3.4 Torsion of an N-linear connection

In this section we study the torsion of an N -linear connection. With respectto a basis adapted to both horizontal and vertical distribution, there are fivecomponents for the torsion of an N -linear connection, [23], [130]. By applyingthese study to the Berwald connection one obtain two components of torsion.The horizontal component is the torsion (2.31) while the vertical componentis the curvature (2.29) of the nonlinear connection. The symmetry of theBerwald connection as an N -linear connection is the same with the symmetryof the nonlinear connection.

For an N -linear connection D we consider the torsion T , defined as usual

T (X,Y ) = DXY −DY X − [X,Y ], ∀X,Y ∈ χ(TM).(3.12)

Theorem 3.4.1 The torsion of an N -linear connection D on TM is com-pletely determined by the following five d-tensor fields:

hT (hX, hY ) = DhXhY −Dh

Y hX − h[hX, hY ], (h)h− torsion;

vT (hX, hY ) = −v[hX, hY ], (v)h− torsion;

hT (hX, vY ) = −DvY hX − h[hX, vY ], (h)hv − torsion;

vT (hX, vY ) = DhXvY − v[hX, vY ], (v)hv − torsion;

vT (vX, vY ) = DvXvY −Dv

Y vX − v[vX, vY ], (v)v − torsion.

(3.13)

3.4. Torsion of an N -linear connection 49

Proof. Each vector field on TM has a horizontal and a vertical component.Therefore, we have that the following decomposition T (X, Y ) = T (hX, hY ) +T (hX, vY )+T (vX, hY )+T (vX, vY ), ∀X,Y ∈ χ(TM). Every vector field fromthe right hand side of the previous equality has a horizontal and a verticalcomponent. From these eight components two are zero because T is skewsymmetric and one is zero because h[vX, vY ] = 0. Since D preserves byparallelism the horizontal and the vertical distributions, the five componentsof torsion are given by formula (3.13). q.e.d.

With respect to the Berwald basis the five components of torsion are givenby

hT

(δ

δxi,

δ

δxj

)=: T k

ji

δ

δxk= (F k

ji − F kij)

δ

δxk;

vT

(δ

δxi,

δ

δxj

)= Rk

ji

∂

∂yk=

(δNk

j

δxi− δNk

i

δxj

)∂

∂yk;

hT

(∂

∂yi,

δ

δxj

)= Ck

ji

δ

δxk;

vT

(∂

∂yi,

δ

δxj

)=: P k

ji

∂

∂yk=

(∂Nk

j

∂yi− F k

ij

)∂

∂yk;

vT

(∂

∂yi,

∂

∂yj

)=: Sk

ji

∂

∂yk= (Ck

ji − Ckij)

∂

∂yk.

(3.14)

An N -linear connection D is said to be symmetric if the (h)h-torsion and(v)v-torsion vanish that is F k

ij = F kji and Ck

ji = Ckij .

The Berwald connection BΓ = (N ij , ∂N i

j/∂yk, 0) induced by a nonlinearconnection has only two nonzero components of torsion:

hT

(δ

δxi,

δ

δxj

)= T k

ji

δ

δxk=

(∂Nk

i

∂yj− ∂Nk

j

∂yi

)δ

δxk, h(h)− torsion;

vT

(δ

δxi,

δ

δxj

)= Rk

ji

∂

∂yk=

(δNk

j

δxi− δNk

i

δxj

)∂

∂yk, v(h)− torsion.

(3.15)

If the nonlinear connection N is symmetric then the h(h)-torsion, T kij ,

vanishes and the only nonzero component of torsion for the Berwald connectionD is a vertical component, Rk

ij , the curvature of the nonlinear connection. Fora symmetric nonlinear connection, all horizontal components of torsion vanish.


3.5 Curvature of an N-linear connection

In this section we study the curvature of an N -linear connection. With respectto the Berwald basis, the curvature of an N -linear connection has three inde-pendent components, [23], [130], while the Berwald connection has only twocomponents. In the previous section we saw that the vertical component oftorsion of the Berwald connection coincides with the torsion of the nonlinearconnection. In this section we show how can we determine the components ofcurvature of the Berwald connection from the curvature of the nonlinear con-nection. If the connection is homogeneous and symmetric we can determinealso the curvature of the nonlinear connection from the horizontal curvatureof the associated Berwald connection.

For an N -linear connection D, as typically, we consider its curvature:

R(X, Y )Z = DXDY Z −DY DXZ −D[X,Y ]Z, ∀X, Y, Z ∈ χ(TM).(3.16)

As D preserves by parallelism the horizontal and the vertical distributions,from (3.16) we have that the operator R(X, Y ) carries horizontal vector fieldsinto horizontal vector fields and vertical vector fields into verticals. Conse-quently we have the following formula:

R(X, Y )Z = hR(X,Y )hZ + vR(X,Y )vZ, ∀X, Y, Z ∈ χ(TM).(3.17)

If we take into account that the operator R(X,Y ) is skew symmetric withrespect to X and Y we have the theorem:

Theorem 3.5.1 The curvature of an N -linear connection D on the tangentbundle TM is completely determined by the following six d-tensor fields:

R(hX, hY )hZ = DhXDh

Y hZ −DhY Dh

XhZ −D[hX,hY ]hZ,

R(hX, hY )vZ = DhXDh

Y vZ −DhY Dh

XvZ −D[hX,hY ]vZ,

R(vX, hY )hZ = DvXDh

Y hZ −DhY Dv

XhZ −D[vX,hY ]hZ,

R(vX, hY )vZ = DvXDh

Y vZ −DhY Dv

XvZ −D[vX,hY ]vZ,

R(vX, vY )hZ = DvXDv

Y hZ −DvY Dv

XhZ −D[vX,vY ]hZ,

R(vX, vY )vZ = DvXDv

Y vZ −DvY Dv

XvZ −D[vX,vY ]vZ.

(3.18)

Since the tangent structure J is absolutely parallel with respect to the N -linear connection D, which means that DJ = 0, we have that JR(X,Y )Z =R(X,Y )JZ, ∀X, Y, Z ∈ χ(TM). Then, the curvature tensor of an N -linearconnection D has, with respect to the Berwald basis, only three different

3.5. Curvature of an N -linear connection 51

components. These three components are given by

R

(δ

δxk,

δ

δxj

)δ

δxh=: Ri

hjk

δ

δxi;

R

(∂

∂yk,

δ

δxj

)δ

δxh=: P i

hjk

δ

δxi;

R

(∂

∂yk,

∂

∂xj

)δ

δxh=: Si

hjk

δ

δxi.

(3.19)

The three components given by expression (3.19) are the components of thefirst, the third and the fifth d-tensors from expression (3.18). The other threed-tensor fields from expression (3.18) have the same local components Ri

hjk,P i

hjk, and Sihjk, which can be obtained as follows:

R

(δ

δxk,

δ

δxj

)∂

∂yh= Ri

hjk

∂

∂yi;

R

(∂

∂yk,

δ

δxj

)∂

∂yh= P i

hjk

∂

∂yi;

R

(∂

∂yk,

∂

∂xj

)∂

∂yh= Si

hjk

∂

∂yi.

(3.20)

Consequently, an N -linear connection DΓ = (N ij , F

ijk, C

ijk) has only three local

components of curvature Rihjk, P i

hjk, and Sihjk, and these components are given

by

Rihjk =

δF ihj

δxk− δF i

hk

δxj+ Fm

hjFimk − Fm

hkFimj + Ci

hmRmjk;

P ihjk =

∂F ihj

∂yk− Ci

hk|j + CihmPm

jk ;

Sihjk =

∂Cihj

∂yk− ∂Ci

hk

∂yj+ Cm

hjCimk − Cm

hkCimj .

(3.21)

Here Cihk|j denotes the h-covariant derivative of the (1,2)-type tensor field Ci

jk.The Berwald connection has only two nonzero components of curvature:

Rihjk =

δF ihj

δxk− δF i

hk

δxj+ Fm

hjFimk − Fm

hkFimj ;

P ihjk =

∂F ihj

∂yk=

∂2N ih

∂yj∂yk=: Di

hjk.

(3.22)

The curvature Rijk of a nonlinear connection N is a (1,2)-type d-tensor field,

while the horizontal component of curvature Rijkl of the Berwald connection is


a (1,3)-type d-tensor field. They cannot coincide as it happens for the case oftorsions of a nonlinear connection and Berwald connection. However, we showhow can we determine one from another. Consider a nonlinear connection Nwith its curvature d-tensor field Rk

ij and the induced Berwald connection D

with its horizontal curvature d-tensor field Rklji. These two tensors are then

related by

Rklij =

∂Rkij

∂yl.

If Xi(x, y) are the components of a d-vector field on TM then from (3.17) wemay derive the Ricci identities of Xi with respect to an N -linear connectionD, although these may be written for an arbitrary tensor field.

Xi|j|k −Xi

|k|j = XmRimjk −Xi

|mTmjk −Xi|mRm

jk,

Xi|j |k −Xi|k|j = XmP i

mjk −Xi|mCm

jk −Xi|mPmjk ,

Xi|j |k −Xi|k|j = XmSimjk −Xi|mSm

jk.

(3.23)

If we use the first set of Ricci identities (3.23) for the Liouville vector field yi

we haveyi|j|k − yi

|k|j = R im jky

m −Rijk − yi

|mTmjk .

Since yi|j = yk(∂N i

k/∂yj) − N ij if the nonlinear connection is symmetric and

homogeneous then we have that yi|j = 0 and consequently one can obtain

the curvature Rijk of the nonlinear connection directly from the horizontal

component Rimjk of the N -linear connection by contraction with ym, which

meansRi

mjkym = Ri

jk.

Consequently, for a symmetric and homogeneous connection, the horizontalcurvature of Berwald connection can be derived from the curvature of thenonlinear connection.

3.6 N-linear connections induced by a complete pa-rallelism

So far we spoke about linear connections on TM that preserve by parallelismthe horizontal and vertical distributions. The parallelism induced by such alinear connection will preserve the direct sum (2.14). In this section we studysuch linear connections we can associate if a field of frames on TM is given,[53].

3.6. N -linear connections induced by a complete parallelism 53

Consider now Haa=1,2n a field of frames on some open submanifold ofTM . If this frame is adapted to the horizontal and the vertical distribu-tions, then with respect to the Berwald basis the frame has the form Ha =H i

α(δ/δxi), V iα(∂/∂yi)α=1,n. Consequently, we have that (Hα(u), Vα(u)) is a

basis of TuTM , adapted to the decomposition (2.14). We call it a nonholo-nomic frame, adapted to the vertical and the horizontal distributions. Thismeans that we have also two nonsingular matrices (H i

α(u)) and (V iα(u)) such

that Hα(u) = H iα(δ/δxi)|u and Vα(u) = V i

α(∂/∂yi)|u. We denote by (Hαi (u))

and (V αi (u)) the inverses of these two matrices, which means

H iαHα

j = δij , Hα

i H iβ = δα

β , V iαV α

j = δij , and V α

i V iβ = δα

β .(3.24)

Next we will deal only with nonholonomic frames for which Vα = J(Hα), thatis V i

α = H iα. This condition means that the frame Ha commutes with the

tangent structure J . Such a nonholonomic frame is called a Finsler frame.Since an N -linear connection D preserves by parallelism the horizontal and

the vertical distributions, and the tangent structure J is absolutely parallelwith respect to it, which means that DJ = 0, then we have the followingformulae:

DHαHβ = F γβαHγ ; DHαVβ = F γ

βαVγ ;

DVαHβ = CγβαHγ ; DVαVβ = Cγ

βαVγ .(3.25)

The set of functions (F γβα, Cγ

βα) are called the nonholonomic coefficients orscalar coefficients of an N -linear connection D with respect to the nonholo-nomic frame (Hα, Vα).

For an N -linear connection DΓ = (N ij , F

ijk, C

ijk) and a nonholonomic frame

H iα(u), the nonholonomic coefficients of D, F γ

βα and Cγβα are given by

F γ

αβ = Hγk Hk

α|iHiβ = −Hγ

k|iHkαH i

β;

Cγαβ = Hγ

k Hkα|iH i

β = −Hγk |iHk

αH iβ.

(3.26)

Theorem 3.6.1 There exists a unique N -linear connection D such that thegiven frame is h- and v-covariant constant. For this N -linear connection Dall components of curvature are zero.

Proof. The nonholonomic horizontal frame Hα is h-covariant constant if for allα ∈ 1, ..., n we have H i

α|j = 0. This is equivalent to δH iα/δxj + F i

mjHmα = 0.

If we solve this for F imj we have

F imj = −δH i

α

δxjHα

m = H iα

δHαm

δxj.


Similarly, the nonholonomic frame Hα is v-covariant constant if for all α ∈1, ..., n we have H i

α|j = 0. This is equivalent to ∂H iα/∂yj + Ci

mjHmα = 0. If

we solve this for Cimj we have

Cimj = −∂H i

α

∂yjHα

m = H iα

∂Hαm

∂yj.

If we use the Ricci identities (3.23) for H iα, we have: Ri

mkjHmα = 0, P i

mkjHmα =

0, and SimkjH

mα = 0, ∀α ∈ 1, ..., n. As Hm

α is invertible we obtain: Rimkj =

P imkj = Si

mkj = 0. q.e.d.

The N -linear connection we have defined in Theorem 3.4. is called theCrystallographic connection of the nonholonomic frame H i

α [18], [19] and thiscorresponds to the complete parallelism on TM induced by the field of frames(Hα, Vα).

3.7 Structure equations of an N-linear connection

Structure equations for an N -linear connection have been studied in [130].In this section we derive the structure equations for an N -linear connectionfollowing the general theory for a linear connection, [65], by specializing it forthe case of an N -linear connection, [53]. The integrability conditions for thesestructure equations are studied also, [53].

Next, we denote by Xa, a ∈ 1, ..., 2n the vector fields of the Berwaldbasis δ/δxi, ∂/∂yi induced by a nonlinear connection N and by θa a ∈1, ..., 2n the dual basis dxi, δyi. For an N -linear connection D, the con-nection 1-forms (ωa

b ), which correspond to this basis are defined as follows:

ωab (X) = θa(DXXb), ∀X ∈ χ(TM).

It is a straightforward calculation to check that the connection 1-forms are

given by ωab =

(ωi

j 00 ωi

j

), where ωi

j = F ijkdxk + Ci

jkδyk.

For a vector field W = W aXa ∈ χ(TM) we have that

DV W = (V (W a) + W bωab (V ))Xa, that is

θa(DV W ) = V (θa(W )) + θb(W )ωab (V ).

Theorem 3.7.1 1) The Cartan’s first structure equations of an N -linear con-nection D are given by:

−dxh ∧ ωih = −Θi,

d(δyi) −δyh ∧ ωih = −Θi,

(3.27)

3.7. Structure equations of an N -linear connection 55

where the 2-forms of torsions Θa = (Θi, Θi) are defined by

Θa(X, Y ) = θa(T (X, Y )), and are given by

Θi =12T i

jkdxj ∧ dxk + Cijkdxj ∧ δyk,

Θi =12Ri

jkdxj ∧ dxk + P ijkdxj ∧ δyk +

12Si

jkδyj ∧ δyk.

(3.28)

2) The Cartan’s second structure equations of an N -linear connection Dare given by

dωij − ωh

j ∧ ωih = −Ωi

j ,(3.29)

where the curvature 2-forms (Ωab ) =

(Ωi

j 00 Ωi

j

), are defined by

Ωab (X,Y ) = θa(R(X, Y )Xb) and Ωi

j are given by

Ωij =

12R i

j khdxk ∧ dxh + P ij khdxk ∧ δyh +

12S i

j khδyk ∧ δyh.(3.30)

Proof. We have that

Θa(X, Y ) = θa(T (X, Y )) = θa(DXY )− θa(DY X)− θa([X,Y ])

= X(θa(Y )) + θb(Y )ωab (X)− Y (θa(X))− θb(X)ωa

b (Y )− θa([X,Y ])

= dθa(X,Y ) + (ωab ∧ θb)(X, Y ).

If we take θa to be dxi and δyi, respectively, then we get the Cartan’s firststructure equations (3.27).

From Ωab (X,Y ) = θa(R(X, Y )Xb) = dωa

b (X, Y ) + (ωac ∧ ωc

b)(X, Y ) we havethe Cartan’s second structure equations (3.29). q.e.d.

The torsion 2-forms Θi contain the horizontal components of the torsion ofthe N -linear connection D. We shall call them the horizontal torsion 2-formsof the N -linear connection D. For the Berwald connection D induced by anonlinear connection, the horizontal torsion 2-forms Θi vanish if and only ifthe nonlinear connection is symmetric. The torsion 2-forms Θi will be calledthe vertical torsion 2-forms.

Proposition 3.7.1 If for an N -linear connection D on TM the curvature2-forms Ωi

j vanish, then there exists a nonholonomic frame (H iα) such that the

local coefficients of the connection D are given by the following formulae:

F ijk = −δH i

α

δxkHα

j = H iα

δHαj

δxk;

Cijk = −∂H i

α

∂ykHα

j = H iα

∂Hαj

∂yk.

(3.31)


Proof. If the curvature 2-forms of D vanish, then the Cartan’s second structureequations are:

dωab + ωa

c ∧ ωcb = 0.

According to the general theory of linear connection, there exists a frame onthe tangent space TM , whose components with respect to the Berwald basisare Ha

b (x, y), such thatdHa

b + ωac Hc

b = 0.(3.32)

The parallelism induced by the N -linear connection D is path independent andperfectly determined by the field of frames Ha

b . As the N -linear connectionpreserves by parallelism the horizontal and the vertical distributions then the

frame Hab has the form Ha

b =

(H i

j 00 H i

j

).

We also have that the tangent structure J is absolutely parallel with respectto the N -linear connection D. This will imply that the frame H and thetangent structure J commute. From this we have that H i

j = H ij

and the field

of frames has the form Hab =

(H i

j 00 H i

j

).

The connection 1-forms of the N -linear connection D are given by ωab =

Hac d(H−1)c

b = −dHac (H−1)c

b. If we take into account the particular form ofthe connection 1-forms ωa

b and the field of frames Hab have, we obtain that

ωij = H i

l d(H−1)lj = −dH i

l (H−1)l

j and consequently, the local coefficients of Dare given by expression (3.31). q.e.d.

The frame Hα = H iα(δ/δxi), Vα = H i

α(∂/∂yi) is said to be holonomic ifthere exist n functions φα on the base manifold M such that Hα

i = ∂φα/∂xi

that is equivalent to say that the 1-forms ηα = Hαi dxi are exact.

Proposition 3.7.2 A frame H iα is holonomic if and only if the horizontal tor-

sion 2-forms Θi, defined by (3.28) of the Crystallographic connection inducedby H i

α, vanish.

Proof. From expressions (3.28) we have that Θi = 0 if and only if T ijk = 0 and

Cijk = 0, where T i

jk = F ikj − F i

jk, and F ijk and Ci

jk are given by (3.31). ButCi

jk = 0 if and only if H iα are functions of (x) only. Then T i

jk = 0 if and only if∂Hα

j /∂xk = ∂Hαk /∂xj and this is equivalent to the fact that Hα

i are gradientsof n functions φα on the base manifold M . q.e.d.

Theorem 3.7.2 Consider D an N -linear connection on TM with local coeffi-cients (F i

jk, Cijk). The horizontal torsion 2-forms Θi and the curvature 2-forms

Ωij vanish if and only if there are local coordinates on the base manifold such

that with respect to the induced coordinates on TM we have F ijk = Ci

jk = 0.

3.8. Geodesics of an N -linear connection 57

Proof. If the curvature 2-forms Ωij of the N -linear connection D vanish then,

according to the Proposition 3.7.1 there is a frame H iα such that the local

coefficients of the N -linear connection D are given by (3.31). From Proposition3.7.2 we have that the frame Hα

i is holonomic, which means that there existn functions φα such that Hα

i = ∂φα/∂xi. Consequently, φα are coordinatefunctions on M . With respect to the induced coordinates on TM the localcoefficients of the N -linear connection D vanish. q.e.d.

If we apply the previous theorem to the Berwald connection induced by asymmetric nonlinear connection N we have the following result:

Proposition 3.7.3 Let N be a symmetric nonlinear connection with local co-efficients N i

j . Then there exists local coordinates (xi) on the base manifold M

such that with respect to the induced coordinates on TM , the functions N ij are

functions of (x) only, if and only if the Berwald connection induced by N haszero curvature.

Proof. The Berwald connection has zero curvature if and only if about everypoint on M there are local coordinates such that with respect to the inducedcoordinates on TM , F i

jk = ∂N ij/∂yk = 0. This is equivalent to the fact that

functions N ij depend on x only. q.e.d.

We shall use this result later on for the case when the nonlinear connectionis determined by a system of second order differential equations and we shallfind conditions in which the system is linearizable with respect to velocitycoordinates.

3.8 Geodesics of an N-linear connection

For this section, we fix N a symmetric nonlinear connection and D an N -linearconnection. For this N -linear connection D, we study its geodesics. Then,we determine necessary and sufficient conditions by which the autoparallelcurves of the nonlinear connection coincide with the geodesics of the N -linearconnection D.

A smooth curve c : t ∈ I 7→ c(t) = (xi(t), yi(t)) ∈ TM is a geodesic of theN -linear connection D if Dcc = 0. Here

c(t) =dxi

dt

∂

∂xi+

dyi

dt

∂

∂yi=

dxi

dt

δ

δxi+

δyi

dt

∂

∂yi

is the tangent vector along c and

δyi

dt:= ∇

(dxi

dt

)=

dyi

dt+ N i

j

(x,

dx

dt

)dxj

dt.


For an N -linear connection D = (N ij , F

ijk, C

ijk) a curve c(t) = (xi(t), yi(t)) is

a geodesic of D if and only if the following equations are satisfied:

d2xi

dt2+ F i

jk

dxj

dt

dxk

dt+ Ci

jk

dxj

dt

δyk

dt= 0,

ddt

(δyi

dt

)+ F i

jk

δyj

dt

dxk

dt+ Ci

jk

δyj

dt

δyk

dt= 0.

(3.33)

From these equations we can see that a horizontal curve, which means thatδyi/dt = 0, is a geodesic of the N -linear connection D if and only if

d2xi

dt2+ F i

jk

dxj

dt

dxk

dt= 0.(3.34)

As we are interested on the geometric properties of the base manifold M , weconsider smooth curves on M , c : t ∈ I 7→ c(t) = (xi(t)) ∈ M . We say thatsuch a curve, c is a geodesic of an N -linear connection D if its natural lift toTM , c(t) = (xi(t),dxi/dt) is a geodesic of D.

Consequently, we have that an autoparallel curve c(t) = (xi(t)) of a nonlin-ear connection is a geodesic of an N -linear connection if and only if equations(3.34) are satisfied.

Theorem 3.8.1 Consider D an N -linear connection such that the horizontaland vertical tensors of deflection satisfy

Dij := yi

|j = F imjy

m −N ij = 0 and di

j := yi|j = δij + Ci

mjym = δi

j .(3.35)

For a curve c on the base manifold M , its complete lift c(t) is an autoparallelcurve for the nonlinear connection N if and only if c is a geodesic for theN -linear connection D.

Proof. As N ij = F i

mjym and Ci

mjym = 0, we have that equations (3.33)

and (3.34) are equivalent. This shows that geodesic curves of the nonlinearconnection D that satisfy (3.35) and autoparallel curves for the nonlinear Ncoincide. q.e.d.

If for an N -linear connection D, the curvature 2-forms Ωij vanish, then

there exists a nonholonomic frame Hα = H iα(δ/δxi), Vα = H i

α(∂/∂yi) suchthat the nonholonomic components of D with respect to it vanish. If wedenote by dxα/dt = Hα

i dxi/dt and δyα/dt = Hαi δyi/dt, the nonholonomic

components of the tangent vector c(t), then the smooth curve c is a geodesicof D if and only if

ddt

(dxα

dt

)= 0,

ddt

(δyα

dt

)= 0.

(3.36)

3.9. Homogeneous Berwald connection 59

If more then that, the torsion 2-forms Θi vanish and the curvature 2-formsΩi of an N -linear connection D vanish, then around every point on the basemanifold M there exists local coordinates (xi) such that the equations ofgeodesics of D are given by:

ddt

(dxi

dt

)= 0,

ddt

(δyi

dt

)= 0.

(3.37)

In such a case, the manifold is flat and the geodesics are straight lines.

3.9 Homogeneous Berwald connection

We have seen in the previous sections that by studying the Berwald connec-tion we obtain information about the nonlinear connection. Consider now ahomogeneous nonlinear connection. For the Berwald connection, its deflec-tion tensors satisfy (3.35) and according to Theorem 3.8.1 the complete liftof a curve from the base manifold is an autoparallel curve for the nonlinearconnection if and only it is a geodesic of the Berwald connection.

For a homogeneous connection N we have that its local coefficients N ij(x, y)

are homogeneous functions of order one with respect to y. We recall here thatfor homogeneous functions we assume that the functions are of C∞ class onTM and only continuous on the null section of the tangent bundle TM . Thisis to avoid the particular case when a homogeneous function of order r isa polynomial function of order r in y. Consider D the Berwald connectionassociated to N . Since N i

j are homogeneous functions of order one, we havethat yk(∂N i

j/∂yk) = N ij , which means that F i

jkyk = N i

j . Then the followingequations

δyi

dt=

d2xi

dt2+ N i

j

(x,

dx

dt

)dxj

dt= 0 and

d2xi

dt2+ F i

jk

(x,

dx

dt

)dxj

dt

dxk

dt= 0

are equivalent. This means that the autoparallel curves of a homogeneousconnection coincide with the geodesics of the induced Berwald connection.

For this particular case, we have that if X = Xi(∂/∂xi) ∈ χ(M) thenthe (1,2)-type tensor field ai

jk(x, y,X) given by expression (2.50) is the Liederivative of F i

jk with respect to the complete lift Xc of X, which means

aijk(x, y,X) = LXc(F i

jk(x, y)).

Consequently, a vector field X ∈ χ(M) is a Lie symmetry of the nonlinearconnection N if and only if

LXc(F ijk(x, y))yk = 0.


For a Lie symmetry X ∈ χ(M) consider φt its one-parameter group. If c is ageodesic of the Berwald connection, then t 7→ φt(c(t)) is again a geodesic ofthe Berwald connection.

Chapter 4

Second Order DifferentialEquations

Viewed as a dynamical system, a system of second order ordinary differentialequations (SODE) is locally defined on the base manifold M by a system (4.1)and globally defined on the tangent bundle TM by a semispray S. Integralcurves of the semispray are solutions of the system (4.1). Global propertiesof this system are derived from the associated semispray S. In this chapterwe study the geometric theory of an SODE following the geometric setupdeveloped in previous chapters. Accordingly, we shall associate to an SODEa nonlinear connection N on TM and an N-linear connection D, namely theBerwald connection, [80], [66]. Cartan’s structure equations of the Berwaldconnection D will give us the geometric invariants of the system, [21]. We alsostudy the symmetries of an SODE, by specializing the results we obtained forthe symmetries of the induced nonlinear connection N .

The geometric theory of an SODE, is named by P.L. Antonelli in [23] theKCC-theory, after its initiators D.D. Kosambi, [103], E. Cartan, [61] and S.S.Chern, [64]. The KCC-theory of an SODE has been intensively studied in [20],[21]. In the next chapter we shall study a metric geometry of an SODE, whichis the geometry of a pair made of a metric structure and an SODE on TM .

4.1 Second order differential vector field

In this section we introduce the geometric theory of a dynamical system de-scribed by a system of second order differential equations, as the geometryone can derive from a special vector field that lives on the tangent bundle of amanifold, [77], [61], [64], [103]. This vector field is a semispray on the tangentbundle of our configuration space.

We start with an n-dimensional manifold M , which is the configuration

62 Chapter 4. Second Order Differential Equations

space of a dynamical system governed by a system of second order ordinarydifferential equations:

d2xi

dt2+ 2Gi

(x,

dx

dt

)= 0.(4.1)

To be more accurate we have to say that each system (4.1) is defined overa local chart on TM . Hence, we have a collection of systems (4.1) on everyinduced local chart on TM that are compatible on the intersection of inducedlocal charts. This compatibility means that under a change (1.1) of localinduced coordinates on TM the left hand side of (4.1) is a d-vector fieldon TM . This is equivalent to say that the functions Gi(x,dx/dt) transformaccording to:

2Gi =∂xi

∂xj2Gj − ∂yi

∂xjyj .(4.2)

Proposition 4.1.1 The vector field S = yi(∂/∂xi)−2Gi(x, y)(∂/∂yi) is glob-ally defined on TM if and only if the functions Gi(x, y), defined on domains ofinduced local charts, satisfy the rule (4.2) under a change of local coordinates(1.1) on TM .

Proof. It is a straightforward calculation to check that under a change ofcoordinates (1.1) on TM we have that

yi ∂

∂xi− 2Gi(x, y)

∂

∂yi= yi ∂

∂xi− 2Gi(x, y)

∂

∂yi

if and only if the functions Gi and Gi are related by (4.2). q.e.d.

Definition 4.1.1 A vector field S ∈ χ(TM) is called a semispray, or a secondorder vector field if JS = C.

Proposition 4.1.2 A vector field S ∈ χ(TM) is a semispray if and only ifon every domain of local charts on TM we have the functions Gi such thatS = yi(∂/∂xi)− 2Gi(x, y)(∂/∂yi).

Proof. A vector field S = Ai(x, y)(∂/∂xi) + Bi(x, y)(∂/∂yi) on TM is asemispray if and only if JS = Ai(x, y)(∂/∂yi) = C = yi(∂/∂yi). If we takeBi = −2Gi we have that S is a semispray if and only if S = yi(∂/∂xi) −2Gi(x, y)(∂/∂yi). q.e.d.

The functions Gi(x, y) are called the local coefficients of the semispray.The functions Gi are supposed to be of C∞-class on TM and only continuouson the null section.

4.2. Nonlinear connections and semisprays 63

Proposition 4.1.3 A vector field S ∈ χ(TM), which is a section of the tan-gent bundle (TTM, τ, TM), is a semispray if and only if S is a section of thebundle (TTM, π∗, TM).

Proof. Let S = Ai(x, y)(∂/∂xi) − 2Gi(x, y)(∂/∂yi) be a vector field on TM .As π∗ : (x, y, X, Y ) ∈ TTM 7→ (x, X) ∈ TM we have that S is a section of π∗if and only if π∗ S = IdTM that is (xi, Ai(x, y)) = (xi, yi) and the proof isfinished. q.e.d.

Definition 4.1.2 A smooth curve c : t ∈ I 7→ c(t) = (xi(t)) ∈ M is said to bea path of a semispray S if its complete lift c : t ∈ I 7→ c(t) = (xi(t),dxi/dt) ∈TM is an integral curve of the vector field S.

If S = yi(∂/∂xi)− 2Gi(x, y)(∂/∂yi) then a smooth curve c(t) = (xi(t)) onM is a path of S if and only if c is a trajectory of (4.1). From Proposition4.1.1 we have seen that a collection of compatible systems (4.1) determine asemispray S with local coefficients Gi.

4.2 Nonlinear connections and semisprays

The theory of connections we developed in the previous chapters can be appliednow to systems (4.1) by means of the associated semispray. In order to do so,one associates to every system (4.1) a nonlinear connection, [80], [66]. Thena solution of a system (4.1) becomes an autoparallel curve for the inducedconnection. Consequently we can use variational equations and symmetrieswe derived for the autoparallel curves of a connection in Section 2.5.

Theorem 4.2.1 If S is a semispray, then P = −LSJ is an almost productstructure on TM that satisfies (2.24), which means that P J = −J andJ P = J .

Proof. We have to prove that the F(TM)-morphism P : χ(TM) → χ(TM),given by P(X) = −(LSJ)(X) = −[S, JX]+J [S,X] satisfies conditions (2.24).First we prove the formula

J [JX, S] = JX, ∀X ∈ χ(TM).(4.3)

Since the Nijenhuis tensor NJ of the tangent structure J , vanishes we have0 = NJ(S, X) = [C, JX]− J [C, X]− J [S, JX]. But the tangent structure J is0-homogeneous that is [C, JX]−J [C, X] = −JX. Consequently, we have thatJ [JX, S] = JX.

Now, we have that JP(X) = −J [S, JX] = JX, ∀X ∈ χ(TM) so, JP = J.Also, PJ(X) = J [S, JX] = −JX, ∀X ∈ χ(TM) and so PJ = −J . We


have then that (2.24) formulae are true and according to Proposition (2.4.3)the almost product structure P determines a nonlinear connection N on TM .Since

P(

∂

∂xi

)= J

[S,

∂

∂xi

]−

[S,

∂

∂yi

]=

∂

∂xi− 2

∂Gj

∂yi

∂

∂yj,

we have that the local coefficients of the induced nonlinear connection areN i

j = ∂Gi/∂yj . We can check this directly. Let Gi be the local coefficientsof a semispray S. Then under a change of local coordinates (1.1) on TM thefunctions Gi satisfy the formula (4.2). Using this we obtain that the functionsN i

j = ∂Gi/∂yj satisfy the formula (2.19) and according to Proposition 2.3.1they are the local coefficients of a nonlinear connection N on TM . q.e.d.

According to Theorem 4.2.1 we have that a semispray determines a nonlin-ear connection with local coefficients N i

j = ∂Gi/∂yj . As ∂N ij/∂yk = ∂N i

k/∂yj

we have that the nonlinear connection is symmetric. We remark here thatfor a (1,1)-type d-tensor field Xi

j(x, y), N ij = ∂Gi/∂yj + Xi

j are also the co-efficients of a nonlinear connection. We shall see later that for a generalizedLagrange space one can find a system of axioms that uniquely determine anonlinear connection associated to a semispray. This nonlinear connectioncoincides with the nonlinear connection determined in Theorem 4.2.1 for theparticular case when the metric structure is Lagrangian.

Let N be the nonlinear connection induced by a semispray S. As wehave seen in Chapter 2, a nonlinear connection N determines a horizontaldistribution that is supplementary to the vertical distribution. This meansthat the following direct sum holds good:

TuTM = HuTM ⊕ VuTM.

The horizontal and the vertical projectors that correspond to the above de-composition are given by:

h(X) =12(X − [S, JX]− J [X, S]),

v(X) =12(X + [S, JX] + J [X, S]).

(4.4)

Theorem 4.2.2 If N is a nonlinear connection on TM and h is the horizontalprojector, then there exists a unique semispray S such that

S = h[C, S].(4.5)

Proof. Let S′ be an arbitrary semispray on TM and denote by S = hS′. ThenS is a semispray on TM , too. Indeed as Jh = J and JS′ = C, then JS = C.More then that, the semispray S does not depend on the semispray S′. That

4.2. Nonlinear connections and semisprays 65

is if S′′ is another semispray on TM , then hS′ = hS′′. This is true because ifS′ and S′′ are two semisprays on TM , then J(S′−S′′) = 0 and their differenceS′ − S′′ is vertical. Consequently hS′ = hS′′.

Now, we prove that S = hS′ = hS satisfies (4.5). From (4.3) if we takeX = S, then J [JS, S] = JS and because JS = C, we have that J [C, S]−C = 0that is equivalent to J([C, S]− S) = 0. Consequently, [C, S] − S is a verticalvector field so (4.5) is true.

If S is a semispray on TM such that equation (4.5) holds true we havethat S′ = [C, S] is a semispray. Consequently, S = h[C, S] = hS′ = S and thetheorem is proved. q.e.d.

In local coordinates the semispray induced by a nonlinear connection Nwith local coefficients N i

j(x, y) is given by

S = yi δ

δxi= yi ∂

∂xi−N i

jyj ∂

∂yi.(4.6)

This means that local coefficients of the induced semispray are given by2Gi(x, y) = N i

j(x, y)yj .For a semispray S = yi(∂/∂xi) − 2Gi(∂/∂yi) we consider the induced

nonlinear connection N with local coefficients N ij = ∂Gi/∂yj . With respect

to this nonlinear connection, we have the following formula:

S = yi δ

δxi− (2Gi −N i

jyj)

∂

∂yi= yi δ

δxi− E i ∂

∂yi= hS − E i ∂

∂yi.(4.7)

The d-vector field E i(x, y) = 2Gi(x, y)−N ij(x, y)yj = 2Gi(x, y)− yj(∂Gi/∂yj)

is called the first invariant of the semispray.

Definition 4.2.1 A semispray S is said to be a spray if the first invariant E i

vanishes.

We have that a semispray S is a spray if and only if the coefficient functionsGi(x, y) are homogeneous of order two, which according to Euler’s Theoremmeans that 2Gi(x, y) = yj(∂Gi/∂yj). This is equivalent to say that S is ahomogeneous vector field of order two on TM . We can express this by sayingthat a semispray S is a spray if and only if LCS = [C, S] = S.

Proposition 4.2.1a) Let S be semispray and N the induced nonlinear connection. Then S is

a spray if and only if it coincides with the semispray induced by N .b) Let N be a symmetric nonlinear connection on TM and S the induced

semispray. The nonlinear connection induced by S coincides with the givennonlinear connection N if and only if this is homogeneous.


Proof.a) Let Gi be the local coefficients of the semispray S. Then the induced

nonlinear connection N has the local coefficients N ij = ∂Gi/∂yj . The semis-

pray S′ induced by N has the local coefficients 2G′i = N ijy

j = yj(∂Gi/∂yj).We have that S = S′ if and only if G′i = Gi, which is equivalent to 2Gi =yj(∂Gi/∂yj) and this means that S is a spray.

b) Let N be a symmetric nonlinear connection with N ij the local coef-

ficients. The symmetry means that ∂N ij/∂yk = ∂N i

k/∂yj . The semisprayS induced by N has the local coefficients 2Gi = N i

jyj . Then, the nonlin-

ear connection induced by S has as local coefficients 2N ′ij = 2∂Gi/∂yj =

N ij + yj(∂N i

k/∂yj) = N ij + yk(∂N i

k/∂yj). The two nonlinear connections coin-cide if and only if N i

j = N ′ij , which is equivalent to N i

j = yk(∂N ij/∂yk), which

means exactly that the nonlinear connection N is homogeneous. q.e.d.

4.3 Berwald connection of a semispray

The Berwald connection has been used to study systems of second order dif-ferential equations by Berwald in [44]. In this section we study necessaryand sufficient conditions in which an SODE is linearizable in velocity. Suchconditions are determined using Cartan structure equations for the Berwaldconnection, [53]. A different approach for this problem can be found in [67].

Consider S a semispray with local coefficients Gi and N the induced nonlin-ear connection with local coefficients N i

j = ∂Gi/∂yj . The Berwald connectionD induced by N is an N -linear connection with the local coefficients

BΓ =(

N ij =

∂Gi

∂yj, F i

jk =∂2Gi

∂yj∂yk, Ci

jk = 0)

.

The connection 1-forms of the Berwald connection D are then given by

ωij = F i

jkdxk =∂2Gi

∂yj∂ykdxk.

Since the nonlinear connection is symmetric then the h(h)-torsion (3.15) ofthe Berwald connection D vanishes and consequently the connection D issymmetric. This connection has only one component of torsion, the v(h)-torsion that is also the curvature of the nonlinear connection

vT

(δ

δxi,

δ

δxj

)= Rk

ji

∂

∂yk=

(δNk

j

δxi− δNk

i

δxj

)∂

∂yk.(4.8)

4.3. Berwald connection of a semispray 67

The horizontal 2-forms of torsion Θi of the Berwald connection vanish then.The vertical 2-forms of torsion of the Berwald connection are given by

Θi =12Ri

jkdxj ∧ dxk.

The two nonzero components of curvature for the Berwald connection D are:

Rihjk =

δF ihj

δxk− δF i

hk

δxj+ Fm

hjFimk − Fm

hkFimj ;

Dihjk =

∂3Gi

∂yh∂yj∂yk.

(4.9)

The curvature 2-forms of the Berwald connection are given by

Ωij =

12Ri

jkhdxk ∧ dxh + Dijkhdxk ∧ δyh.

For the Berwald connection, Theorem 3.7.1 takes the following particular form:

Theorem 4.3.1 The Cartan’s first structure equations of the Berwald con-nection D are given by

− dxh ∧ ωih = 0,

d(δyi) − δyh ∧ ωih = −1

2Ri

jkdxj ∧ dxk.(4.10)

The Cartan’s second structure equations of the Berwald connection D are givenby

dωij − ωh

j ∧ ωih = −1

2Ri

jkhdxk ∧ dxh −Dijkhdxk ∧ δyh.(4.11)

The above Cartan’s structure equations of the Berwald connection D are usefulto determine necessary and sufficient conditions in which the system of secondorder differential equations (4.1) is linearizable in velocities.

Theorem 4.3.2 The Berwald connection of a semispray S has zero curvature(is flat) if and only if about every point p ∈ M there are local coordinates (xi)in M such that with respect to the induced coordinates (xi, yi) on TM , thelocal coefficients of the semispray S have the form

2Gi(x, y) = Aij(x)yj + Bi(x).(4.12)

Proof. If there exist induced coordinates on TM such that the semispray Shas local coefficients 2Gi(x, y) = Ai

j(x)yj +Bi(x) then the local coefficients ofthe Berwald connection D vanish that is F i

jk = ∂2Gi/∂yj∂yk = 0. From (4.9)


we can see that the curvature components of D vanish and consequently theBerwald connection is flat.

Now, we assume that the curvature 2-forms Ωij of the Berwald connection

vanish. Since the horizontal torsion 2-forms Θi are zero, according to Theorem3.7.2 there are induced coordinates on TM with respect to which the localcoefficients of the Berwald connection vanish: F i

jk = 0 and Cijk = 0. But F i

jk =∂2Gi/∂yj∂yk, so with respect to these coordinates we have that 2Gi(x, y) =Ai

j(x)yj + Bi(x). q.e.d.

Let us consider now the system (4.1) we started this section with

d2xi

dt2+ 2Gi

(x,

dx

dt

)= 0.

According to the Theorem 4.3.2, this system is linearizable in velocities, whichmeans that it has the following form

d2xi

dt2+ Ai

j(x)dxj

dt+ Bi(x) = 0(4.13)

if and only if the curvature of the induced Berwald connection vanishes.The systems (4.13) are used to describe models in biology, such systems

are known as Laird’s law in multidimensional growth, [16].

4.4 Jacobi equations of a semispray

A variation for the solution curves of (4.1) can be found in [44], [67] and [20].We defined the dynamical covariant derivative induced by a nonlinear con-

nection in Section 2.7. This covariant derivative was useful to determine aninvariant form (2.40) for the autoparallel curves of a nonlinear connection.We studied also the variational equations of the autoparallel curves and wefound an invariant form (2.43) using this covariant derivative. Now we applyall these considerations for the particular case when the nonlinear connectionis induced by a semispray. This way we can get information about the system(4.1).

If Xi(x, y) is a d-vector field on TM , we define its dynamical covariantderivative by

Xi| = ∇Xi = S(Xi) + N i

jXj .

It can be easily proved that ∇Xi is still a d-vector field. Then we can lookto the dynamical covariant derivative as a map ∇ : χv(TM) −→ χv(TM),defined by

∇(

Xi ∂

∂yi

)=

(∇Xi) ∂

∂yi= Xi

|∂

∂yi.

4.4. Jacobi equations of a semispray 69

We can also view the dynamical covariant derivative as a map ∇ : χ(M) −→χv(TM), defined by ∇(Xi∂/∂xi) = ∇Xi(∂/∂yi). It can be seen that the firstdefinition can be derived from the second one by composition with the verticallift.

Using the dynamical covariant derivative, the system of equations (4.1)takes the form:

∇(

dxi

dt

)= −E i

(x,

dx

dt

).(4.14)

We may remark here that both sides of the equation (4.14) behave like a vectorfield, so they are d-vector fields.

Let c(t) = (xi(t)) be a trajectory of (4.1). If we perform a variation of thistrajectory into nearby ones according to

xi(t) = xi(t) + εξi(t)

as we did for the autoparallel curves of a nonlinear connection we get thevariational equations:

d2ξi

dt2+ 2

∂Gi

∂yj

dξj

dt+ 2

∂Gi

∂xjξj = 0.(4.15)

Theorem 4.4.1 For the variational equations (4.15) we have the equivalentinvariant form (Jacobi equations):

∇2ξi +(

Rijk

dxk

dt+ E i

|j

)ξj = 0.(4.16)

Here E i|j = δE i/δxj + F i

kjEk is the h-covariant derivative (with respect to theBerwald connection) of the first invariant E i. A vector field (ξi(t)) along apath c(t) of the semispray S is called a Jacobi vector field if it satisfies thevariational equations (4.15).

Proof. Denote by

Bij = 2

∂Gi

∂xj− S

(∂Gi

∂yj

)− ∂Gi

∂yr

∂Gr

∂yj.(4.17)

It can be proved that Bij is a (1,1)-type d-tensor field. It has been introduced in

[44], for the homogeneous case. This tensor field is called the second invariantof the given SODE in [103], [61] and [64], the Jacobi endomorphism in [67] orvertical endomorphism.

It is easy to check that the equations (4.15) are equivalent to

∇2ξi + Bijξ

j = 0.(4.18)


All we have to prove now is the following expression of the second invariant:

Bij = Ri

jkyk + E i

|j .(4.19)

Let Xi(x, y) be an arbitrary d-vector field, and consider the vector field

X = Xi ∂

∂xi+ S(Xi)

∂

∂yi

on TM . We have then

[S, X] = (∇2Xi + BijX

j)∂

∂yj.

If we consider the expression of S and X in the Berwald basis we have

S = yi δ

δxi− E i ∂

∂yiand X = Xi δ

δxi+∇Xi ∂

∂yi.

Then, the Lie bracket [S, X] can be expressed as follows:

[S, X] = ∇2Xi + (Rijky

k + E i|j)X

j ∂

∂yi.

If we compare the above two formulae and we take into account that Xi(x, y)is an arbitrary d-vector field, then the second invariant Bi

j can be expressedas in expression (4.19). q.e.d.

4.5 Symmetries of a semispray

We study in this section vector fields whose one-parameter group of transfor-mations preserve the solution curves of the system (4.1). Such vector fieldsare called symmetries of a semispray.

Definition 4.5.11) A Lie symmetry of the semispray S is a vector field X on the base

manifold M such that [S,Xc] = 0, where Xc is the complete lift of X.2) A dynamical symmetry of the semispray S is a vector field X on TM

such that [S, X] = 0.3) A function f ∈ F(TM) is a constant of motion (or a conservation law)

of the semispray S if S(f) = 0.

If X ∈ χ(M) is a Lie symmetry of S then its complete lift Xc is a dynamicalsymmetry of S.

4.5. Symmetries of a semispray 71

Since for a vector field X ∈ χ(M) its complete and vertical lifts are relatedby the following formula: Xc = 2Xh+[S,Xv] we have that X is a Lie symmetryof S if and only if

2[S, Xh] + [S, [S, Xv]] = 2LSXh + LSLSXv = 0.(4.20)

If f is a conservation law for a semispray S, then f is constant along thepaths of S.

Theorem 4.5.11) A vector field X = Xi(x, y)(δ/δxi) + Y i(x, y)(∂/∂yi) is a dynamical

symmetry of S if and only if

Y i = ∇Xi and ∇2Xi + BijX

j = 0.(4.21)

2) If X = Xi(x, y)(δ/δxi)+Y i(x, y)(∂/∂yi) is a dynamical symmetry of thesemispray S and c(t) = (xi(t)) is a path of S, then the restriction of Xi(x, y)along c(t) = (xi(t), dxi/dt) is a Jacobi vector field for S.

Proof.1) If we express the Lie bracket [S, X] using the Berwald basis, we have

[S, X] = (∇Xi − Y i)δ

δxi+ (∇Y i + Bi

jXj)

∂

∂yj.(4.22)

Therefore, X is a dynamical symmetry of S if and only if equations (4.21) aresatisfied.

2) If Xi(x, y) are the horizontal components of a dynamical symmetry X,then ∇2Xi + (Ri

jkyk + E i

|j)Xj = 0. The restriction of this vector field along

the curve c satisfies the equations (4.16) and then Xi is a Jacobi vector fieldalong c. q.e.d.

The Jacobi equations (4.16) are the invariant form of the variational equa-tions (4.15) using the dynamical covariant derivative.

Also, equations (4.21) represent the invariant equations of dynamical sym-metries (or Lie symmetries) in terms of dynamical covariant derivative.

For a vector field X = Xi(∂/∂xi) ∈ χ(M), we consider:

ai(x, y, X) =∂2Xi

∂xj∂xkyjyk − 2Gj ∂Xi

∂xj+ 2

∂Gi

∂xjXj + 2

∂Xj

∂xkyk ∂Gi

∂yj.

The Lie derivative of 2Gi with respect to the complete lift Xc of a vector fieldX = Xi(∂/∂xi) ∈ χ(M) is defined as follows:

LXc(2Gi) = Xc(2Gi)− 2Gj ∂Xi

∂xj+

∂2Xi

∂xj∂xkyjyk.

For every X ∈ χ(M) we have that ai(x, y,X) = LXc(2Gi) is a d-vector fieldon TM .


Proposition 4.5.1a) A vector field X ∈ χ(M) is a Lie symmetry for a semispray S if and

only ifLXc(2Gi(x, y)) = ai(x, y, X) = 0.(4.23)

b) A vector field ξi(t) along a trajectory c(t) = (xi(t)) of (4.1) is a Jacobivector field if and only if

Lξc

(2Gi

(x,

dx

dt

))= ai

(x,

dx

dt, ξ

)= 0.(4.24)

Proof. We have that for every X ∈ χ(M),

[S,Xc] = ai(x, y, X)∂

∂yi= LXc(2Gi)

∂

∂yi.

Then, if we use Theorem (4.5.1) for the particular case of a complete lift of avector field X ∈ χ(M) we obtain imediately that both statements a) and b)of the proposition are true. q.e.d.

4.6 Geometric invariants of an SODE

For a system (4.1) it is important to determine the geometric invariants underthe pseudo-group of transformations xi = xi(xj), rank(∂xi/∂xj) = n.

These geometric invariants where determined by Kosambi [103], Chern[64], and Cartan [61] using the equivalence method. We want to prove nowthat for a given semispray we can determine all these five geometric invariantsusing the associated nonlinear connection and its Berwald connection.

The first invariant is E i and it was defined in (4.7) as the vertical compo-nent of the semispray. The second invariant is Bi

j , the Jacobi endomorphism,defined in (4.17) and it has been used to study the Jacobi equations and thesymmetries of a semispray.

The third, fourth and fifth invariants, as they were defined in [103], [64],[61] are:

Bijk :=

13

(∂Bi

j

∂yk− ∂Bi

k

∂yj

),

Biljk :=

∂Bijk

∂yl,

Dijkl :=

∂F ijk

∂yl=

∂3Gi

∂yj∂yk∂yl.

(4.25)

According to expression (4.9) we have that the tensor Dijkl is one of the nonzero

components of the curvature of the Berwald connection.

4.7. Homogeneous SODE 73

Theorem 4.6.11) The curvature Ri

jk of the nonlinear connection N (or the (v)h-torsionof the Berwald connection D) is the third invariant of the semispray S.

2) The Riemann-Christoffel curvature d-tensor Rijkl of the Berwald con-

nection D is the fourth invariant of the semispray S.

Proof. We have to prove that Rijk = Bi

jk and Rijkl = Bi

jkl. First we prove thatRi

jk and Rijkl satisfy second equation (4.25), that is Ri

ljk = ∂Rijk/∂yl.

From (2.28) we have

Rkji =

δNkj

δxi− δNk

i

δxj, so

∂Rkji

∂yl=

∂

∂yl

(δNk

j

δxi

)− ∂

∂yl

(δNk

i

δxj

).

Since [δ/δxj , ∂/∂yi] = F kji(∂/∂yk), we have that

∂Rkji

∂yl=

δ

δxi

(∂Nk

j

∂yl

)− F p

li

∂Nkj

∂yp− δ

δxj

(∂Nk

i

∂yl

)+ F p

lj

∂Nki

∂yp

=δF k

jl

δxi− δF k

il

δxj+ F p

jlFkpi − F p

ilFkjp = Rk

lji.

According to expression (4.19) the second invariant Bij , has the expression

Bij = Ri

jkyk + E i

|j . Consequently, ∂Bij/∂yk = Ri

jk + Rikljy

l + E i|j |k. Then,

∂Bij

∂yk− ∂Bi

k

∂yj= 2Ri

jk + Rikljy

l + Rijkly

l + E i|j |k − E i

|k|j .

Using the Ricci identities (3.23) for the Berwald connection D and the firstinvariant E i we have that E i

|j |k − E i|k|j = DiljkE l, and E i

|k|j − E i|j|k = DilkjE l.

As the Douglas tensor is symmetric we have that: E i|j |k − E i|k|j = E i

|k|j −E i|j|k. Consequently, we have, E i

|j |k − E i|k|j = E i|j|k − E i|k|j = Ri

lkjyl −Ri

kj .Finally, we have that: ∂Bi

j/∂yk−∂Bik/∂yj = 3Ri

jk +(Riklj +Ri

jkl +Riljk)y

l.Using the Bianchi identity for the Berwald connection D, we have that

Riklj + Ri

jkl + Riljk = 0, so that Ri

jk = 1/3(∂Bij/∂yk − ∂Bi

k/∂yj), and thetheorem is proved. q.e.d.

4.7 Homogeneous SODE

In this section we study systems of second order differential equations:

d2xi

dt2+ 2Gi

(x,

dx

dt

)= 0,(4.26)


where the functions Gi are homogeneous of order two with respect to yi =dxi/dt. Each system of (4.26) is defined on local charts on TM and thesesystems are compatible on the intersections of domains of local charts. Thecompatibility means that if Gi(x, y) and Gi(xi, yi) are defined on π−1(U) andπ−1(V ) respectively, then on π−1(U ∩ V ) we have

2Gi =∂xi

∂xj2Gj − ∂2xi

∂xj∂xkyjyk.(4.27)

From the above formula we can see that the required condition for the func-tions Gi to be homogeneous of order two with respect to yi is chart invariant.Therefore, if Gi are homogeneous of order two, then Gi are also homogeneousof order two. Using the Euler theorem we have that the functions Gi arehomogeneous of order two if and only if yj(∂Gi/∂yj) = 2Gi.

If the systems (4.26) are given on each domain of local chart with thecompatibility conditions (4.27) then we can consider the vector field:

S = yi ∂

∂xi− 2Gi(x, y)

∂

∂yi.(4.28)

Vector field S is globally defined on TM and it is called a spray. Since functionsGi are homogeneous of order two, yi are homogeneous of order one, we obtainthat S is a vector field homogeneous of order two. This is equivalent to saythat LCS = S, where C = yi(∂/∂yi) is the Liouville vector field.

The spray S induces a nonlinear connection N on TM with the localcoefficients N i

j = ∂Gi/∂yj and the horizontal and vertical projectors givenby (2.21). This nonlinear connection is symmetric and homogeneous becausethe local coefficients N i

j are homogeneous functions of order one. This isequivalent, according to Euler’s theorem, to yk(∂N i

j/∂yk) = N ij .

We have seen in Section 2.10 that the homogeneity of a nonlinear con-nection is equivalent with the homogeneity of the induced almost productstructure P, which is equivalent to LCP = 0.

For a spray S, we consider the induced nonlinear connection N with thecorresponding Berwald basis: (δ/δxi, ∂/∂yi). From the homogeneity condi-tion, the first invariant E i = 2Gi−yj(∂Gi/∂yj) = 0 and consequently we havethat the spray S is a horizontal vector field, which means that

S = yi δ

δxi.(4.29)

Also the autoparallel curves of the nonlinear connection N , namely the solu-tions of the system of second order differential equations:

d2xi

dt2+ N i

j

(x,

dx

dt

)dxj

dt= 0(4.30)

4.7. Homogeneous SODE 75

coincide with the paths of the given spray S.Let D be the Berwald connection induced by N . Then the horizontal

coefficients F ijk = ∂2Gi/∂yj∂yk are homogeneous of order zero. This means

that

F ijky

k =∂2Gi

∂yj∂ykyk =

∂Gi

∂yj= N i

j .

Geodesic equations for the Berwald connection are given by

d2xi

dt2+ F i

jk

(x,

dx

dt

)dxj

dt

dxk

dt= 0.(4.31)

Consequently, we have that the geodesics of the Berwald connection D are thesame with the autoparallel curves of the nonlinear connection N and coincidewith the solutions of (4.26), the paths of the given spray S. The systems ofSODE (4.26), (4.30) or (4.31) are equivalent to

∇(

dxi

dt

)= 0.(4.32)

Here ∇ is the covariant derivative induced by the nonlinear connection N orby the Berwald connection D:

∇Xi = S(Xi) + N ijX

j = Xi|jy

j ,(4.33)

where Xi|j is the h-covariant derivation of Xi with respect to the Berwald

connection D.

Proposition 4.7.1 For a spray S, the second, the third and the fourth in-variant are related as follows:

Bij = Ri

jkyk = Ri

mjkymyk,

Rijk = Ri

mjkym.

(4.34)

Proof. The second formula (4.34) is a direct consequence of the Ricci identities(3.23) and the homogeneity of the nonlinear connection N . This homogeneityappears here in the form:

yi|j = F i

jkyk −N i

j =∂N i

j

∂ykyk −N i

j = 0,

which says that local coefficients of the nonlinear connection are homogeneousfunctions of order one. q.e.d.


Proposition 4.7.2a) The Jacobi equations of the homogeneous system of SODE (4.32) have

the form:

∇2ξi + Rikj

dxk

dtξj = 0.(4.35)

b) A vector field X = Xi(∂/∂xi) ∈ χ(M) is a Lie symmetry for a spray Sif and only if

∇2Xi + Rikjy

kXj = 0,(4.36)

or, in the equivalent form,

LXc(2Gi) = LXc(N ij)y

j = LXc(F ijk)y

jyk = 0.(4.37)

For a homogeneous system of SODE, Jacobi equations (4.35) have the sameform as for a linear connection. This is true, if one uses the following identityRi

jk = Rimjky

m.

A vector field X = Xi(∂/∂xi) ∈ χ(M) is a Lie symmetry for a spray Sif and only if it is a symmetry of the induced nonlinear connection, whichis equivalent with the fact that X is a symmetry of the induced Berwaldconnection.

Part II

Finsler-Lagrange geometry

Chapter 5

Finsler Spaces

The geometry of Finsler spaces is an important chapter of modern differentialgeometry of the total space of the tangent bundle. Important contributionsto the geometry of these spaces were obtained by M. Abate and G. Patrizio[1], D. Bao, S.S. Chern and Z. Shen [32], A. Bejancu [39], L. Berwald [43],H. Busemann [57], E. Cartan [62], M. Haimovici [81], M. Matsumoto [114],R. Miron and M. Anastasiei [130], H. Rund [162] and others. Importantapplications of the geometry of these spaces are due to G. Randers [159], P.L.Antonelli, R.S. Ingarden and M. Matsumoto [23], S. Ikeda [175], G.S. Asanov[28].

Finsler spaces and related Finsler objects differ from Riemannian ones bythe fact that at each point they depend also on direction. However, this direc-tional dependence has some restriction. As an example, let us a consider ananisotropic media, through which a signal propagates with a velocity functionV . This velocity function depends on position (if the medium is inhomoge-neous) and on the direction of propagation (if the medium is anisotropic). Ifwe identify our medium with an open submanifold M of R3, then V is a func-tion that lives on the total space TM of the tangent bundle, which means thatV = V (xi, yi) and it is zero homogeneous with respect to y. Consequently, ifsome additional requirements are satisfied F (xi, yi) = (

√δijyiyj)/V (xi, yi) is

the fundamental function of a Finsler space.In this chapter we study the geometry of a Finsler space and its geodesics

using the apparatus we developed in the first part of the book. We shall seethat geodesics of a Finsler space, with natural parameterization, coincide withintegral curves of canonical spray, they coincide with autoparallel curves ofcanonical nonlinear connection and coincide also with geodesics of four N-linear connections to which we shall refer to as Finsler connections. Each ofthese four Finsler connections are uniquely determined by a system of axioms.The Cartan nonlinear connection which is the common nonlinear connection

80 Chapter 5. Finsler Spaces

for all Finsler connections is determined also by a system of two axioms. Con-sequently, we can apply the theory developed in previous chapters to studythese connections, their geodesics and their variation and symmetries. Sym-metries of a Finsler space and Noether-type theorems are studied in sectionsfour and seven. The particular cases of two- and three-dimensional Finslerspaces are studied due to the existence of some frames that one can naturallyassociate to these spaces.

For an anisotropic medium, the velocity function determine a Cartan met-ric, which is the dual via the Legendre transformation of a Finsler metric.Consequently, Finsler and Cartan geometries are the most appropriate ge-ometries of such spaces. Within these geometries, the trajectory that a signalwill follow is then a geodesic, and therefore the geodesic balls are wavefronts.

5.1 Finsler metrics

A Riemannian metric on a manifold M is given by a family of scalar products(gx)x∈M such that the map g : x ∈ M 7→ gx, a scalar product on TxM , issmooth. A Finsler metric on a manifold is given by a family of Minkowskinorms Fx such that the map F : x ∈ M 7→ Fx, a Minkowski norm on TxM , issmooth. We shall see that a Finsler metric reduces to a semi–Riemannian met-ric if and only if the family of Minkowski norms Fx satisfy the parallelogramidentity.

Definition 5.1.1 A Finsler metric on a smooth manifold M is given by apositive function F : TM −→ R such that

F1 : F is of C∞-class on TM and only continuous on the null section of theprojection π : TM −→ M .

F2 : F is positive homogeneous of order one with respect to the fibre coordi-nates, i.e. F (x, λy) = λF (x, y), ∀λ > 0.

F3 : For any (x, y) ∈ TM , the symmetric bilinear form g(x,y) is nondegenerateand has constant signature, where

g(x,y)(v, w) =12

∂2

∂s∂t

[F 2(x, y + sv + tw)

] |s=t=0, y, v, w ∈ TxM.(5.1)

Some authors [32], [170] ask in condition F3 for a stronger requirement,the positive definiteness of the bilinear form g(x,y). If this is the case, then foreach x ∈ M , Fx = F (x, ·) is a norm on TxM . In order to show this one canprove, see [32], that F satisfies the triangular inequality:

F (x, y + v) ≤ F (x, y) + F (x, v),∀y, v ∈ TxM.

5.1. Finsler metrics 81

The equality holds if and only if y = λv, for some λ > 0. In most parts ofthis work we shall not use such property and therefore we shall not ask thebilinear form g(x,y) to be positive definite.

Definition 5.1.2 If F is a Finsler metric on a manifold M , the pair Fn =(M,F ) is called a Finsler space and the bilinear form g(x,y) is called the met-ric (or the fundamental) tensor of the Finsler space. Also F is called thefundamental function of the Finsler space Fn.

The existence of Finsler spaces is assured by the following theorem:

Theorem 5.1.1 If the base manifold M is paracompact, then there exist func-tions F : TM → R, which are the fundamental functions for Finsler spaces.

Proof. There exists a Riemannian metric gx over the manifold M , providedthat M is a paracompact manifold. The positive definiteness of the metricg implies that for every y ∈ TxM , y 6= 0, we have that gx(y, y) > 0. If onedefines F (x, y) =

√gx(y, y), then F is the fundamental function of a Finsler

space. The symmetric bilinear form g(x,y) we associate to a Finsler space, doesnot depend on y and it coincides with the Riemannian metric. q.e.d.

From the homogeneity condition F2 one can see that F 2 is homogeneousof order two with respect to y and using Euler theorem for homogeneousfunctions we have:

g(x,y)(y, v) =12

∂

∂s

[F 2(x, y + sv)

] |s=0, and g(x,y)(y, y) = F 2(x, y).(5.2)

A first example of a Finsler space is provided by the Riemannian spacefrom Theorem 5.1.1. Consider g a Riemannian metric on the manifold M .Define F (x, y) =

√gx(y, y), ∀(x, y) ∈ TM . Then F is a Finsler metric on M .

If the metric tensor (5.1) does not depend on fibre coordinates y, we say thatthe Finsler space is reducible to a semi–Riemannian space (or Riemannian ifgx is positive definite).

Proposition 5.1.1 A Finsler space is reducible to a semi–Riemannian spaceif and only if the fundamental function F of the Finsler space satisfies thefollowing parallelogram identity holds:

F 2(x, y + v) + F 2(x, y − v) = 2F 2(x, y) + 2F 2(x, v), ∀y, v ∈ TxM.(5.3)

Proof. If the metric Finsler F is reducible to a semi–Riemannian metric,then gx given by expression (5.1) is a symmetric bilinear form on TxM andgx(y + v, y + v) + gx(y − v, y − v) = 2gx(y, y) + 2gx(v, v), ∀y, v ∈ TxM . From(5.2), we have that F 2(x, y) = gx(y, y) and then the identity (5.3) is satisfied.


Assume now that the identity (5.3) is satisfied. Then for all y, v, w ∈ TxMand s, t ∈ R we have have the following identity:

F 2(x, y + sv + tw) + F 2(x, y − sv − tw)

= 2F 2(x, y) + 2F 2(x, sv + tw).(5.4)

One can obtain the metric tensor g(x,y) from (5.1) if we replace s and t by −sand −t. If we take now partial derivatives of both sides of identity (5.4) withrespect to s and t and let then s = t = 0, we obtain that the metric tensor gx

does not depend on y. q.e.d.

For each x ∈ M , if one fix a basis eii=1,n of the tangent space TxM , thenmetric tensor (5.1) can be written as

g(x,y)(v, w) =12

∂2F 2

∂yi∂yjviwj , y = yiei, v = viei, w = wiei.(5.5)

As g(x,y) is symmetric, bilinear, nondegenerate and has constant signature weobtain that

gij(x, y) := g(x,y)(ei, ej) =12

∂2F 2

∂yi∂yj(5.6)

are the components of a symmetric (0,2)-type d-tensor field. The fact thatg(x,y) is nondegenerate implies that rank(gij) = n. Denote by gij(x, y) theentries of the inverse matrix of gij(x, y) that is gijg

jk = δki . Usually one

consider the natural basis ei = ∂/∂xi of TxM induced by a coordinate chart(U, φ = (xi)).

5.2 Geometric objects of a Finsler space

In the previous section we have seen that a Finsler space Fn = (M,F (x, y))determine a symmetric, second order fundamental tensor gij(x, y) that is calledthe metric tensor. Next we introduce and study some other important geomet-ric objects that naturally correspond to a Finsler space, [32], [114], [130]. Weshall use the homogeneity condition F2 to determine some natural relationsbetween these geometric objects.

Theorem 5.2.1 For a Finsler space Fn the following properties hold:1) The functions

pi =12

∂F 2

∂yi= F

∂F

∂yi

are the components of a d-covector field on the manifold TM .

5.2. Geometric objects of a Finsler space 83

2) The functions

Cijk =14

∂3F 2

∂yi∂yj∂yk=

12

∂gjk

∂yi(5.7)

are the components of a (0, 3)-type completely symmetric d-tensor field on TM .This tensor field is called the Cartan tensor field of the Finsler space.

3) The 1-form

θ = pidxi =12

∂F 2

∂yidxi = F

∂F

∂yidxi =

12J∗(dF 2)(5.8)

is globally defined on the manifold TM and it is called the Cartan 1-form ofthe Finsler space Fn. Here J∗ is the cotangent structure (1.10).

4) The 2-form

ω = dθ = dpi ∧ dxi =12d(J∗(dF 2))(5.9)

is globally defined on the manifold TM , it is a symplectic structure on TMand it is called the Cartan 2-form of the Finsler space Fn.

5) The tangent structure J and the symplectic structure ω satisfy:

ω(J(X), Y ) + ω(X, J(Y )) = 0, ∀X, Y ∈ χ(TM).(5.10)

Proof. For the first item of the theorem we have

pi =12

∂F 2

∂yi=

∂xr

∂xi

12

∂F 2

∂yr=

∂xr

∂xipr,

which proves that pi are the components of a d-covector field on TM . Since

Cijk =∂xr

∂xi

∂xp

∂xj

∂xq

∂xkCrpq

we have that Cijk are the components of a d-tensor field of (0, 3) type. If weuse now that pi are the components of a d-covector field and (5.8) we havethat the Cartan 1-form θ is globally defined on TM . Consequently, from (5.9)we have that the Cartan 2-form ω is globally defined. In local coordinates wehave that the Cartan 2-form ω is given by

ω =12

∂2F 2

∂yi∂yjdyj ∧ dxi +

12

∂2F 2

∂yi∂xjdxj ∧ dxi,

which is equivalent to

ω = gijdyj ∧ dxi +14

(∂2F 2

∂yi∂xj− ∂2F 2

∂yj∂xi

)dxj ∧ dxi.(5.11)


For the second therm of the right hand side of formula (5.11), the summationis taken over 1 ≤ i < j ≤ n. From this formula we can see that rank(ω) =2rank(gij) = 2n = dimTM , so the Cartan 2-form ω is nondegenerate andconsequently it is a symplectic structure.

Consider two vector fields on TM , X = Xi(∂/∂xi) + X ′i(∂/∂yi) and Y =Y i(∂/∂xi) + Y ′i(∂/∂yi). The left hand side of (5.10) has the form gijX

iY j −gjiX

iY j , which vanishes due to the symmetry of g. q.e.d.

Proposition 5.2.1 For a Finsler space Fn = (M, F (x, y)) the following prop-erties are true:

1) piyi = F 2;

2) yi := gjiyj =: g0i = pi (the subscript “ 0” means contraction with y);

3) C0jh := yiCijh = 0, Cj0h = Cjh0 = 0;

4) F 2(x, y) = gij(x, y)yiyj .

Proof. All the above properties can be proved using the homogeneity conditionof the fundamental function F and the Euler theorem. For the first property,since F 2 is homogenous of order two with respect to y, we have:

12

∂F 2

∂yiyi = F 2.

For the second property we have to use that pi = (1/2)(∂F 2/∂yi) is homoge-neous of order one with respect to y. If we use the Euler theorem, we have:

gjiyj =

∂

∂yj

(12

∂F 2

∂yi

)yj =

12

∂F 2

∂yi.

For the third property, we use that the metric tensor gij is zero homogeneouswith respect to y. Consequently, if we use (5.7), by Euler’s theorem we havethat

C0jh := yiCijh = yi 12

∂gjk

∂yi= 0.

If we combine the first two statements of this theorem we have that

F 2 =12

∂F 2

∂yiyi = gij(x, y)yiyj

and the theorem is proved. q.e.d.

Next, we will present some other consequences of the homogeneity of thefundamental function F regarding the Cartan forms θ and ω, [7].

5.2. Geometric objects of a Finsler space 85

Proposition 5.2.2 For a Finsler space Fn, Cartan forms θ and ω are homo-geneous of order one with respect to y. Following properties are also true:

iCθ = 0, iCω = θ.(5.12)

Proof. Since θ = pidxi and pi are homogeneous of order one with respectto y, we have that θ is homogeneous of order one. Using also the fact thatω = dpi ∧ dxi, dpi are one homogeneous and dxi are zero homogeneous, weobtain that ω is one homogeneous.

From the defining formula (5.8), we have that for every vertical vector fieldX ∈ χ(TM), θ(X) = 0 and then iCθ = 0. Using expression (5.11) and secondproperty from Proposition 5.2.1 we have that ω(C, X) = θ(X), ∀X ∈ χ(TM),which shows that formulae (5.12) are true. q.e.d.

For a Finsler space Fn we consider also:

li =1F

yi =∂F

∂yithe normalized supporting element and

hij = F∂2F

∂yi∂yjthe angular metric.

(5.13)

Proposition 5.2.3 For a Finsler space Fn, the angular metric has rank (n−1) so it is degenerate. The angular metric and the metric tensor of a Finslerspace are related by the following formula:

gij = hij + lilj .(5.14)

Proof. Metric tensor gij of a Finsler space Fn is given by (5.6). We have then

gij =∂

∂yj

(F

∂F

∂yi

)= F

∂2F

∂yi∂yj+

∂F

∂yi

∂F

∂yj.

If we use (5.13), then the above formula is equivalent to (5.14). For the sup-porting element li we have that liy

i = F . If we multiply both sides of formula(5.14) by yi we obtain yj = gijy

i = hijyi + liy

ilj = hijyi + yj . Consequently,

hijyi = 0. Moreover we have that hijA

i = 0 if and only if Ai and yi areproportional, so rank(hij) = n− 1. q.e.d.

We discussed in the previous section that a Finsler space Fn = (M,F (x, y))is reducible to a Riemannian space if its fundamental tensor field gij does notdepend on the directional variables yi. This is equivalent now to the fact∂gij/∂yk = 2Cijk = 0, which does not depend on the coordinates we workwith. Consequently, a Finsler space Fn is reducible to a Riemannian space ifand only if the Cartan tensor Cijk vanishes on the manifold TM .


5.3 Geodesics of a Finsler space

In this section we study the theory of geodesics of a Finsler space Fn =(M,F (x, y)). One can introduce these geodesics by studying the variationalproblem associated to the fundamental function F .

First we shall see that in a Finsler space we can define the notion of ar-clength for a smooth curve, which generalizes the same notion from a Rieman-nian space.

Let c be a parameterized curve on the base manifold M , c : t ∈ [0, 1] 7→c(t) = (xi(t)) ∈ M , (U, (xi)) being a local chart on M . The extension c of c

to TM is defined by the equations

xi = xi(t), yi =dxi

dt(t), t ∈ [0, 1].

Thus, the restriction of the fundamental function F (x, y) to c is F (x(t), dx/dt),t ∈ [0, 1]. We define the length of the curve c with endpoints c(0), c(1) as beingthe number

L(c) =∫ 1

0F

(x(t),

dx

dt(t)

)dt.(5.15)

The number L(c) does not depend on a change of coordinates on TM . Alsobecause of the homogeneity of the fundamental function F , L(c) does notdepend on the parameterization of the curve c. Consequently, L(c) dependson the curve c, only.

We can fix now a canonical parameter on the curve c, given by the arclengthof c. Indeed, the function s = s(t), t ∈ [0, 1], given by

s(t) =∫ t

t0

F

(x(τ),

dx

dτ(τ)

)dτ, t0, t ∈ [0, 1]

is differentiable. This function has the derivative with respect to parameter tgiven by:

ds

dt= F

(x(t),

dx

dt(t)

)> 0, t ∈ [0, 1].

Consequently, the function s = s(t), t ∈ [0, 1], is invertible. Let t = t(s) be itsinverse. The new parameter s is called the arclength of the Finsler space Fn.With respect to the new parameter s we have the property

F

(x(s),

dx

ds(s)

)= 1.(5.16)

If the Finsler metric F is constant along a parameterized curve c(t), then theparameter t and the arclength parameter s are affine related that is t = as+b,where a, b ∈ R.

5.3. Geodesics of a Finsler space 87

Next, we consider the variational problem for the integral (5.15) with fixedendpoints.

Consider the curves

cε : t ∈ [0, 1] 7→ (xi(t) + εV i(t)) ∈ M,(5.17)

which have the same endpoints xi(0), xi(1) as the curve c, V i(t) = V i(x(t))being a regular vector field on the curve c, with the property V i(0) = V i(1) =0, and ε a real number, sufficiently small in absolute value, such that Imcε ⊂ U .

The extension of a curve cε to TM is given by

cε : t ∈ [0, 1] 7→(

xi(t) + εV i(t),dxi

dt+ ε

dV i

dt

)∈ π−1(U).

The length of a curve cε is given by

L(cε) =∫ 1

0F

(x + εV,

dx

dt+ ε

dV

dt

)dt.(5.18)

A necessary condition for L(c) to be an extremal value of L(cε) is

dL(cε)dε

∣∣∣∣ε=0

= 0.(5.19)

Under our conditions of differentiability, the operator d/dε is permuting withthe operator of integration. Consequently, if we use (5.18) then, we obtain

dL(cε)dε

=∫ 1

0

ddε

F

(x + εV,

dx

dt+ ε

dV

dt

)dt.(5.20)

A straightforward calculation leads to

ddε

F

(x + εV,

dx

dt+ ε

dV

dt

)∣∣∣∣ε=0

=∂F

∂xiV i +

∂F

∂yi

dV i

dt

=

∂F

∂xi− d

dt

∂F

∂yi

V i +

ddt

∂F

∂yiV i

, yi =

dxi

dt·

Substituting in expression (5.20) and taking into account the fact that V i(x(t))is arbitrary, we obtain the following theorem.

Theorem 5.3.1 The functional L(c) is an extremal value of L(cε) if the curvec(t) = (xi(t)) satisfies the following Euler-Lagrange equations for F :

Ei(F ) :=∂F

∂xi− d

dt

∂F

∂yi= 0, yi =

dxi

dt·(5.21)


Similarly, as we did for the length function L, one can consider the variationalproblem for the energy function:

E(c) =∫ 1

0F 2

(x(t),

dx

dt

)dt(5.22)

A necessary condition for a curve c(t) = (xi(t)) to be an extremal value forthe energy function (5.22) is to satisfy the following Euler-Lagrange equationsfor F 2:

Ei(F 2) :=∂F 2

∂xi− d

dt

∂F 2

∂yi= 0, yi =

dxi

dt·(5.23)

Inspired by the above formula one can introduce the following differentialoperator:

Ei(f) =∂f

∂xi− d

dt

(∂f

∂yi

),(5.24)

where t is a parameter along a curve c. For the Euler Lagrange operator Ei,given by expression (5.24), we have the following properties:

Theorem 5.3.2 The following properties hold true:1) Ei(f) is a d-covector field, ∀f ∈ F(TM).

2) Ei(f + f ′) = Ei(f) + Ei(f ′), Ei(af) = aEi(f), a ∈ R, f, f ′ ∈ F(TM).

3) Ei(f) = 0 if and only if f ∈ F(TM) is the complete lift of some functionfrom the base manifold.

Proposition 5.3.1 Let Fn = (M, F ) be a Finsler space. For every smoothcurve c(t) = (xi(t)) on the base manifold M the following formulae hold good:

Ei(F 2) = 2FEi(F )− 2dF

dt

∂F

∂yi,

dF 2

dt= −dxi

dtEi(F 2),

dxi

dtEi(F ) = 0.

(5.25)

Proof. First formula (5.25) is a direct consequence of Euler-Lagrange equations(5.21) and (5.23). For the second one, the total derivative with respect to timeof the function F 2 is given by

dF 2

dt=

∂F 2

∂xi

dxi

dt+

∂F 2

∂yi

dyi

dt=

[ddt

(∂F 2

∂yi

)+ Ei(F 2)

]dxi

dt+

∂F 2

∂yi

dyi

dt

=ddt

(∂F 2

∂yiyi

)+ Ei(F 2)

dxi

dt= 2

dF 2

dt+ Ei(F 2)

dxi

dt.

For the last equality in the previous calculations we did use the fact that F 2

is homogeneous of order two with respect to y. For the last formula (5.25),

5.3. Geodesics of a Finsler space 89

we can use similar calculations and the first order homogeneity of the metricfunction F . q.e.d.

Theorem 5.3.31) For a Finsler space Fn = (M, F (x, y)) the energy function F 2 is con-

served along the solution curves c of the Euler-Lagrange equations Ei(F 2) = 0,yi = dxi/dt.

2) If a curve c is a solution of the Euler-Lagrange equations Ei(F 2) = 0,yi = dxi/dt then c is a solution also of the Euler-Lagrange equations Ei(F ) =0, yi = dxi/dt.

3) If a curve c is a solution of the Euler-Lagrange equations Ei(F ) = 0,yi = dxi/dt and c is parameterized by the arclength then c is a solution alsoof the Euler-Lagrange equations Ei(F 2) = 0, yi = dxi/dt.

Proof.1) According to second formula (5.25) if c is a solution curve of the Euler-

Lagrange equations Ei(F 2) = 0, yi = dxi/dt then F 2(xi(t), dxi/dt) =const.2) As we proved above Ei(F 2) = 0 implies F 2 constant, so F is constant

and from first formula (5.25) we have that Ei(F ) = 0.3) If curve c is parameterized by arclength the F is constant along c. If this

is the case, from first formula (5.25) we have that Ei(F ) = 0 and Ei(F 2) = 0are equivalent. q.e.d.

Definition 5.3.1 Solution curves c = (xi(t)), t ∈ [0, 1], of the Euler-Lagrangeequations (5.21) are called geodesics of the Finsler space Fn.

If we use first formula (5.25) and F 2(x, y) = gij(x, y)yiyj , we can write equa-tions (5.21) for the geodesics of a Finsler space in the following equivalentform:

gij

(d2xi

dt2+ 2Gi

(x,

dx

dt

))=

dF

dt

∂F

∂yj, yi =

dxi

dt,(5.26)

whereGi(x, y) =

12γi

jk(x, y)yjyk,(5.27)

and the functions γijk(x, y) are the Christoffel symbols of the fundamental

tensor field gij . These are given by

γijk(x, y) =

12gir(x, y)

(∂grk

∂xj(x, y) +

∂gjr

∂xk(x, y)− ∂gjk

∂xr(x, y)

).(5.28)

We remark here that for equations (5.26), t is an arbitrary parameter. If wechange now to the arclength parameter s, we have F (x,dx/ds) = 1. Then


equations (5.21) and (5.23) are equivalent and we can write the geodesic equa-tions of a Finsler space as:

d2xi

ds2+ γi

jk

(x,

dx

ds

)dxj

ds

dxk

ds= 0.(5.29)

A theorem of existence and uniqueness for the solutions of the system ofsecond order differential equations (5.29) can be formulated.

5.4 Geodesic spray and symmetries

For a Finsler space Fn, the variational problem of the energy function F 2

determines a system of second order differential equations (5.23) on each do-main of local chart. In Section 4.1 we have seen that such systems of secondorder differential equations determines a semispray. In this section we shallsee that for a Finsler space the semispray is homogeneous so it is a spray, andits integral curves are solutions of the Euler-Lagrange equations. This sprayis given by

S = yi ∂

∂xi− 2Gi(x, y)

∂

∂yi.(5.30)

where the local coefficients Gi(x, y) were defined in the previous section byformula (5.27). As the Christoffel symbols γi

jk(x, y) from (5.28) are homoge-neous of order zero with respect to y, using formula (5.27) we can see thatGi(x, y) are homogeneous of order two with respect to y. This is equivalentto say that the vector field S is homogeneous of order two.

Proposition 5.4.1 In a Finsler space Fn = (M,F ) the integral curves of thecanonical spray are the geodesics with the arclength parameterization.

This is the reason why the canonical spray of a Finsler space is called also thegeodesic spray of the Finsler space. Local coefficients Gi(x, y) of the geodesicspray S of a Finsler space Fn are given by (5.27). The second order homogene-ity of the energy function F 2 is essential to express functions Gi(x, y) in termsof local coefficients γi

jk(x, y). Next we shall derive a different expression forfunctions Gi(x, y), which does not use the homogeneity of the metric functionF (x, y).

Theorem 5.4.1 Local coefficients Gi(x, y) of the geodesic spray S are givenby the following formula:

2Gi(x, y) =12gij(x, y)

[∂2F 2

∂xk∂yj(x, y)yk − ∂F 2

∂xj(x, y)

].(5.31)

5.4. Geodesic spray and symmetries 91

Proof. We can write Euler-Lagrange equations (5.23) in the following equiva-lent form:

2gijdyj

dt− ∂F 2

∂xi+

∂2F 2

∂yi∂xkyk = 0, yj =

dxj

dt.(5.32)

If we multiply this system by gjk we write it into the standard form (4.1),with functions Gi given by expression (5.31). q.e.d.

Next, we shall see that one can determine the geodesic spray of a Finslerspace using also the symplectic structure of the space.

Theorem 5.4.2 The geodesic spray of a Finsler space Fn = (M, F (x, y)) isthe only vector field S on TM that satisfies the equation

iSω = −12dF 2(5.33)

Proof. Symplectic structure ω = (1/2)d(J∗(dF 2)) of the Finsler space Fn

determine an isomorphism X ∈ χ(TM) 7→ iXω ∈ Λ1(TM). Consequently,there is a unique vector field S on TM that satisfies equation (5.33). Firstwe have to show that S is a semispray that is J(S) = C. Consider X =Xi(∂/∂xi) + X ′i(∂/∂yi) an arbitrary vector on TM . Using (5.10) we have

ω(JS, X) = −ω(S, JX) =12dF 2(JX) =

12

∂F 2

∂yiXi.

On the other hand we have

ω(C, X) = dθ(C, X) = C(θ(X))−X(θ(C))− θ[C, X] = C(

12

∂F 2

∂yi

)Xi.

We also have that C(∂F 2/∂yi) = ∂F 2/∂yi and consequently ω(J(S), X) =ω(C, X), ∀X ∈ χ(TM). As ω is nondegenerate we obtain that J(S) = Cand S is a semispray. Its expression in local coordinates is given by (5.30).We prove next that equation (5.33) determine the local coefficients Gi andthey are given by (5.31). Consider X = Xi(∂/∂xi) + X ′i(∂/∂yi) an arbitraryvector on TM . Using local expression (5.11), the left hand side of (5.33) canbe written as:

(iSω)(X) = ω(S, X) =12

∂2F 2

∂xi∂yj(yiXj − yjXi) + gij(−2GiXj − yjX ′i).

The right hand side of equation (5.33) can be written as:

−12dF 2(X) = −1

2∂F 2

∂xiXi − 1

2∂F 2

∂yiX ′i.


Using the fact that X is an arbitrary vector field and the above calculation,equation (5.33) is equivalent to

2gijGj =

12

(∂2F 2

∂xj∂yiyj − ∂F 2

∂xi

),

which is equivalent to formula (5.31) and therefore S is the geodesic spray ofthe Finsler space Fn. q.e.d.

For the remaining of this section we consider S the geodesic spray of aFinsler space Fn.

Definition 5.4.1 Consider a Finsler space Fn = (M, F ).1) A vector field X ∈ χ(M) is said to be a Lie symmetry if [S,Xc] = 0,

where Xc is the complete lift of X.2) A vector field X ∈ χ(M) is said to be an invariant vector field if

Xc(F ) = 0.3) A vector field X ∈ χ(TM) is said to be a dynamical symmetry if [S, X] =

0.4) A vector field X ∈ χ(TM) is said to be a Cartan symmetry if X(F ) = 0

and LXω = 0.5) A function f ∈ F(TM) is a constant of motion (or a conservation law)

for Fn if S(f) = 0.

From Definition 5.4.1 we can immediately see that if X ∈ χ(M) is a Liesymmetry then its complete lift Xc is a dynamical symmetry. According toProposition 4.5.1 we have that a vector field X = Xi(∂/∂xi) ∈ χ(M) is a Liesymmetry if an only if


∂xj+

∂2Xi

∂xj∂xkyjyk = 0.

If X is an invariant vector field for a Finsler space Fn, then fundamentalfunction F is constant along the integral curves of the complete lift Xc. Inother words, if φt is the one-parameter group of the vector field X, thenF φc

t = F . In local coordinates, we have that X = Xi(∂/∂xi) ∈ χ(M) is aninvariant vector field for Fn if

Xi ∂F

∂xi+ yj ∂Xi

∂xj

∂F

∂yi= 0.

For a dynamical symmetry X ∈ χ(TM), its flow permutes the integralcurves of the geodesic spray S and then permutes the geodesics of the Finslerspace Fn.

A constant of motion is a function on TM that is constant along thegeodesics of the Finsler space Fn. Such a function is a first integral for theEuler-Lagrange equations (5.23).

5.4. Geodesic spray and symmetries 93

Proposition 5.4.2 The geodesic spray S of a Finsler space Fn is a Cartansymmetry. Consequently, the energy F 2 of a Finsler space is a constant ofmotion.

Proof. Using the skew symmetry of the 2-form ω and the defining equation(5.33) for S we have 0 = iSω(S) = −(1/2)dF 2(S) = −(1/2)S(F 2). Hence,the energy F 2 is a constant of motion. As ω is an exact form we have thatdω = 0 and then LSω = diSω + iSdω = −(1/2)d(dF 2) = 0. Consequently thegeodesic spray S is a Cartan symmetry for the Finsler space Fn. q.e.d.

Proposition 5.4.3 A Cartan symmetry of a Finsler space Fn is a dynamicalsymmetry.

Proof. Consider X ∈ χ(TM) a Cartan symmetry that is X(F 2) = 0 andLXω = 0. As ω is a nondegenerate 2-form and i[X,S]ω = LXiSω − iSLXω =−(1/2)LX(dF 2) = −(1/2)d(X(F 2)) = 0, we have that [X, S] = 0 and there-fore X is a dynamical symmetry. q.e.d.

Since Lie and exterior derivatives commute, for a Cartan symmetry X, wehave

dLXθ = LXdθ = LXω = 0.

Consequently, if X is a Cartan symmetry, then the 1-form LXθ is a closed1-form.

Definition 5.4.2 A Cartan symmetry X is said to be an exact Cartan sym-metry if the 1-form LXθ is exact.

Following two results will prove that there is a one to one correspondencebetween exact Cartan symmetries and constants of motion.

Theorem 5.4.3 (First Noether-type theorem of Finsler geometry) Let X bean exact Cartan symmetry, which means that there is a function f ∈ F(TM)such that LXθ = df . Then the function f − θ(X) is a constant of motion forthe Finsler space Fn.

Proof. We have S(f − θ(X)) = d(f − θ(X))(S) = (LXθ − diX(θ))(S) =iXdθ(S) = iXω(S) = −iSω(X) = (1/2)dF 2(X) = 0. Consequently, thefunction f −θ(X) is constant of motion (or a conservation law) for the Finslerspace Fn. q.e.d.

Proposition 5.4.4 Let f ∈ F(TM) be a conservation law for the Finslerspace Fn. Then the vector field X ∈ χ(TM), which is the unique solution ofthe equation iXω = −df , is an exact Cartan symmetry of the Finsler space.


Proof. Consider X ∈ χ(TM) the unique solution of the equation iXω = −df .Then LXθ = iXω is an exact 1-form. Consequently, 0 = dLXθ = LXdθ =LXω. As f is a constant of motion, we have that 0 = S(f) = df(S) =−iXω(S) = iSω(X) = −(1/2)dF 2(X) = −(1/2)X(F 2) and hence X(F 2) = 0.Therefore, we have proved that X is an exact Cartan symmetry and thisfinishes the proof of the theorem. q.e.d.

Next, we present some other connections between invariant vector fields,exact Cartan symmetries, conservation laws and Lie symmetries.

Theorem 5.4.4 (Second Noether-type theorem of Finsler geometry) If X ∈χ(M) is an invariant vector field for the Finsler space Fn, then its complete liftXc is an exact Cartan symmetry and consequently a Lie symmetry. Functionθ(X) is a constant of motion for the Finsler space Fn.

Proof. Consider X ∈ χ(M) such that Xc(F 2) = 0. We want to prove nowthat LXcθ = 0. This is true because

(LXcθ)(

∂

∂xi

)=

∂

∂yi(Xc(F 2)) = 0, and (LXcθ)

(∂

∂yi

)= 0.

We have that LXcθ is exact, then LXcω = 0 and Xc is an exact Cartansymmetry. According to Proposition 5.4.3 we have that Xc is a dynamicalsymmetry, which means that X is a Lie symmetry. If we use now Theorem5.4.3 we have that

θ(X) =12

∂F 2

∂yiXi

is a constant of motion for the Finsler space Fn. q.e.d.

5.5 Cartan nonlinear connection

Now, applying the theory from Section 4.2 we can derive from the geodesicspray S a canonical nonlinear connection for the Finsler space Fn = (M, F ).This nonlinear connection is metric with respect to the metric tensor of theFinsler space. However, the metric condition does not determine uniquelythe nonlinear connection. In this section we determine the whole family ofnonlinear connections that are metric with respect to the Finsler metric ten-sor. A compatibility condition between a metric nonlinear connection and thesymplectic structure uniquely determine the Cartan nonlinear connection ofthe space, [54]. The problem of Finsler-metrizability for the Euler-Lagrangeequations of a Finsler space was studied by Muzsnay in [150].

Definition 5.5.1 The nonlinear connection determined by the geodesic sprayS of the Finsler space Fn is called Cartan nonlinear connection of the Finslerspace.

5.5. Cartan nonlinear connection 95

Theorem 5.5.1 The Cartan nonlinear connection N has the coefficients

N ij(x, y) = γi

jk(x, y)yk − Cipj(x, y)γp

ks(x, y)ykys,(5.34)

where γijk(x, y) are the Christoffel symbols (5.28) of the metric tensor gij.

Proof. Local coefficients of the geodesic spray S are given by formulae (5.27).The theorem 4.2.1 gives us a nonlinear connection induced by the semisprayS. Local coefficients of this nonlinear connection are given by

N ij =

∂Gi

∂yj=

12

∂(γiksy

kys)∂yj

= γijky

k +12

∂γiks

∂yjykys.

Using formula (5.28) for Christoffel’s symbols, the homogeneity of order zerofor the metric tensor gij , and the total symmetry of the Cartan tensor Cijk =(1/2)∂gij/∂yk we obtain that

12

∂γiks

∂yjykys = −1

2gim ∂gmp

∂yjγp

ksykys = −Ci

pjγpksy

kys.

This implies that the Cartan nonlinear connection has the coefficients givenby formula (5.34). q.e.d.

The Cartan nonlinear connection determines a horizontal distribution,which is supplementary to the vertical distribution. Therefore, the corre-sponding Berwald basis, which is an adapted basis to these two distributions,is given by

δ

δxi=

∂

∂xi−N j

i

∂

∂yj,

∂

∂yi.

The dual basis adapted to the horizontal and vertical distributions HTM andV TM is given by dxi, δyi = dyi + N i

j(x, y)dxj. As the coefficients Gi(x, y)of the geodesic spray are homogenous functions of order two with respect to ywe have that N i

j are homogeneous functions of order one. Consequently, theCartan nonlinear connection of a Finsler space Fn is homogeneous. Moreoverwe have the properties

N ijy

j =∂Gi

∂yjyj = 2Gi = γi

jkyjyk =: γi

00.

Consequently, we have the following theorem:

Theorem 5.5.2 The autoparallel curves of the Cartan nonlinear connectionof a Finsler space Fn coincide with the geodesic curves of the space with thearclength parameterization, and their equations can be written as

dxi

ds= yi,

dyi

ds+ N i

j(x(s), y(s))dxj

ds= 0.(5.35)


For the Cartan nonlinear connection N we consider ∇ the induced dynam-ical covariant derivative, which has been studied in Section 2.7. Its action onvertical vector fields is given by:

∇(

Xi ∂

∂yi

)=

(S(Xi) + N i

jXj) ∂

∂yi.

The corresponding dynamical covariant derivative of the metric tensor g =gijδy

i ⊗ δyj is given by

(∇g)(X,Y ) = S(g(X,Y ))− g(∇X, Y )− g(X,∇Y ),∀X,Y ∈ χv(TM).

In local coordinates, the covariant derivative of metric tensor gij can be writtenas follows:

gij| := (∇g)(

∂

∂yi,

∂

∂yj

)= S(gij)− gmjN

mi − gimNm

j .

Theorem 5.5.3 Metric tensor of a Finsler space Fn is covariant constantwith respect to dynamical covariant derivative induced by Cartan nonlinearconnection.

Proof. We have to prove that geodesic spray S and Cartan nonlinear connec-tion N i

j satisfyS(gij) = gmjN

mi + gimNm

j .(5.36)

We consider for local coefficients Gi of the geodesic spray S expression (5.31).Then the coefficients N i

j of the Cartan nonlinear connection are given by

N ij =

∂Gi

∂yj=

14

∂gip

∂yj

(∂2F 2

∂yp∂xmym − ∂F 2

∂xp

)

+14gip

(∂gjp

∂xmym − ∂2F 2

∂yj∂xp

)+

14gip ∂2F 2

∂yp∂xj.

If we multiply the above formula by gis we obtain:

Nsj := gisNij = −∂gis

∂yjGi +

12

∂gsj

∂xkyk +

14

(∂2F 2

∂ys∂xj− ∂2F 2

∂xs∂yj

),


Nij =12S(gij) +

14

(∂2F 2

∂yi∂xj− ∂2F 2

∂xi∂yj

).(5.37)

From expression (5.37) we can see that (1/2)S(gij) is the symmetric part ofNij . If we denote by N(ij) the symmetric part of Nij then

N(ij) =12(Nij + Nji) =

12S(gij)(5.38)

5.5. Cartan nonlinear connection 97

and, consequently, we have

2N(ij) = gimNmj + gjmNm

i = −2Gk ∂gij

∂yk+ yk ∂gij

∂xk= S(gij).

Consequently, expression (5.36) is true and the Cartan nonlinear connectionis metric with respect to the metric tensor of the Finsler space Fn. q.e.d.

The above theorem shows that Cartan nonlinear connection of a Finslerspace is metric. However, the metric condition determine only the symmetricpart (5.38) of the nonlinear connection. Next, we determine the whole familyof metric nonlinear connections for a Finsler space. First, we introduce thefollowing Obata operators, which can be associated with the metric tensor gij :

Oijkl =

12(δi

kδjl − gijgkl) and O∗ij

kl =12(δi

kδjl + gijgkl).(5.39)

Theorem 5.5.4 The family of all nonlinear connections that are metric withrespect to the metric tensor of a Finsler space is given by

N ij = N ci

j + OkijmXm

k ,(5.40)

where Xmk is an arbitrary (1, 1)-type d-tensor field, and N ci

j are local coeffi-cients of Cartan nonlinear connection.

Proof. The metric condition of both nonlinear connections N cij and N i

j canbe written as S(gij) = gmjN

cmi + gimN cm

j and S(gij) = gmjNmi + gimNm

j .If we subtract these two equations we obtain O∗is

jm(Nmi − N cm

i ) = 0. Usingthe fact that Oij

klO∗kmpj = 0, the solution of this tensorial equation is given by

expression (5.40). q.e.d.

With respect to the Berwald basis of the Cartan nonlinear connection thesymplectic structure of a Finsler space has a simpler form.

Theorem 5.5.5 The Cartan 2-form of a Finsler space Fn has the followingexpression with respect to the adapted cobasis dxi, δyi = dyi + N i

jdxj:

ω = gijδyj ∧ dxi.(5.41)

Proof. If we use expression (5.11) and replace dyi = δyi −N ijdxj we obtain:

ω = gij(δyj −N jkdxk) ∧ dxi +

14

(∂2F 2

∂yi∂xj− ∂2F 2

∂xi∂yj

)dxj ∧ dxi

= gijδyj ∧ dxi +

12

[−Nij + Nji +

12

(∂2F 2

∂yi∂xj− ∂2F 2

∂xi∂yj

)]dxj ∧ dxi.

(5.42)


Denote by N[ij] the skew symmetric part of Nij . According to formula (5.37),this is given by the second term of the right hand side of this formula, whichmeans

N[ij] =12(Nij −Nji) =

14

(∂2F 2

∂yi∂xj− ∂2F 2

∂xi∂yj

).(5.43)

Consequently, the coefficient of dxi ∧ dxj in expression (5.42) vanishes andthen formula (5.41) is true. q.e.d.

Theorem 5.5.6 For a Finsler space Fn, the Cartan nonlinear connection isthe unique nonlinear connection one can associate to the geodesic spray suchthat the following two conditions are true:

C1) ∇g = 0;C2) ω(hX, hY ) = 0, ∀X, Y ∈ χ(TM).

Proof. We have seen that the metric condition C1 is equivalent to (5.38), whichdetermines the symmetric part N(ij) of the nonlinear connection. ConditionC2 is equivalent to the fact that symplectic structure ω can be expressedas in (5.41), which is equivalent to (5.43). Formula (5.43), determine theskew-symmetric part N[ij] of the nonlinear connection. Consequently the localcoefficients N i

j of a nonlinear connection are uniquely determined by C1 andC2 and they are given by:

N ij =

12gim

[S(gim) +

12

(∂2F 2

∂ym∂xj− ∂2F 2

∂xm∂yj

)].(5.44)

According to theorem (5.5.3), the Cartan nonlinear connection satisfies thetwo axioms C1 and C2. Due to the uniqueness of such a nonlinear connectionwe have that local coefficients of Cartan nonlinear connection are given byformulae (5.44). q.e.d.

One can see from expression (5.41) that the symplectic form ω vanishes ifand only if both of its arguments are horizontal vector fields or vertical vec-tor fields. Therefore both horizontal and vertical subbundles are Lagrangiansubbundles for the manifold TM . More than that one can determine a com-patibility between ω and horizontal and vertical projectors.

Proposition 5.5.1 Consider h and v horizontal and vertical projectors thatcorrespond to decomposition (2.14) induced by the Cartan nonlinear connec-tion. Then we have the following equivalent formula of compatibility betweenprojectors h and v and symplectic form ω:

ω(hX, hY ) = 0,

ω(hX, Y ) + ω(X,hY ) = ω(X, Y ),ω(vX, Y ) + ω(X, vY ) = ω(X, Y ),ω(hX, Y ) = ω(X, vY ), ∀X, Y ∈ χ(TM).

(5.45)

5.6. Finsler linear connections 99

Proof. From Theorem 5.5.6 we have that Cartan nonlinear connection satisfiesω(hX, hY ) = 0, ∀X, Y ∈ χ(TM). We have to prove now that conditions (5.45)are equivalent. As ω(vX, vY ) = 0, we have that ω(X, Y ) = −ω(hX, hY ) +ω(hX, Y ) + ω(X, hY ). Consequently first two formulae (5.45) are equivalent.

All formulae (5.45) can also be proved if we use expression (5.41) for thesymplectic form and express vectors X, Y ∈ χ(TM) using the adapted basisof horizontal and vertical distributions. q.e.d.

5.6 Finsler linear connections

In this section we study some N -linear connections one can associate in acanonical way to a Finsler space. There are four such N -linear connections:Cartan, Berwald, Chern-Rund and Hashiguchi connections, [32], [23], [130].For each of these linear connections, N is the Cartan nonlinear connection, thehorizontal tensor of deflection vanishes, while the vertical tensor of deflectionis the identity. We shall refer to such an N -linear connection as to a Finslerconnection. For each Finsler connection there is a system of axioms thatuniquely determine the connection.

For an N -linear connection D = (N ij , F

ijk, C

ijk), we consider the horizontal

and vertical covariant derivatives Dh and Dv we have defined in Section 3.3. Inlocal coordinates, for the metric tensor gij its horizontal and vertical covariantderivatives are given by

gij|k =δgij

δxk− gmjF

mik − gimFm

jk =δgij

δxk− 2F(ij)k,

gij |k =∂gij

∂yk− gmjC

mik − gimCm

jk =∂gij

∂yk− 2C(ij)k,

(5.46)

where F(ij)k is the symmetric part, with respect to first two indices i, j ofFijk := gimFm

jk . We have that gij|k and gij |k are the components of (0,3)-typed-tensor fields.

The torsion of an N -linear connection D has with respect to adapted basisfive components, which are d-tensor fields of (1,2)-type. Among these weconsider the (h)h-torsion T i

jk = F ijk−F i

kj =: 2F i[jk] and the (v)v-torsion Si

jk =Ci

jk − Cikj =: 2Ci

[jk].

Theorem 5.6.1 [114] For a Finsler space Fn = (M,F (x, y)) there is a uniqueN -linear connection D = (N i

j , Fijk, C

ijk) that satisfies the following five axioms:

C1: Dij := yi

|j = 0, i.e. N ij = F i

kjyk,

C2: D is h-metric: gij|k = 0, i.e.δgij

δxk= gmjF

mik + gimFm

jk = 2F(ij)k,


C3: D is h symmetric: T ijk = 0, i.e. F i

jk = F ikj ⇐⇒ Fi[jk] = 0,

C4: D is v-metric: gij |k = 0, i.e.∂gij

∂yk= gmjC

mik + gimCm

jk = 2C(ij)k,

C5: D is v symmetric: Sijk = 0, i.e. Ci

jk = Cikj ⇐⇒ Ci[jk] = 0.

Proof. Suppose there is an N -linear connection that satisfies the five axiomsC1–C5. Then C2 and C3 imply the following form for horizontal coefficientsF i

jk:

F ijk =

12gip

(δgpk

δxj+

δgjp

δxk− δgjk

δxp

).(5.47)

Expression (5.47) for horizontal coefficients F ijk is equivalent with the following

expression for Fijk = gpiFpjk:

Fijk = γijk −NpkCpij −Np

j Cpik −Npi Cpjk.(5.48)

We multiply both sides of (5.48) by yk and sum over k. According to axiomsC1 and C3 we have that Nij = Fikjy

k = Fijkyk. Third part of Proposition

5.2.1 implies that Cpikyk = Cpjky

k = 0 and we have the following formula:

Nij = γij0 −NpkykCpij , where γij0 = γijky

k.(5.49)

We multiply now equation (5.49) by gip and contract again by yk. We obtain

N i0 = N i

kyk = γi

00 = γijky

jyk.

If we substitute this expression in (5.49) we obtain

Nij = γij0 − γp00Cpij or

N ij = γi

j0 − γp00C

ipj

(5.50)

So far we have seen that axioms C1, C2 and C3 determine uniquely N ij given by

(5.50) and horizontal coefficients F ijk given by (5.47). Second formula (5.50)

coincides with (5.34), so N is the Cartan nonlinear connection of the Finslerspace Fn. The only thing we have to show now is that vertical coefficients Ci

jk

of Cartan coefficients are uniquely determined by axioms C4 and C5. This istrue since

Cijk =

12gip

(∂gpk

∂yj+

∂gjp

∂yk− ∂gjk

∂yp

)

=12gip ∂gpk

∂yj=

14gip ∂3F 2

∂yp∂yk∂yj.

(5.51)

The unique N -linear connection D = (N ij , F

ijk, C

ijk) that satisfies axioms C1–

C5 is called the Cartan linear connection of the Finsler space Fn. Here N ij are


coefficients of the Cartan nonlinear connection, they are given by (5.50). F ijk

are the horizontal coefficients of the Cartan connection and they are given by(5.47). Ci

jk are the vertical coefficients of the Cartan connection and they aregiven by expression (5.51). q.e.d.

Theorem 5.6.2 For a Finsler space Fn, there is a unique N -linear connec-tion D = (N i

j , Fijk, C

ijk) that satisfies the following system of axioms:

B1: Dij := yi

|j = 0, i.e. N ij = F i

kjyk,

B2: F 2|i = 0, i.e.

δF 2

δxi= 0,

B3: D is h symmetric: T ijk = 0, i.e. F i


B4: Cijk = 0,

B5: P ijk = 0, i.e. F i

jk =∂N i

j

∂yk.

Proof. Suppose there is an N -linear connection that satisfies the five axiomsB1–B5. Then axioms B4 and B5 imply that the N -linear connection D isthe Berwald connection associated to the nonlinear connection N . The onlything we have to prove now is that the other three axioms B1, B2 and B3

uniquely determine the nonlinear connection and this is the Cartan nonlinearconnection of the Finsler space Fn. Axioms B1 and B3 imply that the nonlinearconnection is symmetric and homogeneous, that is

∂N ij

∂yk=

∂N ik

∂yjand

∂N ij

∂ykyk = N i

j .

The axiom B2 is equivalent to hX(F 2) = 0, ∀X ∈ χ(TM), where h is thehorizontal projector. Consider S the geodesic spray of the Finsler space Fn,which is uniquely determined by (5.33). For a vector field X ∈ χ(TM), wehave

0 = hX(F 2) = dF 2(hX) = −iSω(hX) = ω(hX, S).

According to (5.41), we have that ω(hX, S) = 0 if an only if S is horizontal, thisimplies that local coefficients N i

j(x, y) of the nonlinear connection and localcoefficients 2Gi(x, y) of geodesic spray S are related by 2Gi(x, y) = N i

j(x, y)yj .If we take the derivative with respect to y of this expression and use thesymmetry and the homogeneity of the nonlinear connection we obtain thatN i

j = ∂Gi/∂yj , so N is the Cartan connection of the Finsler space Fn.The unique N -linear connection that satisfies the system of axioms B1–B5

is called the Berwald connection of the Finsler space Fn. As for the Cartanlinear connection, N i

j are coefficients of the Cartan nonlinear connection, they


are given by (5.50). F ijk are the horizontal coefficients of the Berwald connec-

tion and they are determined by B5. The vertical coefficients of the Berwaldconnection are Ci

jk = 0. q.e.d.

We remark here that the axioms C1 and C2 imply the axioms B1 and B2.This can be seen from the following formulae

F 2|k = (gijy

iyj)|k = gij|kyiyj + 2gijyiDj

k.

If Djk = 0 then gij|k = 0 implies F 2

|k = 0.


j , Fijk, C


CR1: Dij := yi

|j = 0, i.e. N ij = F i

kjyk,

CR2: D is h-metric: gij|k = 0, i.e.δgij

δxk= gmjF

mik + gimFm

jk = 2F(ij)k,

CR3: D is h symmetric: T ijk = 0, i.e. F i


CR4: Cijk = 0.

Proof. As we have seen in the proof of the Theorem 5.6.1 the axioms CR1, CR2

and CR3, which are the equivalent with the axioms C1, C2 and C3, uniquelydetermine the Cartan nonlinear connection with the local coefficients N i

j givenby expression (5.50) and the horizontal coefficients F i

jk given by expression(5.47). The axiom CR4 tells us that vertical coefficients are Ci

jk = 0. TheN -linear connection determined by the axioms CR1–CR4 is called the Chern-Rund connection of the Finsler space Fn. In different books it appears asChern connection, [32], or as Rund connection, [130], but it was M. Anastasieiwho showed in [7] that the two connections coincide. q.e.d.

We remark here that the Chern-Rund connection has the same horizontalcoefficients as the Cartan connection, while the vertical coefficients are thesame with those of the Berwald connection.


j , Fijk, C


H1: Dij := yi

|j = 0, i.e. N ij = F i

kjyk,

H2: F 2|i = 0, i.e.

δF 2

δxi= 0,

H3: D is h symmetric: T ijk = 0, i.e. F i


H4: P ijk = 0, i.e. F i

jk =∂N i

j

∂yk,


H5: D is v-metric: gij |k = 0, i.e.∂gij

∂yk= gmjC

mik + gimCm

jk = 2C(ij)k,

H6: D is v symmetric: Sijk = 0, i.e. Ci

jk = Cikj ⇐⇒ Ci[jk] = 0.

Proof. The axioms H1–H4 determine the Cartan nonlinear connection and thehorizontal coefficients of the connection as for the Berwald connection. Theaxioms H5 and H6 determine the vertical coefficients of the connection as forthe Cartan connection. The N -linear connection determined by axioms H1–H6

is called the Hashiguchi connection. q.e.d.

Next, we look for some properties that are true for all four N -linear con-nections we studied above. We refer to such an N -linear connection by aFinsler connection. First we have that the deflection tensor fields of a Finslerconnection D satisfies the following equations:

Dij = yi

|j = 0, dij = yi|j = δi

j .(5.52)

Proposition 5.6.1 The following properties hold true with respect to any ofthe four Finsler connections of a Finsler space Fn = (M, F (x, y)):

1. F|k = 0, F |k =1F

yk,

2. F 2|k = 0, F 2|k = 2yk,

3. yi|k = 0, yi|k = gik.

Proof. For Berwald and Hashiguchi connections, we have by definition thatF 2|k = 0 and then F|k = 0. For the other two Finsler connections, the metric

tensor is h-covariant constant that is gij|k = 0. As F 2 = gijyiyj and yi

|j = 0we have also that F 2

|j = 0. The other properties can be proved in a similarmanner using the axioms that define the Finsler connections. q.e.d.

Proposition 5.6.2 The Ricci identities of a Finsler connection D are givenby

Xi|k|h −Xi

|h|k = XrRirkh −Xi|rRr

kh,

Xi|k|h −Xi|h|k = XrP i

rkh −Xi|rC

rkh −Xi|rP r

kh,

Xi|k|h −Xi|h|k = XrSirkh.

(5.53)

Let us consider the covariant d-tensors of curvature

Rijkh = gjsRsikh, Pijkh = gjsP

sikh, Sijkh = gjsS

sikh.


Proposition 5.6.3 The covariant d-tensors of curvature satisfy the followingidentities:

Rijkh + Rjikh = 0, Pijkh + Pjikh = 0, Sijkh + Sjikh = 0,

Rijkh + Rijhk = 0, Sijkh + Sijhk = 0.

Proposition 5.6.4 A Finsler connection D has the following properties:

Ri0hk = Ri

hk, P i0hk = P i

hk, Si0hk = 0,

Pijk = Cijk|0, (Pijk := gisPsjk),∑

(ijk)(Rijk) = 0, (Rijk := gimRmjk),

(5.54)

where∑

(ijk) stands for the cyclic summation over the indices i, j, k.

Proof. Indeed, by applying the Ricci identities to the Liouville d-vector fieldyi and taking into account that the tensors of deflection satisfy Di

j = yi|j = 0

and dij = yi|j = δi

j , we get the first identities (5.54). For the other identities,we will write the symplectic structure ω in the form ω = gijδy

j∧dxi and writethat its exterior differential vanishes, dω = 0. q.e.d.

For all four linear connections we presented above the nonlinear connectionis the Cartan nonlinear connection with local coefficients N i

j given by (5.28).For all of them the horizontal tensor of deflection vanishes that is F i

jk(x, y)yk =N i

j(x, y). Consequently, one can write autoparallel curves of Cartan nonlinearconnection:

δ

dt

(dxi

dt

):=

d2xi

dt2+ N i

j

(x,

dx

dt

)dxj

dt= 0(5.55)

in the following form:

d2xi

dt2+ F i

jk

(x,

dx

dt

)dxj

dt

dxk

dt= 0.(5.56)

Theorem 5.6.5 A curve c on TM is an autoparallel curve for Cartan non-linear connection if and only if c is a geodesic of a Finsler connection D thatis D is either Cartan, Berwald, Chern-Rund or Hashiguchi connection.

Proof. According to Section 3.8 we have that a curve c(t) = (xi(t), yi(t)) is ageodesic of an N -linear connection D if and only if

d2xi

dt2+ F i

jk

dxj

dt

dxk

dt+ Ci

jk

dxj

dt

δyk

dt= 0,

ddt

(δyi

dt

)+ F i

jk

δyj

dt

dxk

dt+ Ci

jk

δyj

dt

δyk

dt= 0.

(5.57)

5.7. Geodesic deviation and symmetries 105

As for a Finsler connection D we have that Cijky

j = 0, first equations (5.57)are equivalent to equations (5.56). Also for a Finsler connection, the horizontaltensor of deflection vanishes and (5.56) are equivalent to (5.55). This impliesalso that last equations (5.57) are identically satisfied. q.e.d.

We have seen also that the geodesics of a Finsler space are the autoparallelcurves, with arclength parameterization, of the Cartan nonlinear connection.Therefore, from the previous theorem, a curve with arclength parameterizationis a geodesic of the Finsler space if and only if it is a geodesic of one of theFinsler connections. This gives us some freedom for the choice of the Finslerconnection to work with when we study the geodesics of the Finsler space.

5.7 Geodesic deviation and symmetries

According to the previous section, geodesics of a Finsler space with arclengthparameterization coincide with geodesics of a Finsler connection. We havealso that geodesics of a Finsler connection coincide with autoparallel curvesof Cartan nonlinear connection, which are the same with integral curves ofgeodesic spray.

For a Finsler space Fn = (M,F (x, y)) we consider S the geodesic spraywith local coefficients Gi given by (5.27) or (5.31) and Cartan nonlinear con-nection with local coefficients N i

j given by (5.34). Then we have that the sprayS is a horizontal vector field, that is

S = yi δ

δxi.

Consider ∇ the dynamical covariant derivative induced by the Cartan connec-tion N . For a d-vector field Xi, we have

∇Xi = S(Xi) + N ijX

j = yj δXi

δxj+ F i

jkyjXk = Xi

|jyj ,(5.58)

where Xi|j is the h-covariant derivation of Xi with respect to a Finsler con-

nection D studied in the previous section.Using dynamical covariant derivative ∇ one can write the system (5.56) or

(5.55) of second order differential equations in the following invariant form:

∇(

dxi

dt

)=

(dxi

dt

)

|j

dxj

dt= 0(5.59)

Now let c(t) = (xi(t)) be an integral curve of geodesic spray S and consider avariation of it into nearby ones according to

xi(t) = xi(t) + εξi(t).


Here ε denotes a scalar parameter with small value |ε| and ξi(t) are componentsof a contravariant vector field along c(t). If we ask for xi(t) to be also anintegral curve for geodesic spray, we get the so-called variational equations,

d2ξi

dt2+ 2

∂Gi

∂xjξj + 2

∂Gi

∂yj

dξj

dt= 0.(5.60)

For the variational equation (5.60) we have the equivalent invariant form (Ja-cobi equations)

∇2ξi + Bijξ

j = 0; where

Bij := 2

∂Gi

∂xj− S

(∂Gi

∂yj

)− ∂Gi

∂yk

∂Gk

∂yj= Ri

jkyk = Ri

ljkylyk.

(5.61)

Here Rijk is the curvature of the Cartan nonlinear connection, and Ri

ljk is thehorizontal curvature for one of the Finsler connection studied in the previoussection. These two curvature tensors are related by Ri

jk = Riljky

l. The (1,1)-type d-tensor field Bi

j is the Jacobi endomorphism.One can use also equations (5.61) to characterize Lie symmetries of a

Finsler space. We have that a vector field X = Xi(x)(∂/∂xi) ∈ χ(M) isa Lie symmetry of Fn if and only if

∇2Xi + BijX

j = 0.(5.62)

From (5.58) we have that ∇Xi = Xi|jy

j and because Dij = yi

|j = 0 for all N -linear connections studied in previous section, then equation (5.62) is equiva-lent to

(Xi|j|k + Ri

jmkXm)yjyk = 0.(5.63)

If we denote by Bij = gikBkj , then Bij = Rlijky

lyk, and consequently Bij is a(0,2)-type symmetric d-tensor field. Let Rlijk be the h-curvature componentfor one of a Finsler connections. We define, for a d-tensor field Xi(x, y), theflag curvature [32]:

R(x, y, X) =Rlijky

lykXiXj

(glkgij − gligjk)ylykXiXj.(5.64)

Part of the numerator in the above formula can be written as:

(glkgij − gligjk)ylyk = F 2gij − yiyj = F 2(gij − lilj) = F 2hij .

Here yi = gijyj , li = (1/F )yi is the normalized supporting element and hij is

the angular metric (5.13). Then we can rewrite (5.64) as:

Bij(x, y)XiXj = R(x, y, X)F 2(x, y)hij(x, y)XiXj .(5.65)

5.7. Geodesic deviation and symmetries 107

Definition 5.7.1 A Finsler space is said to have scalar-type curvature R(x, y)if the flag curvature (5.64) does not depend on Xi at every point (x, y).

From (5.65) we have that a Finsler space has scalar-type curvature if and onlyif the Jacobi endomorphism can be written as:

Bij = RF 2hij .(5.66)

If we denote by hij = gikhkj , then the Jacobi endomorphism of a Finsler space

with scalar-type curvature can be written as:

Bij = RF 2hi

j .(5.67)

Consequently, the Jacobi equation of a Finsler space with scalar-type curva-ture, has the form

∇2ξi + RF 2hijξ

j = 0.(5.68)

Next we study Lie symmetries of Cartan nonlinear connection.

Definition 5.7.2 A vector field X ∈ χ(M) is said to be a Lie symmetry ofCartan nonlinear connection if LXcY = [Xc, Y ] ∈ χh(TM), ∀Y ∈ χh(TM).

If we use the fact that δ/δxi is a local basis for the horizontal distributionof Cartan nonlinear connection X ∈ χ(M) is a symmetry if and only if

[Xc,

δ

δxi

]= −LXc(N i

j)∂

∂yj∈ χh(TM), ∀i ∈ 1, ..., n.

Consequently, we have that a vector field X ∈ χ(M) is a Lie symmetry ofCartan nonlinear connection if and only if

∂2Xi


∂xkNk

j +∂Xk

∂xjN i

k +∂N i

j

∂xkXk +

∂N ij

∂yk

∂Xk

∂xryr

= LXc(N ij) = 0.

(5.69)

Due to the homogeneity of Cartan nonlinear connection, we have according tofirst equality of (4.37)

LXc(2Gi) = LXc(N ij)y

j

Consequently a Lie symmetry of Cartan nonlinear connection is a Lie symme-try of the Finsler space.


5.8 Two dimensional Finsler space

It is easier to study a scalar form of Jacobi equations (5.62) if one can fix afield of orthonormal and parallel frames. For a two dimensional Finsler spacethere exists such a field of frames, which is called the Berwald frame, [44]. Alsoas a two dimensional Finsler space has scalar-type curvature we shall use theform (5.67) of the Jacobi endomorphism to study the variation of geodesics insuch a space. The scalar form of (5.67) appears also in Rund’s book, [162] buta different treatment of it can be found in [22].

Consider F 2 = (M,F (x, y)) a two-dimensional Finsler space. Denote byC the length of the Cartan vector Ci = gjkCi

jk and mi = (1/C)Ci, the nor-malized Cartan vector. If li = gijlj = (1/F )yi, then (li,mi) are unitary vectorfields and because limi = gijl

imj = 0, then (li, mi) is an orthonormal frame,[44]. We call it the Berwald frame of the two-dimensional Finsler space F 2.With respect to the Berwald frame, the metric tensor gij and the angulartensor hij are given by

gij = lilj + mimj , hij = mimj .

We remark here that the above formula for metric tensor gij uses the assump-tion — not explicitly stated — that the metric is positive definite. We donot explicitly need such an assumption. For a semi-definite metric tensor wewould have the following expression gij = lilj −mimj .

The existence of the Berwald frame means that a two dimensional Finslerspace F 2 is parallelizable.

Proposition 5.8.1 The local coefficients of the Cartan nonlinear connectionand Cartan linear connection of a two dimensional Finsler space are given by

N ij = −ljS(li)−mjS(li),

F ijk = −lj

δli

δxk−mj

δmi

δxk,

Cijk = −lj

∂li

∂yk−mj

∂mi

∂yk+

1F

(ljmi + mjli)mk.

(5.70)

Proof. First we determine the horizontal and vertical covariant derivatives ofBerwald frame with respect to the Cartan N -linear connection. As yi

|j = 0and F|j = 0, we have that li|j = 0. We have also:

li|k =(

1F

yi

)∣∣∣∣k

=−F|kF 2

yi +1F

yi|k

= − 1F

lkli +

1F

δik =

1F

mimj .

5.8. Two dimensional Finsler space 109

Similarly, one can prove that

mi|j = 0 and mi|j = − 1

Flimj .

We can express the h- and v-covariant derivatives of the Berwald frame withrespect to the Cartan N -linear connection as follows:

0 = li|j =δli

δxj+ lkF i

kj , 0 = mi|j =

δmi

δxj+ mkF i

kj

1F

mimj = li|j =∂li

∂yj+ lkCi

kj , −1F

limj = mi|j =∂mi

∂yj+ mkCi

kj .

(5.71)

If we solve these equations for F ijk and Ci

jk, then we get last two expressionsof (5.70). As the horizontal tensor of deflection vanishes we have that N i

j =F i

kjyk. If we multiply both sides of second expression (5.70) by yk and use the

symmetry of F ijk we get first expression (5.70) for N i

j . q.e.d.

Proposition 5.8.2 The dynamical covariant derivative of the Berwald framevanishes identically that is

∇mi = 0, ∇li = 0.(5.72)

Proof. According to (5.58) the dynamical covariant derivative of li and mi aregiven by ∇li = li|jyj and ∇mi = mi|jyj . If we use first equations of (5.71)then we have that (5.72) are true. q.e.d.

Previous proposition says that the Berwald frame is parallel along anyintegral curve of the geodesic spray. Let Xi be a d-vector field, we denote byX(i) its scalar components with respect to the Berwald frame, which meansthat Xi = X(1)li + X(2)mi. Then the dynamical covariant derivative of Xi isgiven by

∇Xi = (X(1))′li + (X(2))′mi, where

(X(1))′ = S(X(1)) and (X(2))′ = S(X(2)).(5.73)

Due to the fact that any two dimensional Finsler space has scalar-type curva-ture we have that the Jacobi endomorphism has the form Bi

j = RF 2hij , where

R = R1212.

Theorem 5.8.1 The scalar form of Jacobi equations (5.61), in a two dimen-sional Finsler space with respect to the Berwald frame is given by

(X(1))′′ = 0,

(X(2))′′ + RF 2X(2) = 0.(5.74)


Proof. Let Xi be a Jacobi vector field and denote by X(i) its scalar compo-nents with respect to the Berwald frame. We have then ∇2Xi = (X(1))′′li +(X(2))′′mi. Also, as Bi

j = RF 2hij , and hi

j = mimj , then Jacobi equations(5.61) are equivalent to

(X(1))′′li + (X(2))′′mi + RF 2mimj(X(1)lj + X(2)mj) = 0.

Using the orthogonality of mjlj = 0, we obtain the following equivalent form

for Jacobi equations (5.61)

(X(1))′′li + (X(2))′′mi + RF 2X(2)mi = 0

and this is equivalent to (5.74). q.e.d.

From (5.74) we can see that the tangential component of every Jacobivector field has the form (at+ b)li, while the orthogonal component is X(2)mi,where X(2) is the solution of second equation (5.74). So, if X is a Jacobi vectorfield, then its squared length is given by

||X(t)||2 = (at + b)2 + (X(2)(t))2,

where X(2) is the solution of second equation (5.74). We can deduce fromthis that if R > 0 then the geodesics are stable (in other words the geodesicrays are bunching together) and if R ≤ 0 then the geodesics are unstable (inother words the geodesic rays are dispersing). Geodesic stability in Finsleriancontext for different models in Ecology and evolution of colonial organismshas been studied by P.L. Antonelli and R.H. Bradbury, [16]. We have toremark here that we can use also (5.74) to find out the Lie symmetries of atwo dimensional Finsler space. So, a vector field Xi is a Lie symmetry if andonly if its scalar components X(i) satisfy (5.74).

5.9 Three dimensional Finsler space

A three dimensional Finsler space has a globally defined orthonormal frame,which is called the Moor frame, [146]. Except the first vector of the frame,the other two are not covariant constant. However, we can determine a scalarexpression of Jacobi equation. We determine also necessary and sufficientconditions for a three dimensional Finsler space to have scalar-type curvature.

The geometry of a three dimensional Finsler space using the Moor frameis studied also by M. Matsumoto in his book [114]. He extends this the-ory to n-dimensional Finsler spaces that are called strongly non-Riemannianspaces. Such spaces have a frame called Miron frame that naturally extendsthe Berwald frame and Moor frame.

5.9. Three dimensional Finsler space 111

Consider F 3 = (M,F (x, y)) a three dimensional Finsler space. As for a twodimensional Finsler space we consider li = (1/F )yi and mi = (1/C)Ci. Con-sider then ni the unitary vector field orthogonal to li and mi. Then (li,mi, ni)is an orthonormal frame, [146], which is the Moor frame of the Finsler space.With respect to it, the metric tensor — assuming that it is positive definite— and the angular metric are expressed as follows:

gij = lilj + mimj + ninj , and hij = mimj + ninj .(5.75)

Denote by (lαi ) = (li,mi, ni) and (liα) = (li, mi, ni), α ∈ 1, 2, 3. Then wehave that

lαi ljα = δji , and liαlβi = δβ

α.(5.76)

Consider the vector fields

Hα = liαδ

δxiand Vα = liα

∂

∂yi.

Denote by Fαβγ and Cα

βγ the horizontal and vertical components of an N -linear connection with respect to the frame (Hα, Vα). We call these the scalarcoefficients of the N -linear connection.

The existence of the Moor frame shows that a three dimensional Finslerspace is parallelizable.

Proposition 5.9.1 Let D be an N -linear connection with (F ijk, C

ijk) its local

coefficients. Then, the scalar coefficients of the N -linear connection D aregiven by

F γαβ = lγk lkα|il

iβ = −lγk|il

kαliβ,

Cγαβ = lγk lkα|iliβ = −lγk |ilkαliβ.

(5.77)

Proof. The scalar coefficients of an N -linear connection D are given by

DHαHβ = F γβαHγ = (F γ

βαljγ)δ

δxj.

Using properties of linear connection D we have also:

DHαHβ = liαD δ

δxi

(ljβ

δ

δxj

)= liαljβ|i

δ

δxj.

If we compare the above two formulae and solve for F γβα we determine the

first formula (5.77). In a similar way, one can determine the vertical scalarcoefficients Cγ

βα of the N -linear connection. Scalar coefficients F γαβ, Cγ

αβ of theN -linear connection D are the nonholonomic coefficients given by expression(3.25) of D with respect to the Moor frame. q.e.d.


If T ij are the components of a d-tensor field, we denote by Tα

β the compo-nents of the tensor with respect to the Moor frame (liα) that is Tα

β = lαi T ij l

jβ

and we call them the scalar components of the d-tensor field. The scalarcomponents of the metric tensor gij are δαβ, that is

gijliαljβ = δαβ .(5.78)

Next we will use expression (5.77) to determine the scalar coefficients of theCartan connection of the Finsler space F 3 with respect to Moor frame. Let Dbe the Cartan connection of the Finsler space, we know that with respect toit li|j = 0, which is the same with li|j = 0. Therefore, we find from expression(5.77) that the scalar components Fα

1β vanish. Let us denote by “′′|α and “|′′αthe scalar h-covariant and v-covariant derivative, respectively. They are givenby

Tαβ|γ = Hγ(Tα

β ) + FαδγT δ

β − F δβγTα

δ , and

Tαβ |γ = V γ(Tα

β ) + CαδγT δ

β − CδβγTα

δ .(5.79)

The scalar covariant derivative and covariant derivative are related by

Tαβ|γ = T i

j|klαi ljβlkγ and Tα

β |γ = T ij |klαi ljβlkγ(5.80)

As the fundamental tensor gij of a Finsler metric is h- and v-covariant constant,according to (5.78) and a corresponding formula for (5.79) we have that

δαβ|γ = gij|klαi ljβlkγ = 0 and δαβ |γ = gij |klαi ljβlkγ = 0.(5.81)

From δαβ|γ = 0 we have that δηβF ηαγ + δαηF

ηβγ = 0 so Fα

βγ is skew symmetricwith respect to α and β. Since we already have that Fα

1β = 0, then F 1αβ = 0

and Fααβ = 0. The only nonzero scalar coefficients of the Cartan connection

are F 32γ = −F 2

3γ . Denote by

hj = F 32γlγj the h-connection covector.

The d-covector field hj has been introduced by M. Matsumoto in [114]. Wedenote by T , the scalar FF 3

21, so T = FF 321.

Proposition 5.9.2 The dynamical covariant derivative of the Moor frame isgiven by

∇li = 0,∇mi = Tni and ∇ni = −Tmi.(5.82)

Proof. The h-covariant derivative of the Moor frame with respect to the Cartanconnection is given by [114]

li|j = 0,mi|j = nihj and ni

|j = −mihj .

Since hiyi = F 3

2γlγi Fli1 and lγj lj1 = δγ1 , we have that hjy

j = FF 321 = T and the

equations (5.82) are true. q.e.d.

5.9. Three dimensional Finsler space 113

Theorem 5.9.1 A three dimensional Finsler space has scalar-type curvatureif and only if

Bijmimj = Bijn

inj and Bijminj = 0.(5.83)

Proof. From (5.66) we can see that the Finsler space has scalar-type curvatureif and only if Bij = RF 2hij . As Bij is symmetric and Bijy

j = 0, then withrespect to Moor frame we have that

Bij = Amimj + B(minj + mjni) + Cninj , where

A = Bijmimj , B = Bijm

inj and C = Bijninj .

(5.84)

Angular metric hij has with respect to Moor frame the following expressionhij = mimj + ninj , then a three dimensional Finsler space has scalar-typecurvature if and only if

Bij = RF 2(mimj + ninj).(5.85)

If we compare (5.84) with (5.85) the Finsler space has scalar-type curvatureif and only if A = C = RF 2 and B = 0, which means that (5.83) is true.

Conversely if (5.83) is true then, Bij = A(mimj + ninj) = Ahij . So, theFinsler space has scalar-type curvature R = (1/F 2)Bijm

imj . q.e.d.

Next, we will look for a scalar version (with respect to Moor frame) of theJacobi equations (5.61). Consider Xi a d-vector field with scalar componentsX(i) that is: Xi = X(1)li + X(2)mi + X(3)ni. According to Proposition 5.9.2we have that the dynamical covariant derivative of Xi is given by

∇Xi = (X(1))′li + [(X(2))′ − T ]mi + [(X(3))′ + T ]ni.(5.86)

The second covariant derivative of Xi is given by

∇2Xi = (X(1))′′li +[(X(2))′′ − T ′ − T (X(3))′ − T 2

]mi+

[(X(3))′′ + T ′ + T (X(2))′ − T 2

]ni.

(5.87)

Theorem 5.9.2 The scalar form of the Jacobi equations (5.61), in a threedimensional Finsler space, with respect to Moor frame is given by

(X(1))′′ = 0;

(X(2))′′ − T (X(3))′ + AX(2) + BX(3) − T ′ − T 2 = 0;

(X(3))′′ + T (X(2))′ + BX(2) + CX(3) + T ′ − T 2 = 0.

(5.88)

Exactly, as it happened for two dimensional Finsler spaces, we can see thatthe tangential component of a Jacobi vector field is a solution of first equation


(5.88) that is (at + b)li. If the scalar component F 321 of the Cartan connection

vanishes, then T = 0 and equations (5.88) have a simpler form

(X(1))′′ = 0;

(X(2))′′ + AX(2) + BX(3) = 0;

(X(3))′′ + BX(2) + CX(3) = 0.

(5.89)

If the Finsler space has scalar-type curvature R and F 321 = 0, then we have

the following scalar version of the Jacobi equation (5.61):

(X(1))′′ = 0;

(X(2))′′ + RF 2X(2) = 0;

(X(3))′′ + RF 2X(3) = 0.

(5.90)

We can discuss now the geodesic stability depending on the sign of scalar-typecurvature R. As for the two dimensional case we have that for a positivescalar-type curvature the geodesics are stable and for a negative scalar-typecurvature the geodesics are unstable. Also we can use the equations (5.88) todetermine the Lie symmetries of a three dimensional Finsler space.

5.10 Randers spaces

Randers spaces were introduced in 1941 by G. Randers, [159], in order to unifythe gravitational field — given by a semi-Riemannian metric — and an elec-tromagnetic field. But the metric structure he considered is not Riemanniananymore and it was R.S. Ingarden who studied this metric using Finsleriantechniques. In fact, R.S. Ingarden [93] was first to point out that the Lorentzforce law, in this case, can be written as a geodesic equation on a Finsler spacecalled Randers space. This will determine some geometric entities, which de-pend on the electromagnetic field (vector potential), particle (velocity) andbackground space-time parameters. The Finsler structure implies the exis-tence of a global nonholonomic frame, which in turn yields a connection withtorsion and vanishing curvature.

P.R. Holland studies a unified formalism, which uses a nonholonomic frameon space-time, a sort of plastic deformation, arising from consideration of acharged particle moving in an external electromagnetic field in the backgroundspace-time viewed as a strained medium, [87], [88].

In this section, we determine a nonholonomic frame that connects the twometrics a Randers space has, the Riemannian one and the Finsler one, [18].

Let aij be a semi-Riemannian metric on the base manifold M , considerα2(x, y) = aij(x)yiyj and bi(x)dxi a 1-form on M . Then β(x, y) = bi(x)yi isa scalar function on TM .

5.10. Randers spaces 115

The function F : TM −→ R, defined by

F (x, y) = α(x, y) + β(x, y)

is the fundamental function of a Finsler space, [122]. The pair (M, F ) is calleda Randers space. Consider

gij =12

∂2F 2

∂yi∂yj,

the fundamental tensor of the Randers space (M, F ). Taking into account thehomogeneity of α and F we have the following formulae:

pi :=1α

yi = aij ∂α

∂yj; pi := aijp

j =∂α

∂yi;

li :=1F

yi = gij ∂F

∂yj; li := gijl

j =∂F

∂yi= pi + bi;

li =α

Fpi; lili = pipi = 1; lipi =

α

F;

pili =F

α; bip

i =β

α; bil

i =β

F.

(5.91)

The metric tensors (aij) and (gij) are related by

gij =F

αaij + bipj + pibj + bibj − β

αpipj =

F

α(aij − pipj) + lilj .(5.92)

Theorem 5.10.1 For a Randers space (M, F ) consider the matrix with theentries:

Y iγ =

√α

F

(δiγ − lilγ +

√α

Fpipγ

)(5.93)

defined on TM . Then Yγ = Y iγ (∂/∂yi), γ ∈ 1, ..., n is an nonholonomic

frame.

Proof. Consider also the coframe

Y γj =

√F

α

(δγj +

√F

αlγlj − pγpj

).(5.94)

We have to check that Y iγY γ

j = δij and Y i

γY βi = δβ

γ . Let us verify the former

Y iγY γ

j =

(δiγ +

√F

αlilγ − pipγ

)(δγj − lγlj +

√α

Fpγpj

)

= δij − lilj +

√α

Fpipj +

√F

αlilj −

√F

αlilj + lilγpγpj

−pipj + pipγlγlj −√

α

Fpipj = δi

j .


Consequently, each Randers space has a globally defined nonholonomic frame.Such frame has been determine first by P.R. Holland for the particular caseof a four dimensional manifold, when aij is the Lorentz metric, [87], [88].Therefore, we call this frame the Holland frame of the Randers space. Thisframe is homogeneous of order zero in y and a conformal invariant in the sensethat the transformation F 7→ eφ(x) · F leaves Y i

γ fixed. q.e.d.

Theorem 5.10.2 With respect to Holland’s frame the holonomic componentsof the Riemannian metric tensor (aαβ) are given by the components of theRanders metric (gij), that is:

gij = Y αi Y β

j aαβ .(5.95)

Proof. We have

Y βj aαβ =

√F

α

(δβj +

√F

αlβlj − pβpj

)aαβ

=

√F

αaαj + pαlj −

√F

αpαpj .

If we multiply the above expression by Y βj , we have

Y αi Y β

j aαβ =

√F

α

(δαi +

√F

αlαli − pαpi

)·(√

F

αaαj + pαlj −

√F

αpαpj

)

=F

αaij +

√F

αpilj − F

αpipj +

F

α· α

F

√F

αpjli

+F

α· α

Flilj − F

α·√

F

α

α

Flipj − F

αpjpi −

√F

αpilj +

F

αpipj

=F

α(aij − pipj) + lilj = gij .

Formula (5.95) shows also that the two metric tensors, the Riemannian oneand the Finslerian one have the same signature. q.e.d.

5.11 Ingarden spaces

Consider a Randers space Fn = (M, F = α + β), where α2(x, y) = aij(x)yiyj

is a pseudo-Riemannian metric, while β(x, y) = bi(x)yi is a 1-form. For thisFinsler space there are different nonlinear connections one can associate toit. One is the Cartan nonlinear connection and the horizontal curves of this

5.11. Ingarden spaces 117

nonlinear connection coincide with the geodesics of the Finsler space parame-terized by the arclength. If we consider the variational problem of this Finslerspace and work with the parameterization given by the pseudo-Riemannianmetric aij , we obtain the so-called Lorentz equations. Solution curves ofLorentz equations are horizontal curves of a new nonlinear connection thatwe shall call Lorentz nonlinear connection. This section is based on R.Miron’spaper [127]

Consider the length of the a curve c : t ∈ [0, 1] −→ (xi(t)) ∈ U ⊂ M givenby

L(c) =∫ 1

0

[α

(x,

dx

dt

)+ β

(x,

dx

dt

)]dt.(5.96)

The variational problem of the functional L(c) leads to the Euler-Lagrangeequations:

∂(α + β)∂xi

− ddt

∂(α + β)∂yi

= 0, yi =dxi

dt,(5.97)

which can be written in the form

∂α

∂xi− d

dt

(∂α

∂yi

)=

(∂bi

∂xj− ∂bj

∂xi

)yj , yi =

dxi

dt,

or in the equivalent form:

12α

[∂α2

∂xi− d

dt(∂α2

∂yi)]− 1

2dα−1

dt

∂α2

∂yi=

(∂bi

∂xj− ∂bj

∂xi

)yj .(5.98)

The function σ : t ∈ [0, 1] −→ σ(t) ∈ R defined by

σ(t) =∫ t

0α

(x,

dx

dτ

)dτ(5.99)

is invertible on the interval [0,1]. It allows us to consider σ as a parameteron the curve c. This is the arclength of the pseudo-Riemannian structure aij .We obtain

α

(x,

dx

dσ

)= 1.(5.100)

In this parametrization of the curve c, equations (5.98) become:

∂α2

∂xi− d

dσ

(∂α2

∂yi

)= 2Fij(x)yj , yi =

dxi

dσ,(5.101)

where

Fij =12

(∂bi

∂xj− ∂bj

∂xi

), F i

j (x) = aisFsj(5.102)

is the electromagnetic tensor field of the Finsler space Fn.


If we denote by γijk(x) the Christoffel symbols of the pseudo-Riemannian

structure aij(x)dxi ⊗ dxj then, the Euler-Lagrange equations (5.101) can bewritten as follows:

d2xi

dσ2+ γi

jk

dxj

dσ

dxk

dσ= F i

j

dxj

dσ.(5.103)

Previous equations are called the Lorentz equations of electrodynamics. As itis known, its solution curves are the integral curves of a semispray S on TM

S = yi ∂

∂xi− 2Gi(x, y)

∂

∂yi(5.104)

with coefficients:2Gi(x, y) = γi

jk(x)yjyk − F ij (x)yj .(5.105)

We have that the vector field S depends on the fundamental Randers metricα + β, only. Evidently, this vector field is not homogeneous with respect to yi

and therefore it is a semispray and not the geodesic spray of the Finsler space.

This semispray determines a remarkable nonlinear connection N with thecoefficients:

N ij =

∂Gi

∂yj= γi

jkyk − F i

j , F ij(x) =

12F i

j (x).(5.106)

The nonlinear connection with coefficients given by (5.106) is called the Lorentznonlinear connection of the Finsler space Fn = (M, α + β).

Next we present some properties of the Lorentz nonlinear connection1) The nonlinear connection N is not a homogeneous, since its coefficients

N ij from (5.106) are not homogeneous functions with respect to y.

2) The Lorentz nonlinear connection N gives rise to another splitting ofthe tangent bundle T (TM)

Tu(TM) = HuTM ⊕ VuTM, ∀u ∈ TM.(5.107)

An adapted basis(δ/δxi, ∂/∂yi

)(i = 1, .., n) to the decomposition (5.107) has

the vector fields δ/δxi given as follows

δ

δxi=

∂

∂xi−N j

i

∂

∂yj=

δ

δxi+ F j

i

∂

∂yj,(5.108)

withδ

δxi=

∂

∂xi− N j

i

∂

∂yj, N j

i = γjiky

k.(5.109)

Here N ji are local coefficients of Cartan nonlinear connection. Therefore:


3) The weak torsion

tijk =∂N i

j

∂yk− ∂N i

k

∂yj

of the nonlinear connection N vanishes and therefore the Lorentz nonlinearconnection is symmetric.

4) The curvature tensor

Rijk =

δN ij

δxk− δN i

k

δxj

of the nonlinear connection N is given by

Rijk = ysRi

sjk − (∇kFij − ∇jF

ik),(5.110)

where Risjk(x) is the curvature tensor of the pseudo-Riemann structure aij

and ∇k is the covariant derivation with respect to the Levi-Civita connectionof aij . Consequently, from expression (5.110) we obtain the following result.

Theorem 5.11.1 The Lorentz nonlinear connection N is integrable if, andonly if, the Riemann space Rn = (M, aij) is locally flat and ∇kF

ij = 0.

Now we can give the following definition.

Definition 5.11.1 The Finsler space Fn = (M,α + β) equipped with Lorentznonlinear connection N is called an Ingarden space. It is denoted by IFn =(M,α + β,N).

An important result regarding the Ingarden space IFn is expressed in thefollowing theorem.

Theorem 5.11.2 For an Ingarden space IFn the Berwald connection BΓ(N)has the coefficients (γi

jk, 0).

Proof. For the Berwald connection BΓ(N) the horizontal coefficients are givenby

∂N ij

∂yk= γi

jk.

The vertical coefficients are zero by definition. q.e.d.

Of course, there is a substantial difference between the Randers spaces Fn

and the Ingarden space IFn because:1) The Cartan nonlinear connection of the Randers space is different from

Lorentz nonlinear connection of Ingarden space.


2) The geodesics of the Ingarden spaces are given by the Lorentz equations(5.103), while the geodesics of Randers spaces are given by

d2xi

ds2+ N i

j

(x,

dx

ds

)dxj

ds= 0.

Here N ij are local coefficients of Cartan nonlinear connection of the Randers

space.3) One can prove that the N− and N -metrical connections of these two

spaces are different, too.4) The geometrical model on TM of a Randers space is an almost Kahlerian

space. The corresponding model of an Ingarden space is more general. It isan almost Hermitian space on TM.

5) On TM, the Randers Fn determines a symplectic structure, while IFn

determines a nonintegrable almost symplectic structure.All these inconveniences can be eliminated if we change the nonlinear con-

nection. For the remaining of this section consider a nonlinear connection Nwith local coefficients given by

N ij = γi

jkyk − βF i

j .

Thus we have a homogeneous nonlinear connection that depends only on thefundamental function F = α + β. The pair (α + β, N) can be consideredas an Ingarden space with homogeneous nonlinear connection. Such a spacewill be denoted by IFn = (M, α + β, N), [127]. Next we shall present someremarkable results regarding the Ingarden space IFn. Consider the adaptedbasis

(δ/δxi, ∂/∂yi

)to horizontal and vertical distributions HTM and V TM .

Then the canonical metric N -linear connection has local coefficients F ijk and

Cijk given as usually by:

F ijk =

12gis

(δgsk

δxj+

δgjs

δxk− δgjk

δxs

)

Cijk =

12gis

(∂gsk

∂yj+

∂gjs

∂yk− ∂gjk

∂ys

).

(5.111)

Theorem 5.11.3 Local coefficients of the canonical metric N -linear connec-tion can be expressed into the following equivalent form:

F ijk = γi

jk +12gir(Dh

kgjr + Dhj grk − Dh

rgjk)

+βgir(F skCsjr + F s

j Csrk − F sr Csjk),

Cijk =

12α

gsr[(αbk − βlk)hjr + (αbj − βlj)hrk − (αbr − βlr)hjk],

(5.112)


where hij = gij − lilj is the angular metric of the Finsler space Fn.

The h-covariant derivative Dh and the v-covariant derivative Dv determinethe (1,1)-type deflection tensors Di

j and dij and by lowering the indices the

covariant tensors of deflection Dij and dij .If we denote by Fij and fij the horizontal and the vertical electromagnetic

tensor fields, we obtain that they can be expressed as follows:

Fij =12β(gisF

sj − gjsF

si ) +

12yr(Dh

j gir − Dhi gjr),

fij = 0.

(5.113)

In order to derive the Maxwell equations of the space IFn, we consider theelectromagnetic 2-form:

F = Fijdxi ∧ dxj

and the Cartan forms:

θ =12

∂F 2

∂yidxi and ω = gij δy

i ∧ dxj .

Now we can prove the following theorem:

Theorem 5.11.4 The electromagnetic 2-form F and the Cartan forms θ andω depend only on the fundamental function F = α+β. These forms are relatedby the following formulae:

dθ = ω + F and dω = dF .(5.114)

Proof. If we take the exterior derivative of the Cartan 1-form θ, we obtain

dθ =14

(δ

δxr

∂F 2

∂yi− δ

δxi

∂F 2

∂yr

)dxr ∧ dxi + gij δy

i ∧ dxj .

If one use the formula F 2(x, y) = gij(x, y)yiyj we obtain the formulae (5.114)and the theorem is proved. q.e.d.

If one use the formulae (5.114) and the fact that dω = dF we obtainanother important result regarding the electromagnetic tensor field.

Theorem 5.11.5 The h-electromagnetic tensor field Fij of the Ingarden spaceIFn satisfies the following generalized Maxwell equations:

Σ(ijk)

(Dh

kFij + Rijk

)= 0, Σ(ijk)D

vkFij = 0.


If the Lorentz homogeneous nonlinear connection N of the Ingarden spaceIFn is integrable then the electromagnetic tensor Fij satisfies the followinggeneralized Maxwell equations:

Σ(ijk)DhkFij = 0, Σ(ijk)D

vkFij = 0.

We remark here that one can study also the Einstein equations of the spaceIFn.

5.12 Anisotropic inhomogeneous media

It has been shown in several studies, [26], [15], [173], that Finsler geometryprovides a fruitful platform for the study of seismic ray theory in anisotropicinhomogeneous media. In this section, we show that the Finsler metric, dis-cussed in the above mentioned work, provides a natural context for the studyof rays and wavefronts in such media.

In general, in anisotropic media, rays and wavefronts are not orthogonalto each other in the sense of Euclidean geometry. However, in this section werigorously show that rays and wavefronts are orthogonal to each other in thesense of the geometry imposed by the properties of the medium, which arestated in the context of a given phase-velocity function. This section is basedon the paper [56]. Some geometric aspects of this were generalized by Dahl in[74].

In order to investigate rays and wavefronts, we shall work in this section onthe cotangent bundle of a manifold. The presence of a phase-velocity functionwill determine a Hamiltonian function and an associated metric. With respectto this metric we prove the orthogonality between rays and wavefronts inanisotropic inhomogeneous media.

The geometry of an elastic medium is the geometry of the triple (M, ρ, c),where M is an open subset of the Euclidean three-dimensional space R3, ρ isthe mass-density function, and c is an elasticity tensor. Consider (T ∗M, π∗,M)the cotangent bundle of the manifold M . We note that T ∗M is often calledthe momentum space in classical mechanics or the phase space in Hamiltonianmechanics.

Our geometry must be associated with T ∗M since the key entities that arediscussed below — such as the phase slowness, the Hamiltonian, Hamilton’sequations and the Hamiltonian metric — are defined in this space. The co-ordinates of T ∗M are denoted by (xi, pi), where i = 1, 2, 3. If we remove thezero section of the cotangent bundle, we denote the remaining space by T ∗M .

Let us now return to the triple (M, ρ, c). At each point x of M , the elas-ticity tensor is a (4, 0)-type tensor field whose components, in our coordinatesystem, are cijkl(x) and satisfies the following properties:

5.12. Anisotropic inhomogeneous media 123

1) cijkl(x) = cjikl(x) = cklij(x), ∀x ∈ M ,

2) cijklξiξjηkηl ≥ 0, ∀ξ, η ∈ R3 and cijklξiξjηkηl = 0 if and only if ξ = 0and η = 0.

Using standard methods for solving equations of motion in anisotropicinhomogeneous media, which employ an asymptotic trial solution, [173], weobtain the following system of equations:

[Γij(x, p)− δij

]Aj(x) = 0,(5.115)

where A(x) is the wavefront amplitude and Γ(x, p) is the matrix whose entriesare

Γij(x, p) =cikjl(x)ρ(x)

pkpl,(5.116)

with ρ(x) being the mass density. Matrix Γij(x, p) is called the Christoffel’smatrix. Due to the properties of the elasticity tensor, cijkl(x), Christoffel’smatrix is symmetric and positive-definite. Consequently, the three eigenvaluesof this matrix are real and positive. We denote them by Gα(x, p), whereα ∈ 1, 2, 3. It can be shown that the eigenvalues, Gα, are homogeneous of order2 in p, [63]. Also, functions Gα are differentiable on some open submanifoldof T ∗M . We wish to consider a given eigenvalue, Gα. Let us refer to it as Gand, for convenience, let

H(x, p) :=12G(x, p),(5.117)

to which we shall refer as the ray-theory Hamiltonian. In view of the propertiesof G, we see that H is also homogeneous of order 2 in p. Since H is homo-geneous of order 2 in pi, by factoring p2 = δijpipj , we rewrite Hamiltonian(5.117) as

H(xi, pi) =12p2v2(xi, pi),(5.118)

where, as we immediately see,

v2(xi, pi) :=2H(xi, pi)

p2.(5.119)

The function v(xi, pi) is homogeneous of order 0 in pi. This property meansthat the value of v depends on direction of pi but not on the magnitude of pi.

Expression (5.118) is the form of our Hamiltonian to be used in the re-mainder of this section. We note that our Hamiltonian is a product of twofunctions, namely, p2 and v2(xi, pi). The former function accounts for thehomogeneity of H, while the latter one, which is homogeneous of order 0 in


pi, contains information about the medium. For instance, if v does not de-pend on pi, the medium is isotropic, while if v depends on pi, the medium isanisotropic.

Having phase-slowness covector, pi, phase-velocity function v, and the ray-theory Hamiltonian, H, which are all associated with T ∗M , we are now readyto formulate the equations that will allow us to discuss wavefronts and rays.

To discuss the wavefronts, we invoke equation (5.115) and study the eigen-values that are associated with this equation. For nontrivial solvability ofequation (5.115), we require

det[Γij(x, p)− δij

]= 0,(5.120)

which is an eigenvalue equation. Hence — in the context of equation (5.115)— each of the three corresponding eigenvalues of matrix (5.116) gives us anequation, which can be written as

G(xi, pi) = 1.(5.121)

This is an eikonal equation allowing us to describe the wavefronts. Followingexpressions (5.117) and (5.118), we can restate eikonal equation (5.121) as

p2 =1

v2(xi, pi).(5.122)

In view of equation (5.122) and since wavefronts are loci of constant phase,we refer to v, defined in expression (5.119), as the phase-velocity function.Equations (5.121) and (5.122) are valid for anisotropic inhomogeneous mediasince, considering wavefronts and their normals, we see that xi refers to thedependence on position and pi refers to the dependence on direction.

We wish to explicitly state the eikonal equation as a differential equation.Then, we can write equation (5.122) as

3∑

i=1

(∂ψ

∂xi

)2

=1

v2

(xi,

∂ψ

∂xi

) .(5.123)

This is a first-order nonlinear partial differential equation for function ψ. Tostate equation (5.123) in terms of our Hamiltonian, H, in view of expression(5.118), we can write

2H(xi, pi) ≡ 2H

(xi,

∂ψ

∂xi

)= 1,(5.124)

which is akin to equation (5.121) and is a standard form of the Hamilton-Jacobi equation. The solution of equation (5.124) is function ψ, whose levelsets are the wavefronts.


Solving the eikonal equation by the method of characteristics, [73], weobtain Hamilton’s ray equations, namely,

dxi

dt=

∂H

∂pi,

dpi

dt= −∂H

∂xi.

(5.125)

Rays are curves whose components, x1(t), x2(t), x3(t), are solutions of system(5.125).

Having formulated the equations describing wavefronts and rays, we arenow ready to study their geometrical relation, which is a function of a metricthat characterizes a given geometry. Thus, we begin by formulating pertinentmetrics.

In general, a given elastic medium exhibits two metric structures that are ofinterest in our study. They are the Euclidean metric δij and metric gij(x, p),to which we refer to as the Hamiltonian metric. The Euclidean metric isintrinsically associated with the physical space, while the Hamiltonian metricis induced by the phase-velocity function that corresponds to a given type ofwaves that propagate in the medium. Since, in general, there are three distinctphase-velocity functions, which correspond to the three types of waves thatpropagate in an anisotropic inhomogeneous medium, at every point of themedium there are three Hamiltonian metrics.

Now, we wish to obtain the Hamiltonian metric that corresponds to theproperties of a given medium.

Consider the ray-theory Hamiltonian given by expression (5.118). Usingthis Hamiltonian, we write a convenient metric whose components are givenby

gij(x, p) :=∂2H

∂pi∂pj(x, p).(5.126)

Inserting expression (5.118) into expression (5.126), we can obtain an explicitexpression for the components of this metric, namely,

gij = v2δij + 2v

(pi ∂v

∂pj+ pj ∂v

∂pi

)+ p2

(v

∂2v

∂pi∂pj+

∂v

∂pi

∂v

∂pj

).(5.127)

Thus, our Hamiltonian metric naturally emerges from the properties of a givenmedium, which are contained in the phase-velocity function, v.

We require that H be regular, which means that the matrix with en-tries given in expression (5.126) be nondegenerate on T ∗M . In other words,det

[gij

] 6= 0. We note that, physically, det[gij

]= 0 corresponds to the in-

flection points of a phase-slowness surface. In this section, however, we do not


study these singular points. In view of expression (5.118), the assumption ofdifferentiability of H is equivalent to the assumption of differentiability of v.

Examining expression (5.126), we observe the following properties of theHamiltonian metric. Since H is homogeneous of order 2 in the pi, it followsthat the components of the Hamiltonian metric are homogeneous of order 0in the pi. Furthermore, since det

[gij

] 6= 0, we also have the inverse of theHamiltonian metric, which we denote by gij(x, p).

Examining expression (5.127), we recognize that we can also view thisexpression as the relation between the Hamiltonian metric, gij , and the Eu-clidean metric, δij . We note that for an isotropic medium, where v is inde-pendent of p, equation (5.127) reduces to gij = v2δij . This means that in anisotropic medium the two metrics differ by a multiplicative scalar factor; inother words, they are conformal to one another. This also justifies our choiceof metric (5.126).

Having formulated the Hamiltonian metric, we are now ready to completeour study of a geometrical relation between rays and wavefronts, namely, theirorthogonality.

To discuss orthogonality, we use the fact that a gradient of a function withrespect to a given metric is a vector that is orthogonal to the level sets of thisfunction with respect to this metric. For function ψ on M , we can define itsgradient as being either the vector whose components are

(∇ψ)i = δij ∂ψ

∂xj,(5.128)

with respect to the Euclidean metric, or as the vector whose components are

(∇gψ)i = gij ∂ψ

∂xj,(5.129)

with respect to the Hamiltonian metric. In other words, expression (5.128)defines the Euclidean gradient, while expression (5.129) defines the Hamilto-nian gradient. Consequently, ∇ψ is orthogonal to the level sets of ψ withrespect to the Euclidean metric, while ∇gψ is orthogonal to the level sets of ψwith respect to the Hamiltonian metric. For conciseness of terminology, whendealing with the latter case, we will refer to it as the Hamilton-orthogonality.

In expression (5.127), we obtained the relation between the Euclidean andHamiltonian metrics. Using this relation we can also derive an analyticalexpression relating the components of the two corresponding gradients, asfollows. In view of expressions (5.127), (5.128) and (5.129), we can write

(∇gψ)i = v2 (∇ψ)i + vp2 ∂v

∂pi.(5.130)

Thus, in general, the two gradients are distinct from one another.


We show now that vector ∇gψ is Hamilton-orthogonal to the wavefronts.This is equivalent to saying that ∇gψ is Hamilton-orthogonal to any curvethat belongs to a level set of ψ.

Consider such a curve described by our coordinate functions as

ψ(x(t)) = C,(5.131)

where C denotes a constant. Taking the derivative of both sides of equation(5.131) with respect to t, we obtain

∂ψ

∂xi

dxi

dt= 0.(5.132)

To state expression (5.132) in terms of the Hamiltonian gradient, we canrewrite expression (5.132) as

gjigjk ∂ψ

∂xk

dxi

dt= 0,(5.133)


gji(∇gψ)j dxi

dt= 0.(5.134)

Equation (5.134) states that the scalar product — with respect to theHamiltonian metric — of the Hamiltonian gradient and the vector tangentto the wavefront vanishes. Thus, the Hamiltonian gradient is Hamilton-orthogonal to the wavefront.

To show the Hamilton-orthogonality of rays and wavefronts, it now sufficesto show that the vector tangent to the ray coincides with the Hamiltonian-gradient vector ∇gψ.

Consider a ray described in our coordinates as a curve given by x(t). Thevector tangent to this ray can be written as dx/dt. In view of the first equationof system (5.125), we can write the components of dx/dt as

dxi

dt=

∂H

∂pi.(5.135)

We want to show that ∂H/∂pi are the components of the Hamiltonian gradientof ψ.

Since H is homogeneous of order 2 in the pi, by Euler’s homogeneous-function theorem, we have

2H(x, p) = gij(x, p)pipj and∂H

∂pi= gijpj(5.136)

Using expression (5.136), we can now write equation (5.135) as

dxi

dt= gijpj ,(5.137)


which is equivalent todxi

dt= (∇gψ)i .(5.138)

Equation (5.138) states that the vector tangent to the ray coincides withthe Hamiltonian gradient of ψ, whose level sets are the wavefronts. Hence,the proof of the statement that rays and wavefronts are Hamilton-orthogonalto each other is complete.

This section proves the Hamilton-orthogonality between rays and wave-fronts in anisotropic inhomogeneous media. In other words, the Euclideanorthogonality associated with isotropic media is extended to the anisotropicones. This can be explained in the following way. The Hamiltonian metric,which is derived from the angle-dependent phase velocity, contains this angu-lar dependence. This means that the anisotropy has been accounted for bythe metric itself.

In addition, this study allows us to illustrate the following ray-theory prop-erties.

The results we have shown include, in particular, the case of isotropy. Ifa medium is isotropic, then the phase-velocity function depends on positiononly, namely, v = v(x). In such a case, expression (5.130) can be written as

∇gψ = v2∇ψ.(5.139)

Also, in such a case, expression (5.127) reduces to

gij = v2δij ,(5.140)

which means that, for isotropy, the Hamiltonian and Euclidean metrics areconformal to one another. From equations (5.139) and (5.140), we see that, inisotropic media, the Hamilton-orthogonality reduces to the standard Euclideanorthogonality. The pair Cn = (M,H(x, p)), with H a regular Hamiltonian,2-homogeneous with respect to momenta pi is called a Cartan space, [138].Cartan space Cn is dual to a Finsler space Fn via the Legendre transformation.

Chapter 6

Lagrange Spaces

The geometry of a Lagrange space over a real, finite-dimensional manifold Mhas been introduced and studied as a subgeometry of the geometry of thetangent bundle TM by R. Miron. This geometry was developed together withhis collaborators in [119], [121], [130]. Geometric problems derived from thevariational problem of a Lagrangian were previously studied by J. Kern [96].In this geometry three important geometric structures are studied. These are:a metric structure, a symplectic structure and a semispray with the inducednonlinear connection. Compared to Finsler geometry, in Lagrange geometrywe loose the assumption of homogeneity. A first consequence is that solutionsof Euler-Lagrange equations, which are integral curves for canonical semispray,do not coincide with the autoparallel curves of Cartan nonlinear connection.However, Finsler geometry is a particular case of Lagrange geometry imposedby the homogeneity of the fundamental function.

For a regular Lagrangian of order one, solutions of the Euler-Lagrangeequations are integral curves of the canonical semispray, which is a second or-der vector field. The canonical semispray of a Lagrange space is uniquely de-termined by its symplectic structure and the energy of the space. This point ofview allows for studying first integrals, Cartan symmetries, and Noether typetheorems and it has been investigated by J. Klein [100], C. Godbillon [79], R.Abraham and J. Marsden [2], V.I. Arnold [27], M. Crampin and F.A.E. Pirani[65], M. de Leon and P.R. Rodrigues [110], O. Krupkova [105], R. Miron andM. Anastasiei [130]. For a regular Lagrangian, the canonical nonlinear connec-tion is the only connection that is metric and compatible with the symplecticstructure, as it has been shown in [54].

In this chapter we develop the geometry of Lagrange spaces, using the fun-damental concepts from Analytical Mechanics as: the integral of action, theEuler-Lagrange equations, the law of conservation of energy, Noether sym-metries, symplectic form, etc. The geometry of a Lagrange space is mostly

130 Chapter 6. Lagrange Spaces

derived from the Euler-Lagrange equations. As these equations will determinea semispray we shall develop the geometry of a Lagrange space using thiscanonical semispray, following the methods given in the first part. We givehere conditions that uniquely determine the canonical semispray, canonicalnonlinear connection, and N -linear connection. We can state then that thegeometry of a Lagrange space is determined by the geometry of the inducedcanonical semispray.

6.1 Lagrange metrics

In this section we introduce first the notion of a regular, differentiable La-grangian over the tangent manifolds TM and TM , where M is a differentiable,real manifold, of dimension n. Then, if we ask for the Lagrangian to be regularand homogeneous with respect to velocities we have the definition of a Finslerspace.

Definition 6.1.1 A differentiable Lagrangian is a mapp L : (x, y) ∈ TM 7→L(x, y) ∈ R, of class C∞ on the manifold TM and only continuous on the nullsection 0 : M → TM of the projection π : TM → M.

The Hessian of a differentiable Lagrangian L, with respect to yi, has theelements

gij(x, y) =12

∂2L

∂yi∂yj(x, y).(6.1)

The set of functions gij(x, y) are the components of a d-tensor field, covariantof order 2, symmetric. This means that under a change of coordinates (1.1)on TM , they transform according to the following rule:

gij(x, y) =∂xr

∂xi

∂xs

∂xjgrs(x, y).

To prove the above formula we have to use that

∂L

∂yi=

∂xr

∂xi

∂L

∂yrand

12

∂2L

∂yi∂yj=

∂xr

∂xi

∂xs

∂xj

12

∂2L

∂yr∂ys.

Definition 6.1.2 A differentiable Lagrangian L(x, y) is called regular if thefollowing condition holds:

rank(gij(x, y)) = n,∀(x, y) ∈ TM.(6.2)

Now we can give the definition of a Lagrange space.

6.1. Lagrange metrics 131

Definition 6.1.3 A Lagrange space is a pair Ln = (M, L(x, y)) formed bya smooth, real, n-dimensional manifold M and a regular differentiable La-grangian L(x, y) for which the d-tensor field gij has a constant signature overthe manifold TM.

For the Lagrange space Ln = (M,L(x, y)) we say that L(x, y) is the fun-damental function and gij(x, y) is the fundamental (or metric) tensor. We willdenote by gij the inverse matrix of gij . This means that

gijgjk = δk

i and gijgjk = δik.

Definition 6.1.4 We say that a Lagrange space Ln = (M, L(x, y)) is reducibleto a Finsler space if there is a scalar function F : TM → R such that

1) L(x, y) = F 2(x, y);

2) F (x, y) is a positive function, it is differentiable on TM and only con-tinuous on the null section;

3) F is positively homogeneous of order one on the fibres of the tangentbundle TM , which means that F (x, λy) = λF (x, y), ∀λ > 0.

As we know, for a Finsler space Fn = (M, F (x, y)), the metric tensor (orthe fundamental tensor) (6.1) is given by

gij(x, y) =12

∂2F 2

∂yi∂yj(x, y).(6.3)

Next, we give some examples of Lagrange spaces:1) Every Riemannian space (M, gij(x)) determines a Finsler space Fn =

(M,F (x, y)) and consequently a Lagrange space Ln = (M, F 2(x, y)), where

F (x, y) =√

gij(x)yiyj .(6.4)

The fundamental tensor of this Finsler space coincides to the metric tensorgij(x) of the Riemannian manifold (M, gij(x)).

2) Let us consider the function

F (x, y) = 4√

(y1)4 + · · ·+ (yn)4(6.5)

defined in a preferential local system of coordinates on TM . The pair Fn =(M,F (x, y)), with F defined in (6.5) is a Finsler space. The fundamentaltensor field gij can be easily calculated. This was the first example of Finslerspace from the literature of the subject. It was given by B. Riemann in 1854.


3) Antonelli-Shimada’s ecological metric is given, in a preferential localsystem of coordinate on TM, by

F (x, y) = eφL, φ = αixi (αi are positive constants),

where L = (y1)m + (y2)m + · · ·+ (yn)m1/m, m ≥ 3 is a given integer.4) A Randers metric is determined by the function

F (x, y) = α(x, y) + β(x, y),

where α2(x, y) := aij(x)yiyj is a Riemannian metric and β(x, y) := bi(x)yi isa differentiable linear function in yi. We studied this metric in Section 5.10and we have seen that F (x, y) is the fundamental metric of a Finsler metricwhich is called the Randers metric and has been introduced by G. Randers in[159]. The fundamental tensor of the Randers space is given by [114]:

gij =α + β

αhij + lilj , hij := aij − pipj , li := bi + pi, pi :=

∂α

∂yi.

It can be shown that the fundamental tensor field gij is positive definite underthe condition b2 = aijbibj < 1 (see the book [32]).

5) General Randers space. Let F be the fundamental function of a Finslerspace Fn = (M,F ) and β as in the previous example. Then the function

F(x, y) = F (x, y) + β(x, y),

is the fundamental function of a Finsler space. This was considered and studiedby R. Miron in [122]. The pair (M,F) is called a general Randers space. Thefundamental tensor of the general Randers space (M,F) is given by

gij =F + β

Fkij + lilj , li :=

∂F∂yi

, kij :=1F

∂li∂yj

.

All examples we presented above are Lagrange spaces that are reducibleto Finsler spaces.

Next we give examples of Lagrange spaces, which are not reducible toFinsler spaces.

1) The following Lagrangian from electrodynamics

L(x, y) = mcγij(x)yiyj +2em

Ai(x)yi + U(x)(6.6)

where γij(x) is a pseudo-Riemannian metric, Ai(x) a covector field and U(x) asmooth function, m, c, e are the well-known constants from physics, determinea Lagrange space Ln.

6.2. Geometric objects of a Lagrange space 133

2) Consider the Lagrangian function

L(x, y) = F 2(x, y) + Ai(x)yi + U(x),(6.7)

where F (x, y) is the fundamental function of a Finsler space, Ai(x) are thecomponents of a covector field and U(x) a smooth function gives rise to aremarkable Lagrange space, called the Almost Finsler-Lagrange space (shortlyAFL–space).

The above two examples reduce to a Finsler space if and only if Ai(x) = 0and U(x) = 0.

3) Let Fn = (M, F (x, y)) be a Finsler space and let ϕ : R+ → R a C∞-classfunction. The composition L := ϕ(F 2) defines a differentiable Lagrangian.This was called by P.L. Antonelli and D. Hrimiuc [25] the ϕ-Lagrangian as-sociated to the Finsler space Fn. They showed that if the function ϕ has theproperties

ϕ′(t) 6= 0, and ϕ

′(t) + 2tϕ

′′(t) 6= 0, for every t ∈ Im(F 2),(6.8)

then L is a regular Lagrangian and thus Ln = (M, L(x, y)) is a Lagrange space.The case ϕ(t) = tm/2(m ≥ 3) of the m-homogeneous Lagrangians L(x, y) =

Fm(x, y) was previously studied by M. Anastasiei and P.L. Antonelli, [8].There are two another interesting cases: ϕ(t) = log t and ϕ(t) = et. We shallsee later that the ϕ-Lagrangian L generates the same spray as the fundamentalFinsler function F . In a similar way, we can derive from a regular Lagrangiana fundamental Finsler function. Indeed, if Ln = (M, L(x, y)) is a Lagrangespace and ψ : R → R+ is a function of C∞-class such that ψ(t) 6= 0, ψ′(t) 6=0, ∀t ∈ Im(L) and the function F (x, y) = ψ(L(x, y)) is positively homogeneousof order 1 in y, then Fn = (M,F (x, y)) is a Finsler space.

4) Using the previous notations one can consider the following regularLagrangian:

L(x, y) = ϕ(F 2) + Ai(x)yi + U(x).

A rigorous study of this Lagrangian has not been done yet. We notice thatstarting with a Lagrange space Ln = (M, L(x, y)), the general Randers metricF(x, y) = ψ(L(x, y))+β(x, y) can be considered. This general Randers metricdeserves also a detailed study.

6.2 Geometric objects of a Lagrange space

In the previous section we have seen that a Lagrange space Ln = (M, L(x, y))determines a symmetric, second order d-tensor field with components gij(x, y),which is called the metric tensor. In this section we introduce and study someother important geometric objects that naturally correspond to a Lagrangespace. The most important are the Cartan 1- and 2-forms.


Theorem 6.2.1 For a Lagrange space Ln the following properties hold:1) The functions

pi =12

∂L

∂yi

are the components of a d-covector field on the manifold TM .2) The functions

Cijk =14

∂3L

∂yi∂yj∂yk=

12

∂gjk

∂yi(6.9)

are the components of a third order symmetric d-tensor field. This tensor fieldis called the Cartan tensor field of the Lagrange space.

3) The 1-form

θ = pidxi =12

∂L

∂yidxi =

12J∗(dL)(6.10)

is globally defined on the manifold TM and it is called the Cartan 1-form ofthe Lagrange space Ln. Here J∗ is the cotangent structure (1.10).

4) The 2-form

ω = dθ = dpi ∧ dxi =12d(J∗(dL))(6.11)

is globally defined on the manifold TM , it is a symplectic structure on TMand it is called the Cartan 2-form of the Lagrange space Ln.

5) The tangent structure J and the symplectic structure ω satisfy

ω(J(X), Y ) + ω(X, J(Y )) = 0, ∀X, Y ∈ χ(TM).(6.12)

Proof. For the first item of the theorem we have

pi =12

∂L

∂yi=

∂xr

∂xi

12

∂L

∂yr=

∂xr

∂xipr,

which proves that pi are the components of a d-covector field on TM . As

Cijk =∂xr

∂xi

∂xp

∂xj

∂xq

∂xkCrpq

we have that Cijk are the components of a d-tensor field of (0,3) type. If we usenow that pi are the components of a d-covector field and the expression (6.10)we have that the Cartan 1-form θ is globally defined on TM . Consequently,from (6.11) we have that the Cartan 2-form ω is globally defined. In localcoordinates we have that the Cartan 2-form ω is given by

ω = gijdyj ∧ dxi +14

(∂2L

∂yi∂xj− ∂2L

∂xi∂yj

)dxj ∧ dxi.(6.13)

6.3. Variational problem 135

From this formula we can see that rank(ω) = 2rank(gij) = 2n = dimTM ,so the Cartan 2-form ω is nondegenerate, dω = 0 and consequently ω is asymplectic form.

Consider two vector fields on the total space TM of the tangent manifold:X = Xi(∂/∂xi) + X ′i(∂/∂yi) and Y = Y i(∂/∂xi) + Y ′i(∂/∂yi). The left handside of (6.12) has the form gijX

iY j − gjiXiY j , which vanishes due to the

symmetry of g. q.e.d.

Definition 6.2.1 We say that a Lagrange space Ln = (M, L(x, y)) is reducibleto a Riemannian space if its fundamental tensor field gij does not depend onthe directional variables yi.

The previous definition has a geometric meaning, since the ∂gij/∂yk = 2Cijk =0 does not depend on change of local coordinates. Consequently, a Lagrangespace Ln is reducible to a Riemannian space if and only if the Cartan tensorCijk vanishes on the manifold TM .

6.3 Variational problem

The variational problem can be formulated for differentiable Lagrangians andcan be solved in the case when we consider the parameterized curves, even ifthe integral of action depends on the parameterization of the considered curve.Here we consider the variational problem with fixed endpoints, [78].

Let L : TM → R be a regular Lagrangian and c : t ∈ [0, 1] 7→ (xi(t)) ∈U ⊂ M be a regular curve (with a fixed parameterization) having the image inthe domain of a local chart U of the manifold M . The curve c can be extendedto π−1(U) ⊂ TM by

c : t ∈ [0, 1] 7→(

xi(t),dxi

dt(t)

)∈ π−1(U).

Since the vector field (dxi/dt)(t), t ∈ [0, 1], does not vanish, the image of themapping c belongs to TM.

The integral of action of the Lagrangian L on the curve c is given by thefunctional

I(c) =∫ 1

0L

(x,

dx

dt

)dt.(6.14)

Consider the curves

cε : t ∈ [0, 1] 7→ (xi(t) + εV i(t)) ∈ M,(6.15)

which have the same end points xi(0), xi(1) as the curve c. The vector fieldV i(t) = V i(x(t)) is a regular vector field along the curve c, with the property


V i(0) = V i(1) = 0 and ε a real number, sufficiently small in absolute value,so that Imcε ⊂ U . The extension of a curve cε to TM is given by

cε : t ∈ [0, 1] 7→(

xi(t) + εV i(t),dxi

dt+ ε

dV i

dt

)∈ π−1(U).

The integral of action of the Lagrangian L on the curve cε is given by

I(cε) =∫ 1

0L

(x + εV,

dx

dt+ ε

dV

dt

)dt.(6.16)

A necessary condition for I(c) to be an extremal value of I(cε) is

dI(cε)dε

∣∣∣∣ε=0

= 0.(6.17)

Under our condition of differentiability, the operator d/dε commutes with theoperator of integration. From (6.16) we obtain

dI(ε)dε

=∫ 1

0

ddε

L

(x + εV,

dx

dt+ ε

dV

dt

)dt.(6.18)

A straightforward calculation leads to

ddε

L

(x + εV,

dx

dt+ ε

dV

dt

)∣∣∣∣ε=0

=∂L

∂xiV i +

∂L

∂yi

dV i

dt

=

∂L

∂xi− d

dt

∂L

∂yi

V i +

ddt

∂L

∂yiV i

, yi =

dxi

dt·

Substituting in (6.18) and taking into account the fact that V i(x(t)) is arbi-trary, we obtain the following theorem.

Theorem 6.3.1 In order for the functional I(c) to be an extremal value ofI(cε) it is necessary for the curve c(t) = (xi(t)) to satisfy the Euler-Lagrangeequations:

Ei(L) :=∂L

∂xi− d

dt

∂L

∂yi= 0, yi =

dxi

dt·(6.19)

For the Euler-Lagrange operator Ei(L) defined by expression (6.19) wehave the following properties.

Theorem 6.3.2 The following properties hold true:1) Ei(L) is a d-covector field.

2) Ei(L + L′) = Ei(L) + Ei(L′), Ei(aL) = aEi(L), a ∈ R.

3) Ei(dF/dt) = 0, ∀F ∈ F(TM) with ∂F/∂yi = 0.

6.4. Canonical semispray 137

The notion of energy of a Lagrangian L can be introduced as in TheoreticalMechanics [166], by

EL = yi ∂L

∂yi− L = C(L)− L.(6.20)

Theorem 6.3.3 For every smooth curve c on the base manifold M the fol-lowing formula holds:

dEL

dt= −Ei(L)

dxi

dt, yi =

dxi

dt.(6.21)

Proof. The total derivative with respect to time of the energy function (6.20)is given by

dEL

dt=

dyi

dt

∂L

∂yi+ yi d

dt

(∂L

∂yi

)− yi ∂L

∂xi− dyi

dt

∂L

∂yi

= −[

∂L

∂xi− d

dt

(∂L

∂yi

)]yi, where yi =

dxi

dt,

which proves that formula (6.21) is true. q.e.d.

Theorem 6.3.4 For a differentiable Lagrangian L(x, y) its energy functionEL is conserved along the solution curves c of the Euler-Lagrange equationsEi(L) = 0, yi = dxi/dt.

Let us point out now that if L is 2-homogeneous with respect to y, thenthe energy EL coincides with the fundamental function L. This is the casewhen the Lagrange space reduces to a Finsler space, which means that L = F 2

and the energy is EL = F 2.

6.4 Canonical semispray

In this section we show that we can associate to a Lagrange space a canonicalsemispray, using the variational problem. This semispray is uniquely deter-mined by the symplectic structure and the energy of the Lagrange space.

For a Lagrange space Ln, the variational problem of the Lagrangian func-tion L determine the Euler-Lagrange equations (6.19). These equations deter-mine a system of n second order ordinary differential equations, which can bewritten as

d2xi

dt2+

12gij

[∂2L

∂yj∂xkyk − ∂L

∂xj

]= 0.(6.22)

In Section 4.1 we have seen that such systems of second order differentialequations determine a semispray. We refer to this semispray as the canonical


semispray of the Lagrange space Ln since it is given only by the LagrangianL. The canonical semispray is expressed as

S = yi ∂

∂xi− 2Gi(x, y)

∂

∂yi.(6.23)

where the local coefficients Gi(x, y) are given by the following formula:

2Gi(x, y) =12gij

[∂2L

∂xk∂yjyk − ∂L

∂xj

].(6.24)

Consequently, we have that the solution curves of Euler-Lagrange equations(6.19) coincide with the integral curves of the canonical semispray S.

Next, we will see that the canonical semispray of a Lagrange space isuniquely determined by the symplectic structure ω of Ln.

Theorem 6.4.1 For a Lagrange space Ln = (M, L(x, y)) its canonical semis-pray is the only vector field S on TM that satisfies the following equation:

iSω = −12dEL,(6.25)

where EL is the energy function (6.20).

Proof. The symplectic structure ω = (1/2)d(J∗(dL)) of the Lagrange spaceLn determine an isomorphism X ∈ χ(TM) 7→ iXω ∈ Λ1(TM). Consequently,there is a unique vector field S on TM that satisfies equation (6.25). We wantto prove now that S is a semispray, which means that J(S) = C. ConsiderX = Xi(∂/∂xi) + X ′i(∂/∂yi) an arbitrary vector on TM . Using (6.12) wehave:

ω(J(S), X) = −ω(S, J(X)) =12dEL(X) =

12

∂EL

∂yiXi.

On the other hand we have

ω(C, X) = dθ(C, X) = C(θ(X))−X(θ(C))− θ[C, X] = C(

12

∂L

∂yi

)Xi.

We also have that C(∂L/∂yi) = ∂EL/∂yi and consequently ω(J(S), X) =ω(C, X), ∀X ∈ χ(TM). As ω is nondegenerate we obtain that J(S) = Cand S is a semispray. Its expression in local coordinates is given by (6.23).We prove next that equation (6.25) determine the local coefficients Gi andthey are given by (6.24). Consider X = Xi(∂/∂xi) + X ′i(∂/∂yi) an arbitraryvector field on TM . Using the local expression of the symplectic form ω, theleft hand side of (6.25) can be written as

(iSω)(X) = ω(S, X) =12

∂2L

∂xi∂yj(yiXj − yjXi) + gij(−2GiXj − yjX ′i).

6.4. Canonical semispray 139

The right hand side of equation (6.25) can be written as

−12dEL(X) = −1

2∂EL

∂xiXi − 1

2∂EL

∂yiX ′i

= −12

∂2L

∂xi∂yjyjXi +

12

∂L

∂xiXi − gijy

jX ′i.

Using the fact that X is an arbitrary vector field and the above calculation,equation (6.25) is equivalent to

2gijGj =

12

[∂2L

∂xj∂yiyj − ∂L

∂xi

],

which is equivalent to expression (6.24) and therefore S is the canonical semis-pray of the Lagrange space Ln. q.e.d.

Theorem 6.4.2 For a Lagrange space Ln = (M, L), consider EL its energyfunction, ωL its symplectic form, and SL its canonical semispray. If L′n =(M,L′) is another Lagrange space, then

1) EL = EL′ if and only if L−L′ is homogeneous of order one with respectto y.

2) The two symplectic forms coincide, which means that ωL = ωL′ if andonly if L − L′ = ϕc + ψv, where ϕc is the complete lift and ψv is the verticallift of some functions ϕ and ψ from the base manifold M .

3) If L− L′ is the complete lift of a function ϕ ∈ F(M) then SL = SL′.

Proof.1) We have that the energies of the two Lagrangians L and L′ coincide,

which means that EL = EL′ , if and only if

yi ∂L

∂yi− L = yi ∂L′

∂yi− L′ ⇐⇒ C(L− L′) = L− L′.

Using Euler’s theorem, we have that C(L− L′) = L− L′ if and only if L− L′

is a homogeneous function of order one. An example when this happens is ifL− L′ = ϕc, ϕ ∈ F(M).

2) The symplectic 2-forms ωL and ωL′ coincide if and only if dθL = dθL′ ,which means that θL − θL′ is a closed 1-form, where θL and θL′ are the corre-sponding Cartan 1-forms. As

θL − θL′ =12J∗d(L− L′),

we have that ωL = ωL′ if and only if dJ∗d(L− L′) = 0. According to Propo-sition 1.3 this happens if and only if there are two functions ϕ and ψ on thebase manifold M such that L− L′ = ϕc + ψv.


3) If L − L′ is the complete lift of a function f ∈ F(M) then EL = EL′

and ωL = ωL′ . Using then equation (6.25) we have that SL = SL′ . q.e.d.

Definition 6.4.1 Consider S an arbitrary semispray on TM . We define theS-differential of a function f ∈ F(TM) through

dS(f) =[

∂f

∂xi− S

(∂f

∂yi

)]dxi.(6.26)

Proposition 6.4.1 The S-differential dS has the following properties:1) dS(f + g) = dSf + dSg, dS(af) = adS(f), ∀f, g ∈ F(TM), ∀a ∈ R;

2) dS(f c) = 0, ∀f ∈ F(M);

3) dS(L) = 0 if and only if S is the canonical semispray of the Lagrangespace Ln = (M, L).

Proof. First property is a consequence of the linearity of the operators thatdefine dS . Consider now f a function on the base manifold and let f c be itscomplete lift. Then

∂f c

∂xi= S

(∂f c

∂yi

)=

∂2f

∂xi∂xjyj

and consequently dS(f c) = 0.The equation dS(L) uniquely determine the coefficients Gi of the canonical

semispray given by expression (6.24). q.e.d.

6.5 Symmetries and Noether type theorems

In this section we study different symmetries of a Lagrange space and theirrelations with conservation laws. Here we follow notations and some resultsfrom the book [65].

Consider Ln = (M, L) a Lagrange space and S the canonical semispraydetermined in the previous section.

Definition 6.5.11) A vector field X ∈ χ(M) is said to be a Lie symmetry for the Lagrange

space Ln if [S, Xc] = 0, where Xc is the complete lift of X.2) A vector field X ∈ χ(M) is said to be an invariant vector field for the

Lagrange space Ln if Xc(L) = 0.3) A vector field X ∈ χ(TM) is said to be a dynamical symmetry for the

Lagrange space Ln if [S,X] = 0.4) A vector field X ∈ χ(TM) is said to be a Cartan symmetry for the

Lagrange space Ln if X(EL) = 0 and LXω = 0.5) A function f ∈ F(TM) is a constant of motion (or a conservation law)

for the Lagrange space Ln if S(f) = 0.

6.5. Symmetries and Noether type theorems 141

From Definition 6.5.1 one can immediately see that X ∈ χ(M) is a Liesymmetry if and only if its complete lift Xc is a dynamical symmetry.

According to Proposition 4.5.1 we have that a vector field X = Xi(∂/∂xi)on the base manifold M is a Lie symmetry if an only if


∂xj+

∂2Xi

∂xj∂xkyjyk = 0.

If X is an invariant vector field for a Lagrange space Ln, then the funda-mental function L is constant along the integral curves of the complete liftXc. In other words, if φt is the one-parameter group of the vector field X,then L φc

t = L. In local coordinates, we have that X = Xi(∂/∂xi) ∈ χ(M)is an invariant vector for Ln if

Xi ∂L

∂xi+ yj ∂Xi

∂xj

∂L

∂yi= 0.

For a dynamical symmetry X ∈ χ(TM), its flow permutes the integralcurves of the canonical semispray S and then permutes the solutions of theEuler-Lagrange equations of the Lagrange space Ln.

A constant of motion is a function on TM that is constant along the solu-tions of the Euler-Lagrange equations of the Lagrange space Ln. Consequently,such a function is a first integral for the Euler-Lagrange equations (6.19).

Proposition 6.5.1 The canonical semispray S of a Lagrange space Ln is aCartan symmetry.

Proof. Using the skew symmetry of the 2-form ω and the defining equation(6.25) for S we have 0 = 2iSω(S) = −dEL(S) = −S(EL). As ω is an exact 2-form we have that dω = 0 and then LSω = diSω+ iSdω = −(1/2)d(dEL) = 0.Consequently, the canonical semispray S is a Cartan symmetry. q.e.d.

Proposition 6.5.2 A Cartan symmetry of a Lagrange space Ln is a dynam-ical symmetry.

Proof. Consider X ∈ χ(TM) a Cartan symmetry, which means that X(EL) =0 and LXω = 0. Since ω is a nondegenerate 2-form and i[X,S]ω = LXiSω −iXLSω = −(1/2)LX(dEL) = −(1/2)X(EL) = 0, we have that [X, S] = 0 andX is a dynamical symmetry. q.e.d.

Since Lie and exterior derivatives commute, for a Cartan symmetry X, wehave

dLXθ = LXdθ = LXω = 0.

Consequently, for a Cartan symmetry X, the 1-form LXθ is a closed 1-form.


Definition 6.5.2 A Cartan symmetry X is said to be an exact Cartan sym-metry if the 1-form LXθ is exact.

The following two Noether type theorems will prove that for a Lagrangespace, there is a one to one correspondence between exact Cartan symmetriesand constants of motion.

Theorem 6.5.1 Let X be an exact Cartan symmetry, which implies that thereis a function f ∈ F(TM) such that LXθ = df . Then f − θ(X) is a constantof motion for the Lagrange space Ln.

Proof. We have S(f − θ(X)) = d(f − θ(X))(S) = (LXθ − diX(θ))(S) =iXdθ(S) = iXω(S) = −iSω(X) = (1/2)dEL(X) = 0. Consequently, f − θ(X)is a conservation law for the Lagrange space Ln. q.e.d.

Theorem 6.5.2 Let f ∈ F(TM) be a conservation law for the Lagrange spaceLn. Then, the vector field X ∈ χ(TM), solution of the equation iXω = −df ,is an exact Cartan symmetry of the Lagrange space.

Proof. Consider X ∈ χ(TM) the unique solution of the equation iXω = −df .Then, LXθ = iXω is an exact 1-form. Consequently, 0 = dLXθ = LXdθ =LXω. Since f is a constant of motion, we have that 0 = S(f) = df(S) =−iXω(S) = iSω(X) = −(1/2)dEL(X) = −(1/2)X(EL). Therefore X(EL) =0 and X is an exact Cartan symmetry. q.e.d.

Next, we present the connection between invariant vector fields, exact Car-tan symmetries, conservation laws and Lie symmetries.

Theorem 6.5.3 If X ∈ χ(M) is an invariant vector field for the Lagrangespace Ln, then its complete lift Xc is an exact Cartan symmetry and con-sequently a Lie symmetry. Function θ(X) is a constant of motion for theLagrange space Ln.

Proof. Consider X ∈ χ(M) an invariant vector field for L, which means thatXc(L) = 0. We want to prove now that LXcθ = 0. This is true because:

(LXcθ)(

∂

∂xi

)=

∂

∂yi(Xc(L)) = 0, and (LXcθ)

(∂

∂yi

)= 0.

We have that if LXcθ is exact, then LXcω = 0 and Xc is an exact Cartansymmetry. According to Proposition 6.5.2 we have that Xc is a dynamicalsymmetry, which means that X is a Lie symmetry. If we use now Theorem6.5.1 we have that

θ(X) =∂L

∂yiXi

is a constant of motion for the Lagrange space Ln. q.e.d.

6.6. Canonical nonlinear connection 143

The above theorem is a version of Noether theorem, which usually appearsin the following form. Let (ϕt)t∈R be a one-parameter group of diffeomor-phisms of the base manifold that preserves the Lagrangian function L, whichmeans that L ϕc

t = L. Then the following function

I(x, y) =∂L

∂yi

dϕit

dt

∣∣∣∣t=0

is a first integral for Euler-Lagrange equations (6.19), this form appears in[27]. One can see that first integral I coincides with θ(X) from the previoustheorem.

6.6 Canonical nonlinear connection

The importance of a nonlinear connection for the geometry of the tangentbundle of a manifold M has been discussed in the previous chapters. Inthis section we prove that a Lagrange space Ln = (M,L(x, y)) determinesa canonical nonlinear connection, which depends only on the fundamentalfunction L(x, y). We have seen in a previous section that there is a canonicalsemispray S determined only by the fundamental function L(x, y) of the givenLagrange space Ln. This semispray is uniquely determined by expression(6.25). According to Theorem 4.2.1, every semispray S determines a nonlinearconnection. Now we determine the nonlinear connection one can associate toa semispray as we did in Theorem 4.2.1. A system of axioms that uniquelydetermine this nonlinear connection is given.

Theorem 6.6.1 Every Lagrange space Ln = (M, L) has a canonical nonlinearconnections, which depends only on the fundamental function L. The localcoefficients of this nonlinear connection are given by

N ij =

∂Gi

∂yj=

14

∂

∂yj

gik

(∂2L

∂yk∂xhyh − ∂L

∂xk

).(6.27)

Proposition 6.6.1 The nonlinear connection N with local coefficients N ij ,

given by expression (6.27), is invariant with respect to the Caratheodory trans-formations,

L = L′ + ϕc,(6.28)

where ϕc is the complete lift of an arbitrary smooth function ϕ on the basemanifold M .

Proof. Suppose that L − L′ is the complete lift of a function ϕ from thebase manifold. According to Theorem 6.4.2 the canonical semisprays SL and


SL′ of the two Lagrange spaces coincide. Then the two canonical nonlinearconnections determined in previous theorem will coincide. q.e.d.

The nonlinear connection N , whose coefficients are given by (6.27) is calledthe canonical nonlinear connection of the Lagrange space Ln.Example. Let us consider the Lagrange space of electrodynamics, Ln =(M,L(x, y)), with the Lagrangian L(x, y) given by (6.6), with U(x) = 0. Thecanonical semispray has the local coefficients given by

Gi(x, y) =12γi

jk(x)yjyk − gij(x)Fjk(x)yk,(6.29)

where γijk(x) are the Christoffel symbols of the metric tensor gij(x) = mcγij(x)

of the space Ln and Fjk is the electromagnetic tensor

Fjk =c

2m

(∂Ak

∂xj− ∂Aj

∂xk

).(6.30)

Therefore, the integral curves of the Euler-Lagrange equation are given by thesolution curves of the Lorentz equations:

d2xi

dt2+ γi

jk(x)dxj

dt

dxk

dt= gij(x)Fjk(x)

dxk

dt.(6.31)

According to expression (6.27), the canonical nonlinear connection of the La-grange space of electrodynamics Ln has the local coefficients given by

N ij(x, y) = γi

jk(x)yk − gik(x)Fkj(x).(6.32)

It is remarkable that the coefficients N ij of the canonical nonlinear connection

N of the Lagrange spaces of electrodynamics are affine with respect to yi.This fact has some consequences:

1) The Berwald connection of the space, has the horizontal coefficientsγi

jk(x).2) The solution curves of Euler-Lagrange equations and the autoparal-

lel curves of the canonical nonlinear connection N are given by the Lorentzequation (6.31).

For a nonlinear connection, a curve c(t) = (xi(t)) on the base manifoldis called an autoparallel curve of the nonlinear connection if and only if itscomplete lift c(t) = (xi(t), (dxi)/(dt)) is horizontal.

Theorem 6.6.2 The autoparallel curves of the canonical nonlinear connec-tion N are given by the following system of differential equations:

d2xi

dt2+ N i

j

(x,

dx

dt

)dxj

dt= 0,(6.33)

where N ij are given by expression (6.27).


For the canonical nonlinear connection N we consider ∇ the induced dy-namical covariant derivative we studied in Section 2.7. Its action on verticalvector fields is given by

∇(

Xi ∂

∂yi

)= (S(Xi) + N i

jXj)

∂

∂yi.

Dynamical covariant derivative of the metric tensor g = gijδyi ⊗ δyj is given

by

(∇g)(X,Y ) = S(g(X,Y ))− g(∇X, Y )− g(X,∇Y ),∀X,Y ∈ χv(TM).

In local coordinates covariant derivative of metric tensor gij can be written asfollows:

gij| := (∇g)(

∂

∂yi,

∂

∂yj

)= S(gij)− gmjN

mi − gimNm

j .

Theorem 6.6.3 For a Lagrange space Ln there is a unique nonlinear connec-tion N such that

1) The horizontal subbundle HTM is a Lagrangian subbundle of TTM ,which means that

ω(hX, hY ) = 0, ∀X,Y ∈ χ(TM).(6.34)

2) The metric tensor gij of the Lagrange space is covariant constant withrespect to the dynamical covariant derivative induced by HTM , which is equiv-alent to

∇g = 0.(6.35)

Proof. First we prove that conditions (6.34) and (6.35) uniquely determine anonlinear connection. Then, we show that this nonlinear connection is pre-cisely the one determined by Theorem 6.6.1.

Consider N a nonlinear connection with local coefficients N ij . We want

to express the Cartan form ω using the adapted cobasis dxi, δyi. If we useexpression (6.13) and replace dyi = δyi −N i

jdxj we obtain

ω = gij(δyj −N jkdxk) ∧ dxi +

14

(∂2L

∂yi∂xj− ∂2L

∂xi∂yj

)dxj ∧ dxi

= gijδyj ∧ dxi +

12

[−Nij + Nji +

12

(∂2L

∂yi∂xj− ∂2L

∂xi∂yj

)]dxj ∧ dxi.

(6.36)

We have that (6.34) is true if and only if the second term of the right hand sideof (6.36) vanishes. Denote by N[ij] the skew symmetric part of Nij := gikN

kj .

According to formula (6.36), we have that (6.34) is true if and only if

N[ij] =12(Nij −Nji) =

14

(∂2L

∂yi∂xj− ∂2L

∂xi∂yj

).(6.37)


We have seen that the skew symmetric part of Nij is perfectly determinedby (6.34). Next, we shall prove that the symmetric part of Nij is perfectlydetermined by (6.35). In local coordinates, condition (6.35) is equivalent to

S(gij) = gmjNmi + gimNm

j = Nij + Nji = 2N(ij).(6.38)

Using (6.37) and (6.38) we have that local coefficients N ij of the nonlinear

connection N that satisfies (6.34) and (6.35) are given by

N ij = gikNkj = gik(N(kj) + N[kj])

=12gik

[S(gkj) +

12

(∂2L

∂yk∂xj− ∂2L

∂xk∂yj

)].

(6.39)

We prove now that the nonlinear connection with local coefficients given byexpression (6.27) satisfies the equations (6.39). The coefficients N i

j of thecanonical nonlinear connection (6.27) of a Lagrange space can be written as

N ij =

∂Gi

∂yj=

14

∂gip

∂yj

(∂2L

∂yp∂xmym − ∂L2

∂xp

)

+14gip

(2∂gjp

∂xmym − ∂2L

∂yj∂xp

)+

14gip ∂2L

∂yp∂xj.

If we multiply the above formula by gis we obtain

Nsj := gsiNij = −∂gis

∂yjGi +

12

∂gsj

∂xiyi +

14

(∂2L

∂ys∂xj− ∂2L

∂xs∂yj

),

which is equivalent to:

Nij =12S(gij) +

14

(∂2L

∂yi∂xj− ∂2L

∂xi∂yj

).(6.40)

We can see that (6.40) is equivalent to (6.39). From expression (6.40) itfollows that the canonical nonlinear connection of a Lagrange space with localcoefficients (6.27) satisfies the two axioms of the theorem. q.e.d.

With respect to the adapted cobasis dxi, δyi = dyi+N ijdxj of the canon-

ical nonlinear connection, the symplectic form ω of a Lagrange space Ln hasa simpler form:

ω = gijδyj ∧ dxi.(6.41)

Expression (6.41) is equivalent to (6.34), which says that symplectic form ωvanishes if both of its arguments are horizontal vector fields. One can see alsofrom (6.13) that ω(X,Y ) = 0 if both vectors X and Y are vertical vector fields.Therefore both horizontal and vertical subbundles are Lagrangian subbundlesfor the manifold TM . More than that one can determine a compatibilitybetween ω and horizontal and vertical projectors.


Proposition 6.6.2 Consider h and v horizontal and vertical projectors, P thealmost tangent structure and F the almost complex structure that correspondcanonical nonlinear connection N . Then following formula of compatibilitybetween these structures and symplectic form ω are true:

ω(hX, Y ) + ω(X, hY ) = ω(X,Y ),

ω(vX, Y ) + ω(X, vY ) = ω(X, Y ),

ω(hX, Y ) = ω(X, vY ),

ω(PX, Y ) + ω(X,PY ) = 0,

ω(PX,PY ) = −ω(X, Y ),

ω(FX,FY ) = ω(X, Y ), ∀X, Y ∈ χ(TM).

(6.42)

Proof. All formulae (6.42) can be proved if we use expression (6.41) for thesymplectic form and express vectors X,Y ∈ χ(TM) using the adapted basis ofhorizontal and vertical distributions. One can prove that first three formulae(6.42) are equivalent to ω(hX, hY ) = 0 if one use ω(vX, vY ) = 0. For the lasttwo formulae one can use P = h− v. q.e.d.

Theorem (6.6.3) shows that canonical nonlinear connection of a Lagrangespace is metric. Next, we shall determine the family of all metric nonlinearconnections for a Lagrange space. First we shall introduce the following Obataoperators:

Oijkl =

12(δi


kl =12(δi


Theorem 6.6.4 The family of all nonlinear connections that are metric withrespect to the metric tensor of a Lagrange space is given by

N ij = N ci

j + OkijmXm

k .(6.44)

Here Xmk is an arbitrary (1, 1)-type d-tensor field, and N ci

j are local coefficientsof the canonical nonlinear connection of the Lagrange space.

Proof. The condition that both nonlinear connections N cij and N i

j are metricwith respect to the metric tensor g, can be written as S(gij) = gmjN

cmi +

gimN cmj and S(gij) = gmjN

mi +gimNm

j . If we subtract these two equations weobtain O∗is

jm(Nmi − N cm

i ) = 0. Using the fact that OijklO

∗kmpj = 0, the solution

of this tensorial equation is given by expression (6.44). q.e.d.

Next we study symmetries of canonical nonlinear connection.

Definition 6.6.1 A vector field X ∈ χ(M) is said to be a Cartan symmetryof the nonlinear connection if LXcY = [Xc, Y ] ∈ χh(TM), ∀Y ∈ χh(TM).


If we use the fact that δ/δxi is a local basis for the horizontal distributionof canonical nonlinear connection X ∈ χ(M) is a symmetry if and only if

[Xc,

δ

δxi

]= −LXc(N i

j)∂

∂yj∈ χh(TM), ∀i ∈ 1, ..., n.

Consequently, we have that a vector field X ∈ χ(M) is a symmetry of canonicalnonlinear connection if and only if

0 = LXc(N ij)

=∂2Xi


∂xkNk

j +∂Xk

∂xjN i

k +∂N i

j

∂xkXk +

∂N ij

∂yk

∂Xk

∂xryr.

(6.45)

6.7 Almost Kahlerian model of a Lagrange space

We have seen in the previous section that a Lagrange space Ln induces acanonical nonlinear connection N . But the existence of a nonlinear connectionon the tangent bundle TM is equivalent to the existence of some geometricstructures on the manifold TM . As we have seen in Section 2.4 an impor-tant geometric structure on the manifold TM whose existence is equivalent tothe existence of the canonical nonlinear connection N is the almost complexstructure. This is given by the F(TM)-linear mapping F : χ(TM) −→ χ(TM)

F(

δ

δxi

)= − ∂

∂yi; F

(∂

∂yi

)=

δ

δxi(i = 1, ..., n),(6.46)

or by the tensor field

F = − ∂

∂yi⊗ dxi +

δ

δxi⊗ δyi.(6.47)

The almost complex structure F satisfies the property F F = −Id. Since thecanonical nonlinear connection N of a Lagrange space Ln is symmetric, thenaccording to Theorem 2.6.1, we have the following result.

Theorem 6.7.1 The almost complex structure F is a complex structure, whichmeans that F is integrable, if and only if the canonical nonlinear connectionN is integrable.

Proof. The Nijenhuis tensor NF of the almost complex structure F is given by

NF =12tijkδy

k ∧ δyj ⊗ δ

δxi+

12Ri

jkδyk ∧ δyj ⊗ ∂

∂yi.

Then, the Nijenhuis tensor NF vanishes if and only if:1) The horizontal distribution HTM is integrable, i.e., Ri

jk = 0 and

6.7. Almost Kahlerian model of a Lagrange space 149

2) The weak torsion of the nonlinear connection vanishes, which means thattijk = ∂N i

j/∂yk − ∂N ik/∂yj = 0 and the nonlinear connection is symmetric.

Since the local coefficients of the canonical nonlinear connection are givenby N i

j = ∂Gi/∂yj , it follows that tijk = 0, which means that the nonlinearconnection is symmetric. Consequently the integrability of the nonlinear con-nection is equivalent with the integrability of the associated almost complexstructure. q.e.d.

The metric tensor gij(x, y) of the Lagrange space Ln = (M, L(x, y)) andits canonical nonlinear connection N with local coefficients N i

j allow us tointroduce a pseudo-Riemannian structure G on the manifold TM . This isgiven by the so called Sasaki lift of the tensor metric gij :

G(x, y) = gij(x, y)dxi ⊗ dxj + gij(x, y)δyi ⊗ δyj .(6.48)

Theorem 6.7.21) G is a pseudo-Riemannian structure on the manifold TM determined

only by the fundamental function L(x, y).2) The horizontal and the vertical distributions HTM and V TM are or-

thogonal with respect to the metric G.

Proof.1) The tensorial character of gij , dxi and δyi will imply that G does not

depend on the transformation of induced local coordinates on TM. Also asrank(gij(x, y)) = n on TM we have that rankG = 2n on TM .

2) Since G(δ/δxi, ∂/∂yj) = 0, (i, j = 1, ..., n) and δ/δxi, ∂/∂yj generatesthe horizontal and the vertical distributions, we have that the two distributionsare orthogonal with respect to G. q.e.d.

Theorem 6.7.31) The pair (G,F) is an almost Hermitian structure on TM , determined

only by the fundamental function L(x, y).2) The symplectic structure associated to the pair (G,F) is the Cartan 2-

form ω, which is given byω = gijδy

i ∧ dxj .(6.49)

3) The space (TM,G,F) is almost Kahlerian.

Proof.1) Using the defining formula (6.46) of the almost complex structure F

and the defining formula (6.48) of the Sasaki metric G, we have by a directcalculation that G(FX,FY ) = G(X, Y ), ∀X, Y ∈ χ(TM). So, the pair (G,F)is an almost Hermitian structure on TM .


2) All equations that give the almost complex structure, (6.46), the Sasakilift G, (6.48) and the Cartan 2-form ω, (6.49) are expressed with respectthe adapted basis. Then it is easy to prove, using the adapted basis, thatthe following formula is true ω(X, Y ) = G(FX, Y ). We have seen in theprevious section that horizontal and vertical subbundles HTM and V TM areLagrangian submanifolds of TTM . As ω(X, Y ) = G(FX, Y ) we have that thisis true if and only if HTM and V TM = F(HTM) are orthogonal.

3) As ω is a symplectic structure on TM we have that space (TM,G,F)is almost Kahlerian. q.e.d.

The space K2n = (TM,G,F) is called the almost Kahlerian model of theLagrange space Ln = (M,L(x, y)). We can use it to study the geometry ofLagrange space Ln. For instance, the Einstein equations of the Riemannianspace (TM,G) can be considered as the Einstein equations of the Lagrangespace Ln.

G.S. Asanov showed [28] that the metric G given by the lift (6.49) does notsatisfy the principles of the Post-Newtonian calculus. This is due to the factthat the horizontal and the vertical terms of G do not have the same physicaldimensions. This is the reason for R. Miron to introduce a new lift [126] thatcan be used in gauge theory.

Let us consider the scalar field

||y||2 = gij(x, y)yiyj ,(6.50)

which is determined only by L(x, y). We assume that ||y|| > 0 on TM . Thenwe consider the following lift of the fundamental tensor field gij(x, y):

G(x, y) = gij(x, y)dxi ⊗ dxj +a2

||y||2 gij(x, y)δyi ⊗ δyj ,(6.51)

where a > 0 is a constant, imposed by applications in theoretical physics. Thisis to preserve the physical dimensions of the both terms of G.

Let us consider also the tensor field on TM

F = −||y||a

∂

∂yi⊗ dxj +

a

||y||δ

δxi⊗ δyi,(6.52)

and the 2-formω =

a

||y||ω,(6.53)

where ω is the symplectic form of the Lagrange space Ln given by (6.49).

Theorem 6.7.41) The pair (G, F) is an almost Hermitian structure on TM that depends

only on the fundamental function L(x, y).

6.8. Metric N -linear connections 151

2) The 2-form ω given by (6.53) is the symplectic structure that correspondto the pair (G, F).

3) As the symplectic structure ω is conformal to the symplectic structureω, then the pair (G, F) is a conformal almost Kahlerian structure.

We can remark now that the conformal almost Kahlerian space K2n =(TM, G, F) can be used for applications in gauge theories, which implies thenotion of the regular Lagrangian.

6.8 Metric N-linear connections

In this section we determine some N -linear connections on the tangent bundleTM that are compatible with the Riemannian metric G given by (6.48). TheseN -linear connections will be canonically associated with the Lagrange spaceLn. So what we prove next is that on the manifold TM there exist linearconnections D, which satisfy the axioms:

1) D is a metric connection with respect to G i.e.

DG = 0.(6.54)

2) D preserves by parallelism the horizontal distribution HTM of thecanonical nonlinear connection N .

3) The tangent structure J is absolutely parallel with respect to D, whichmeans that DJ = 0.

We have seen that for a Finsler space, using some of the previous threeaxioms and some other axioms, we obtain one of the well known N -linearconnections: Cartan, Chern-Rund, Berwald and Hashiguchi. For a differentpoint of view about N -linear connections one can associate to a Lagrangespace see [181].

If we consider Dh and Dv the h- and v-covariant derivative induced bythe N -linear connection D then we have that the metric condition DG = 0 isequivalent to the following conditions

DhXG = 0, Dv

XG = 0, ∀X ∈ χ(TM).(6.55)

Moreover, the second and the third axioms can be translated as follows:

v(DhXh(Y )) = 0, h(Dh

Xv(Y )) = 0,

v(DvXh(Y )) = 0, h(Dv

Xv(Y )) = 0,

DhXJ = 0, Dv

XJ = 0, ∀X, Y ∈ χ(TM).

(6.56)

Consequently, the linear connection D is an N -linear connection and has thelocal coefficients DΓ(N) = (Li

jk, Cijk), which verify the following tensorial


equations:gij|k = 0, gij |k = 0,(6.57)

where “|” and “|” are the h- and v-covariant derivatives with respect to connec-tion D. Conversely, if an N -linear connection D with the coefficients Li

jk, Cijk

verifies the properties (6.56) and (6.57) then it is metric with respect to G.This means that the equations (6.55) are verified and consequently the equa-tion DG = 0 is also verified.

Next, we determine the general solution (Lijk, C

ijk) of the tensorial equa-

tions (6.57).

Theorem 6.8.11) There exists a unique N -linear connection D, with local coefficients

Lijk, C

ijk, which verifies the following axioms:

A1 : N is the canonical nonlinear connection of the Lagrange space Ln;A2 : gij|k = 0 (it is h-metric);A3 : gij |k = 0 (it is v-metric);A4 : T i

jk = 0 (it is h-torsion free, or h-symmetric);A5 : Si

jk = 0 (it is v-torsion free, or v-symmetric).2) The coefficients (Li

jk, Cijk) are expressed by the following generalized

Christoffel symbols:

Lijk =

12gir

(δgrk

δxj+

δgrj

δxk− δgjk

δxr

),

Cijk =

12gir

(∂grk

∂yj+

∂grj

∂yk− ∂gjk

∂yr

).

(6.58)

3) This connection depends only on the fundamental function L(x, y) ofthe Lagrange space Ln.

Proof. Axioms A2 and A4, uniquely determine the horizontal coefficients ofan N -linear connection. Indeed, the metric condition A2 implies

0 =δgij

δxk− gmjL

mik − gimLm

jk.

If we permute the indices i, j, k, we get

0 =δgki

δxj− gmiL

mkj − gkmLm

ij ,

0 =δgjk

δxi− gmkL

mji − gjmLm

ki.

If we add the first two equations, subtract the third one, and take into accountthe symmetry axiom A4: Li

jk = Likj then we obtain first formula (6.58).


Second formula (6.58), which gives the vertical coefficients Cijk of the

canonical N -linear connection, can be obtained in a similar manner usingthe two axioms A3 and A5. q.e.d.

The metric N -linear connection (6.58) is called the canonical N -linearconnection or the Cartan connection of the Lagrange space and will be denotedby CΓ(N).

Now we can study the geometry of the Lagrange space Ln using the canon-ical metric connection or using a general metric connections, which satisfy theaxioms A1, A2, A3.

For the canonical metric N -linear connection we study now Cartan’s struc-ture equations. The connection 1-forms ωi

j of the N -linear connection CΓ(N)are given by

ωij = Li

jkdxk + Cijkδy

k,(6.59)

where Lijk, Ci

jk are given by (6.58).

Theorem 6.8.2 The connection 1-forms ωij of the canonical metric N -linear

connection CΓ(N) satisfy the following structure equations:

d(dxi)− dxk ∧ ωik = −

(0)

Ω i,

d(δyi)− δyk ∧ ωik = −

(1)

Ω i,

(6.60)

andωi

j − ωkj ∧ ωi

k = −Ωij(6.61)

Here the 2-forms of torsion(0)

Ω i and(1)

Ω i are given by the following formulae:

(0)

Ω i = Cijkdxj ∧ δyk,

(1)

Ω i =12Ri

jkdxj ∧ dxk + P ijkdxj ∧ δyk.

(6.62)

The 2-forms of curvature Ωij are given by

Ωij =

12Ri

jkhdxk ∧ dxh + P ijkhdxk ∧ δyh +

12Si

jkhδyk ∧ δyh.(6.63)

The d-tensors of curvature Rijkh, P i

jkh, Sijkh and the d-tensors of torsion Ri

jk,P i

jk are given by expressions (3.19) and (3.14). Differentiating in expression(6.60) and using them in the result one obtains the Bianchi identities.


We remark that besides the canonical metric N -linear connection CΓ(N) =(Li

jk, Cijk), there are three other remarkable connections like the Berwald con-

nection BΓ(N), the Chern-Rund connection CRΓ(N) and the Hashiguchi con-nection HΓ(N) these connections have the local coefficients given by and

BΓ(N) =

(∂N i

j

∂yk, 0

), CRΓ(N) = (Li

jk, 0), HΓ(N) =

(∂N i

j

∂yk, Ci

jk

).

Next we present some properties of the canonical metric connection CΓ(N).

Proposition 6.8.1 We have the following properties of the canonical nonlin-ear connection CΓ(N):

1)∑

(ijk) Rijk = 0, and Pijk = gisPsjk is totally symmetric.

2) Cijk =14

∂3L

∂yi∂yj∂yk= gjsC

sik.

3) The covariant d-tensors of curvature Rijkh, Pijkh, Sijkh (with Rijkh =gjsR

sihk) are skew-symmetric in the first two indices.

4) Sijkh = CiksCsjh − CihsC

sjk.

Proof. These properties can be proved using the fact that dω = 0, where ω isthe symplectic structure of the Lagrange space, the Ricci identities applied tothe metric tensor gij of the space and the h- and v-metricity conditions givenby the tensorial equations gij|k = 0 and gij |k = 0. q.e.d.

Using similar methods we can consider and study a metric N -linear con-nection DΓ(N) = (Li

jk, Cijk), which satisfies the axioms A1, A2,A3 and has a

priori given d-tensors of torsion T ijk and Si

jk.

Theorem 6.8.31) There exists only one N -linear connection DΓ(N) = (Li

jk, Cijk), which

satisfies the following axioms:

A′1 : The nonlinear connection N is the canonical nonlinear connection ofthe space Lagrange Ln;

A′2 : The N -linear connection (D) is h-metric, i.e. gij|k = 0;A′3 : The N -linear connection (D) is v-metric, i.e. gij |k = 0;A′4 : The (h)h-tensor of torsion T i

jk is a priori given;A′5 : The (v)v-tensor of torsion Si

jk is a priori given.2) The coefficients (Li

jk, Cijk) of the previous connection are given by the

following formulae:

Lijk = Li

jk +12gih(gjrT

rkh + gkrT

rjh − ghrT

rkj),

Cijk = Ci

jk +12gih(gjrS

rkh + gkrS

rjh − ghrS

rkj).

(6.64)

6.9. Almost Finslerian Lagrange spaces 155

Here (Lijk, C

ijk) are the local coefficients of the canonical metric N -linear

connection CΓ(N).From now on T i

jk, Sijk will be denoted simply by T i

jk, Sijk.

Proposition 6.8.2 For a metric N -linear connection DΓ(N) the Ricci iden-tities are given by

Xi|j|k −Xi|k|j = XmRimjk −Xi|mTm

jk −Xi|mRmjk,

Xi|j |k −Xi|k|j = XmP imjk −Xi|mCm

jk −Xi|mPmjk ,

Xi|j |k −Xi|k|j = XmSimjk −Xi|mSm

jk.

(6.65)

The previous identities can be extended to a d-tensor field of type (r, s).The horizontal and the vertical deflection tensor fields Di

j and dij are de-

fined byDi

j = yi|j , dij = yi|j .(6.66)

The deflection tensor fields can be expressed as follows:

Dij = ysLi

sj −N ij ; di

j = δij + ysCi

sj .(6.67)

Applying the Ricci identities (6.65) to the Liouville vector field yi we obtain:

Theorem 6.8.4 If DΓ(N) is a metric N -linear connection then the followingidentities hold

Dij|k −Di

k|j = ymRimjk −Di

mTmjk − di

mRmjk,

Dij |k − di

k|j = ymP imjk −Di

mCmjk − di

mPmjk ,

dij |k − di

k|j = ymSimjk − di

mSmjk.

(6.68)

We shall apply this theory next section where we shall use the canonical metricN -linear connection CΓ(N), for which T i

jk = 0 and Sijk = 0. Of course, a theory

of parallelism of vector fields with respect to the connection DΓ(N) can bedeveloped.

6.9 Almost Finslerian Lagrange spaces

In this section we apply the theory developed until now in a study of theLagrange spaces whose Lagrangians are given by

L(x, y) = F 2(x, y) + Ai(x)yi + U(x),(6.69)

where F (x, y) is the fundamental function of a Finsler space Fn, Ai(x) arecomponents of a covector field and U(x) is a smooth function.


These Lagrange spaces are known as Almost Finsler Lagrange Spaces(shortly AFL-spaces), see chapter IX in [130]. We associate to the AFL-spaces Ln = (M,L(x, y)), with L(x, y) given by (6.69), the Finsler spaceFn = (M,F (x, y)) and we shall indicate by a circle “′′ placed over all ge-ometric objects related to it. For example,

gij(x, y) =12

∂2F 2

∂yi∂yj(6.70)

is the fundamental metric tensor of Fn, while its Cartan linear connection isCΓ(N) = (F i

jk, Cijk), where N is the Cartan nonlinear connection of Fn.

And we have that

Cijk =14

∂3F 2

∂yi∂yj∂yk, N i

j =12

∂

∂yj(γi

rsyrys),(6.71)

gij|k = 0, gij |k = 0, F 2|k = 0, Di

j = 0, dij = δi

j .(6.72)

The energy of the Finsler space Fn is E(x, y) = gij(x, y)yiyj = F 2(x, y).By a direct calculation one finds that the metric tensor of the AFL-space

Ln coincides with the metric tensor of the Finsler space Fn, that is

gij(x, y) = gij(x, y).(6.73)

Therefore, the absolute energy E(x, y) = gij(x, y)yiyj is equal with the energyof of the Finsler space Fn.

From expression (6.73) it follows that the metric tensor gij(x, y) is posi-tively homogeneous of order zero in y. This property provides a local charac-terization of AFL-spaces. Indeed, we have

Theorem 6.9.1 A Lagrange space Ln = (M, L(x, y)) is locally an AFL-spaceif and only if its metric tensor

gij(x, y) =12

∂2L

∂yi∂yj(x, y)

is positively homogeneous of order zero in y.

Proof. Assume that gij(x, y) is positively homogeneous of order zero in y andconsider the absolute energy E(x, y) = gij(x, y)yiyj of Ln. This function has allthe properties of a fundamental function of a Finsler space Fn = (M, E(x, y)).The fundamental metric tensor of Fn is gij(x, y). Thus we have

∂2(L− E)∂yi∂yj

= 0.

6.9. Almost Finslerian Lagrange spaces 157

Consequently, L− E = Ai(x)yi + U(x). Therefore, the Lagrangian L has theform (6.69). The converse has been presented above. q.e.d.

The canonical semispray of the AFL-space Ln has the local coefficientsgiven by

Gi = Gi − gij

(Fjky

k +14

∂U

∂xj

),(6.74)

where Gi are the local coefficients of the canonical spray of Fn and

Fjk(x) =12

(∂Ak

∂xj− ∂Aj

∂xk

)(6.75)

is the electromagnetic tensor field of potentials Ai(x).The integral curves of the Euler-Lagrange equations are the solution curves

of the equations

d2xi

dt2+ γi

jk

(x,

dx

dt

)dxj

dt

dxk

dt= 2F i

k(x)dxk

dt+

12gij ∂U

∂xj(6.76)

that generalize the Lorentz equations (6.31).Applying the formulae (6.27) for Gi given by expression (6.74), one obtains

that the canonical nonlinear connection of the AFL-space Ln has the followinglocal coefficients:

N ij = N i

j − F ij + 2Ci

jsFskyk +

12Ci

jsgsh ∂U

∂xh.(6.77)

The raising and lowering of indices is done by the metric tensor gij(x, y).Let us introduce the d-tensor field

Bij = F i

j − 2CijsF

skyk − 1

2Ci

jsgsh ∂U

∂xh.(6.78)

If we contract with yj , we have

Bijy

j = F ijy

j(6.79)

since Cijsy

j = 0. And, therefore expression (6.77) reduces to

N ij = N i

j −Bij .(6.80)

According to Theorem 6.6.2 we have that the autoparallel curves of the non-linear connection of the AFL-space Ln are given by the following differentialequation:

d2xi

dt2+ γi

jk

(x,

dx

dt

)dxj

dt

dxk

dt= F i

j (x)dxj

dt,(6.81)


that is also a generalized Lorentz equation.This has to be compared with equations (6.76). It is confirmed that for a

general Lagrange space the solution curves of the Euler-Lagrange equation donot coincide with the autoparallel curves of the canonical nonlinear connection.

The covariant d-tensor Bij = gikBkj has the form

Bij = Fij − 2CijsFshyh − 1

2Cijsg

sh ∂U

∂xh.(6.82)

This form shows that Fij(x) is the skew symmetric part of Bij , that is

Fij =12(Bij −Bji).(6.83)

Theorem 6.9.2 The following two assertions are equivalent:

Ai(x) is a gradient,(6.84)

Bij(x) = −12Ck

ij

∂U

∂xk.(6.85)

Proof. If Ai(x) is a gradient, then Fij = 0 and by (6.82), we have that (6.84)holds good. Conversely, (6.84) and (6.82) imply Fij − 2CijsF

shyh = 0. A

contraction by yj gives Fij(x)yj = 0, because of Cijsyj = 0. This implies

Fij = 0, hence Ai(x) is a gradient. q.e.d.

Theorem 6.9.3 The canonical N -linear connection CΓ = (N ij , L

ijk, C

ijk) of

the AFL-space Ln has the coefficients

Lijk = F i

jk + CijsB

sk + Bi

jk, Cijk = Ci

jk,(6.86)

where the d-tensor Bijk is given by

Bijk = gih(CkhsB

sj − CjksB

sh).(6.87)

Proof. The second equality in (6.86) follows from the equality gij(x, y) =gij(x,y). Taking into account this equality and inserting in (6.58) the coefficientsN i

j(x, y) given by (6.80), we obtain first

δgij

δxk=

δgij

δxk+ 2CijsB

sk

and then, by a direct calculation, the form of Lijk from (6.86) with Bi

jk givenby (6.87) is obtained. q.e.d.

Let Dij = yi

|j and dij = yi|j be the h-deflection tensor field and the v-

deflection tensor field, respectively, of the AFL-space Ln. Their expanded

6.10. Geometry of ϕ-Lagrangians 159

form are given in (6.67). The second equality in (6.86) shows that dij = δi

j .Making use of the form of Li

jk in (6.86), we obtain Dij = Di

j + Bikjy

k + Bij ,

where Dij is the h-deflection tensor field of the Finsler space Fn. By (5.52)

this vanishes. Using Bijk from (6.87), Di

j takes the form

Dij = Bi

j + CijhBh

kyk.(6.88)

The equalities Cijk = Ci

jk and Cijky

k = 0 were repeatedly used.The equation (6.88) is equivalent with

Dij = Bij + CijhBhkyk.(6.89)

At this point we notice the following consequence of (6.83):

Fij(x) =12(Dij(x, y)−Dji(x, y)).(6.90)

The equation (6.90) provides a reason to define the h-electromagnetic ten-sor field for any Lagrange space as the d-tensor field (1/2)(Dij − Dji), withDij = gikD

kj , where Di

j is its h-deflection tensor field. On the other hand inany Lagrange space we have also the v-deflection tensor field di

j . Thus we maydefine the v-electromagnetic tensor field of any Lagrange space as the d-tensorfield (1/2)(dij − dji). It is easy to check that the tensor field Fij(x) given by(6.75) satisfies the following Maxwell equations:

Fij|k + Fjk|i + Fki|j = 0, Fij |k + Fjk|i + Fki|j = 0.(6.91)

We will see later that for a Lagrange space, the d-tensor fields (1/2)(Dij −Dji) and (1/2)(dij − dji) satisfy some equations that generalize the Maxwellequations (6.91).

6.10 Geometry of ϕ-Lagrangians

Among the examples of regular Lagrangians we gave in Section 1 of this chap-ter we included the so-called ϕ-Lagrangians. In this section we treat theseLagrangians from a pure geometrical point of view following P.L. Antonelliand D. Hrimiuc, [24], [25]. For a view on their applications we refer to themonograph [23]. We recall that a ϕ-Lagrangian L is associated to a Finslerspace Fn = (M, F ) by means of a C∞-class function ϕ : R+ → R using thecomposition L = ϕ(F 2). The following conditions for ϕ:

(i) ϕ′(t) 6= 0,


(ii) ϕ′(t) + 2tϕ′′(t) 6= 0,

for any t ∈ Im(F 2), assure that the Lagrangian L = ϕ(F 2) is regular and so thepair Ln = (M, L = ϕ(F 2)) is a Lagrange space, called hereafter a ϕ-Lagrangespace. We put

gij =12

∂2F 2

∂yi∂yjand aij =

12

∂2L

∂yi∂yj.

As∂L

∂yi= ϕ′(F 2)

∂F 2

∂yi,

we have that the two metric structures aij and gij are related by

aij = ϕ′(F 2)gij + 2F 2ϕ′′(F 2

) ∂F

∂yi

∂F

∂yj.

As in the previous section we shall refer to all geometric objects associated toFn, by putting a “” over them. Then,

yi := F∂F

∂yi=

12

∂F 2

∂yi=

12

∂(gikyjyk)

∂yi= giky

k.

With this notation, the metric tensor aij of the ϕ-Lagrange space is given by

aij = ϕ′(

gij + 2ϕ′′

ϕ′yiyj

).(6.92)

If conditions (i) and (ii) for ϕ are satisfied, then the matrix (aij) is invertibleand its inverse is given by

aij =1ϕ′

(gij − 2

ϕ′′

ϕ′ + 2F 2ϕ′′yiyj

).(6.93)

Let N ij(x, y) be the coefficients of the Cartan nonlinear connection of the

Finsler space Fn and N ij be the coefficients of the canonical nonlinear connec-

tion of the ϕ-Lagrange space Ln. We have N ij = ∂Gi/∂yj , where Gi are the

coefficients of the canonical semispray of L as well as N ij = ∂Gi/∂yj , where

Gi are the coefficients of the geodesic spray derived from F 2.

Theorem 6.10.1 For a ϕ-Lagrange space Ln = (M,L = ϕ(F 2)) the canon-ical nonlinear connection with local coefficients N i

j coincides with the Cartannonlinear connection with local coefficients N i

j of the associated Finsler spaceFn = (M, F ).

6.10. Geometry of ϕ-Lagrangians 161

Proof. One can check the equality N ij = N i

j by a direct calculation. Here is asimpler argument. Let

s(t) =∫ t

0F

(x(τ),

dx

dτ

)dτ

be the Finsler arclength of a curve xi = xi(t) in M . We take s as a pathparameter. Then F (x(s), dx/ds) = 1. For L = ϕ(F ), where ϕ(t) = ϕ(t2) weget

dds

(∂L

∂yi

)− ∂L

∂xi= ϕ′(F )

(dds

(∂F

∂yi

)− ∂F

∂xi

).

Thus, the solution curves of the Euler-Lagrange equation for L, called some-times the extremes of Ln, coincide with the geodesics of Fn and consequentlyGi = Gi. It follows N i

j = N ij . q.e.d.

In the proof of the Theorem 6.10.1 we have seen that the semispray definedby a ϕ-Lagrangian is in fact a spray, namely that defined by F 2. The truth isthat the ϕ-Lagrangians were discovered looking for Lagrangians that are notin the form F 2, which do generate a spray and not a semispray. The problemarises from Biology. The first class of such Lagrangians was provided by thehomogeneous Lagrangians of order m ≥ 2 that are obtained for ϕ(t) = tm/2.These Lagrangians were studied by M. Anastasiei and P.L. Antonelli, [8].

Let CΓ(N) = (Lijk, C

ijk) be the canonical metric N -linear connection of

the ϕ-Lagrange space Ln and let CΓ(N) = (Lijk, C

ijk) be the Cartan linear

connection of the Finsler space Fn. Their coefficients are given by (6.58) withthe obvious choices of the fundamental metrics.

Theorem 6.10.2 Let Ln = (M,L(x, y)) be a ϕ-Lagrange space associated tothe Finsler space Fn = (M, F (x, y)). The coefficients of the canonical metricN -linear connections CΓ(N) and CΓ(N) of these spaces are related as follows:

Lijk = Li

jk, Cijk = Ci

jk +ϕ′′

ϕ′(δi

j yk + δikyj)

+ϕ′′

ϕ′ + 2F 2ϕ′′gjky

i +2(ϕ′′′ϕ′ − 2(ϕ′′)2)ϕ′(ϕ′ + 2F 2ϕ′′)

yj ykyi.

(6.94)

Proof. We put together the formulae to be used in the calculations, leaving outthe algebraic manipulations. First we notice that for any C∞-class functionψ : R+ → R, putting f(x, y) = ψ(F 2(x, y)) we get

δf

δxk= f ′F 2

|k = 0,∂f

∂yk= f ′

∂F 2

∂yk= 2f ′yk(6.95)


by the Proposition 5.6.1. The same proposition tells us that

δyi

δxk= yi|k + Lh

ikyh = Lhikyh,

∂yi

∂yk= gik.(6.96)

Using the form (6.92) of aij one gets

δaij

δxk= ϕ′

δgij

δxk+ 2ϕ′′yh(Lh

ikyj + Lhjkyi),

∂aij

∂yk= ϕ′

∂gij

∂yk+ 2ϕ′′(gij yk + gjkyi + gkiyj) + 2ϕ′′′yiyj yk.

(6.97)

Using (6.97) in (6.58) and taking into account (6.93) by some algebra one finds(6.94). q.e.d.

Theorem 6.10.3 The canonical metric N -linear connection CΓ(N) of a ϕ-Lagrange space has the following properties:

1) Dik := yi

|k = 0, yi|k = 0;

2) dik := yi|k =

ϕ′ + ϕ′′F 2

ϕ′δik + Byky

i, where yi|k = aik + Cikhyh and

B = 2ϕ′ϕ′′ + F 2(ϕ′′′ϕ′ − (ϕ′′)2)

ϕ′(ϕ′ + 2F 2ϕ′′);

3) L|k = 0, L|k =2ϕ′

ϕ′ + 2F 2ϕ′′yk, where yi := aijy

j;

4) For ϕ = atα + b, a 6= 0, α /∈ 0, 1/2 we have dik = αδi

k, and for ϕ =a ln t + b, a 6= 0, we have di

k = 0.

Proof.1) The h-deflection tensor field Di

k is equal with the h-deflection tensorfield Di

k of Fn. The deflection tensor of the Finsler space Fn satisfies Dik = 0.

From yi = aihyh it follows yi|k = aih|kyh + aihyh|k = 0 as both aih|k and yh

|k arezero.

2) The v-deflection tensor field can be expressed as follows: yi|k = δik +

Cikhyh. Here Ci

khyh is calculated taking into account that Cikhyh = 0 and

yhyh = F 2.3) L|k = 0 by (6.95). Again by (6.95) we get L|k = 2ϕ′(F 2)yk. On the

other hand, yi = aijyj = (ϕ′ + 2F 2ϕ′′)yi. Hence

L|k =2ϕ′

ϕ′ + 2F 2ϕ′′yk.

4) It follows by a direct calculation based on 2). q.e.d.

6.11. Gravitational and electromagnetic fields 163

The vanishing of the h-deflection tensor field of the N -linear connectionCΓ(N) means that N i

j(x, y) = Lijk(x, y)yk. From Theorem 6.6.2 we obtain

that the autoparallel curves of the canonical nonlinear connection of a ϕ-Lagrange space are given by the following system of differential equations:

d2xi

ds2+ Li

jk

(x,

dx

ds

)dxj

ds

dxk

ds= 0.(6.98)

The same system gives the geodesics of the Finsler space Fn.In a ϕ-Lagrange space the components of the torsion of the canonical

metric N -linear connection CΓ(N) are given by

T ijk = Li

kj − Lijk = 0, Si

jk = Cikj − Ci

jk = 0,

P ijk =

∂N ij

∂yk− Li

kj , Rijk =

δN ij

δxk− δN i

k

δxj,

are the same as those of the associated Finsler space, except Cijk, which is

different, given in Theorem 6.10.2.

Theorem 6.10.4 In a ϕ-Lagrange space the following properties hold true:

Ri0kh = (1 + A)Ri

kh, P i0kh = (1 + A)P i

kh,

Si0kh = AB(δi

kyh − δihyk), where

A :=ϕ′′

ϕ′F 2 and B = 2

ϕ′ϕ′′ + F 2(ϕ′′′ϕ′ − (ϕ′′)2)ϕ′(ϕ′ + 2F 2ϕ′′)

.

In particular, if ϕ(t) = atα + b, α /∈ 0, 1/2, a 6= 0, we get

Ri0kh = αRi

kh, P i0kh = αP i

kh, Si0kh = 0.(6.99)

If ϕ(t) = a ln t + b, a 6= 0, we get

Ri0kh = 0, P i

0kh = 0, Si0kh = 0.(6.100)

These identities have to be compared with those from (5.54).

6.11 Gravitational and electromagnetic fields

Let us consider a Lagrange space Ln = (M,L(x, y)) endowed with the canon-ical metric N -linear connection CΓ(N) = (Li

jk, Cijk).

The covariant deflection tensors fields Dij and dij can be introduced asfollows

Dij = gisDsj , dij = gisds

j .(6.101)


As the metric tensor gij is h- and v- covariant constant, then the h- covariantderivatives of these two d-tensor fields is given by:

Dij|k = gisDsj|k, dij|k = gisds

j|k

and similar formulae hold for the v-covariant derivation.

Theorem 6.11.1 The covariant deflection tensor fields Dij and dij of thecanonical metric N -linear connection DΓ(N) satisfy the identities:

Dij|k −Dik|j = ysRsijk − disRsjk,

Dij |k − dik|j = ysPsijk −DisCsjk − disP

sjk,

dij |k − dik|j = ysSsijk.

(6.102)

Proof. These identities follow from (6.68) with T ijk = 0, Si

jk = 0. q.e.d.

At the end of the Section 6.9 we discussed about a reason for the followingdefinition.

Definition 6.11.1 The tensor fields

Fij =12(Dij −Dji), fij =

12(dij − dji)(6.103)

are called the h- and v-electromagnetic tensor field of the Lagrange spaces Ln,respectively.

It is obvious that with respect to the canonical metric N -linear connectionCΓ(N), the v- electromagnetic tensor fij vanishes.

The terms of h- and v-electromagnetic fields are motivated not only by theLagrangian of electrodynamic or by the almost Finsler Lagrangians but alsoby the following result

Theorem 6.11.2 The following generalized Maxwell equations hold true:

Fij|k + Fjk|i + Fki|j = −∑(ijk) Ci0sR

sjk,

Fij |k + Fjk|i + Fki|j = 0,(6.104)

where Ci0j = ysCisj .

Proof. We have2

∑

(ijk)

Fij|k =∑

(ijk)

(Dij|k −Dik|j)

and by (6.102) we get

2∑

(ijk)

Fij|k =∑

(ijk)

(R0ijk −Rijk − C0isRsjk),

6.11. Gravitational and electromagnetic fields 165

because dij = gij + C0ij .According to formula (5.54) we have that

∑(ijk) Rijk = 0. Now, we recall

a Bianchi identity (chapter VIII in [130]):∑

(hjk)(Rshjk − Rr

jkCsrk) = 0. This

has the following equivalent form:∑

(hjk)

(R0hjk + C0ksRshj) = 0,

where the identity Rihjk + Rhijk = 0 was used. Thus we get∑

(ijk) R0ijk =−∑

(ijk) C0isRsjk. Using the above formula, the first identity from (6.104) is

obtained.Let us prove the second one. Again by (6.102), we get

2∑

(ijk)

Fij |k =∑

(ijk)

[(P0ijk − P0ikj) + (dik|j − dij|k)− (disPsjk − disP

skj)].

Now, we make use of dik|j = C0ik|j , a consequence of gik|j = 0 and of Dij = 0,

as well as of disPsjk = Pijk + C0ihP h

jk and take into account that Pijk is totallysymmetric. We obtain

2∑

(ijk)

Fij|k =∑

(ijk)

A[jk](P0ijk + C0ik|j − C0ihP hjk),

where A[jk] denotes a substraction after the interchange of the indices j andk. The right hand side of the later identity can be put in the form

yh∑

(ijk)

A[jk](Cihk|j + Ci

hsPskj − P i

hkj)

and it is zero because of the Bianchi identity

A[jk](Cihk|j + Ci

hsPskj − P i

hkj) = 0.

Consequently, Maxwell equations (6.104) are true. q.e.d.

Remarks.1) If the Lagrange space Ln reduces to a Finsler space Fn then Ci0s = 0

and equations (6.104) are much simpler. Equations Ci0s = 0 are also true foran AFL-space.

2) If the Cartan nonlinear connection N is flat, which means that thehorizontal distribution HTM is integrable that is equivalent to the fact thatthe curvature tensor vanishes, Rh

ij = 0, then the previous equations have alsoa simpler form.

3) For the electrodynamic Lagrange space the generalized Maxwell equa-tions coincide withe the classical ones.


If we putF ij = gisgjrFsr and

hJ i = F ij |j , vJ i = F ij |j ,(6.105)

then it is easy to prove the following theorem:

Theorem 6.11.3 The following laws of conservation hold

hJ i|i =

12F ij(Rij −Rji) + F ij |rRr

ij,vJ i|i = 0,

(6.106)

where Rij is the Ricci tensor of the curvature tensor Rhijk.

The electromagnetic tensor fields Fij , fij and the Maxwell equations wereintroduced by R. Miron and M. Radivoiovici, [130].

6.12 Einstein equations of Lagrange spaces

Let Ln = (M, L) be a Lagrange space and let us consider its almost Kahlerianmodel (TM, G, F ) constructed in Section 6.7. As we said in this section, theEinstein equations of the pseudo-Riemannian space (TM, G) can be consideredas “the Einstein equations” of the Lagrange space Ln. Here we explain thisidea in much more details. The equations to be obtained are basic in whatit can be called a Lagrangian theory of relativity, thought of as an extensionof some Finslerian theories of relativity, [119]. The construction, which wepresent here, following the theory presented in [119] and [175] is only a shortand introductory contact with a subject of large interest in general relativityand gauge theories, [28], [47], [35], [93], [187], [189], [191].

On the pseudo-Riemannian manifold (TM, G) we consider and denote byDΓ(N) = (Li

jk, Cijk) the metric N -linear connection with the torsion compo-

nents T ijk and Si

jk a priori given. For the canonical nonlinear connection weuse the notation DΓ(N) = (Li

jk, Cijk). The coefficients of DΓ(N) are given by

(6.64) with an obvious change in notations. The Einstein equation written forthe connection DΓ(N) on the pseudo-Riemannian manifold (TM,G) is

Ric(D)− 12Sc(D)G = kT(6.107)

where Ric(D) is the Ricci tensor field and Sc(D) is the scalar curvature ofDΓ(N), k is a constant and T is the energy-momentum tensor field. We shallexpress (6.107) in the basis (δ/δxi, ∂/∂yi) that is adapted to the decompositionof TuTM, u ∈ TM into horizontal and vertical subspaces. Recall that suchdecomposition is produced by the nonlinear connection N derived from L.

6.12. Einstein equations of Lagrange spaces 167

In order to do this we set (Xα) = (Xi, X(i)), where Xi = δ/δxi and X(i) =∂/∂yi. The Greek indices will run from 1 to 2n and the indices (i), (j), ... willrun from n + 1 to 2n. The local vector fields (Xα) provide a nonholonomicbasis that introduces the nonholonomic coefficients given by

[Xβ, Xγ ] = WαβγXα.(6.108)

They satisfy the following Vranceanu identities [190]∑

(αβγ)

[Xα(W δβγ) + Wϕ

αβW δγϕ] = 0.(6.109)

Let DXγXβ = ΓαβγXα. Then in the basis (Xα) the torsion T of the N -linear

connection D has the components

Tαβγ = Γα

βγ − Γαγβ + Wα

βγ .(6.110)

In the basis (Xα) the curvature R of the N -linear connection D has the com-ponents

Rαβγδ = XδΓα

βγ −XγΓαβδ + Γϕ

βγΓαϕδ − Γϕ

βδΓαϕγ + Γα

βϕWϕγδ.(6.111)

These components are given by

T(Xγ , Xβ) = TαβγXα, R(Xδ, Xγ)Xβ = Rα

βγδXα.(6.112)

In the adapted basis (Xα) the Bianchi identities of D take the form∑

(αβγ)

(DαRϕδβγ + Rϕ

δαψTψβγ) = 0,(6.113)

∑

(αβγ)

(DαTδβγ + Tϕ

αβTδϕγ − Rδ

αβγ) = 0,(6.114)

where Dα := DXα .If in the preceding equations the components with respect to Xi = δ/δxi

and X(i) = ∂/∂yi are separated, it comes out that among the coefficients Γαβγ

we haveΓi

jk = Lijk, Γ(i)

(j)(k) = Cijk,(6.115)

the other coefficients, coming in six groups, being zero.This is a great advantage created by the choice of the basis (Xα) as well

as by the fact that D is an N -linear connection.The set of nonholonomic coefficients Wα

βγ splits in eight groups, five ofthem are zero and the other three are given by

W(i)jk = Ri

jk, W(i)(j)k = −∂N i

k

∂yj, W

(i)j(k) =

∂N ij

∂yk.(6.116)


The set of components Tαβγ of the torsion field T splits into following eight

groups:

Tijk = T i

jk, Ti(j)k = −Ci

jk, Tij(k) = −Ci

jk, Ti(j)(k) = 0,

T(i)jk = Ri

jk, T(i)(j)k = −P i

kj , T(i)j(k) = P i

jk, T(i)(j)(k) = 0.

(6.117)

The components Rαβγδ of the curvature tensor field R, when separated,

appear in sixteen groups. Only the following ones have non zero elements:

Rijkh = R(i)

(j)kh = Rijkh,

Rijk(h) = −Ri

j(k)h = R(i)(j)k(h) = −R(i)

(j)(k)h = P ijkh,

Rij(k)(h) = R(i)

(j)(k)(h) = Sijkh.

(6.118)

We setRijkh = gjsR

sikh, Pijkh = gjsP

sikh, Sijkh = gjsS

sikh,

Rij = Rsijs, Sij = Ss

ijs,1P ij= P s

ijs,2P ij= P s

isj ,

R = gijRij , S = gijSij .

(6.119)

With respect to the basis (Xα), the Ricci tensor field of the N -linear connectionDΓ(N) has the components

Rij = Rij , R(i)j =1P ij , Ri(j) = −

2P ij , R(i)(j) = Sij .(6.120)

The pseudo-Riemannian metric G has the components Gαβ given by

Gij = gij , Gi(j) = 0, G(i)j = 0, G(i)(j) = gij(6.121)

and its inverse Gαβ has the following components:

Gij = gij , Gi(j) = 0, G(i)j = 0, G(i)(j) = gij .(6.122)

Thus, the tensor field Rαβ = GαγRγβ and the scalar curvature Sc(D) have in

the frame Xα the components

Rij = Ri

j , R(i)j =

1P i

j , Rij =

2P i

j , R(i)(j) = Si

j , Sc(D) = R + S,(6.123)

where R = gijRij , S = gijSij .


Theorem 6.12.1 The Einstein equations of the Lagrange space Ln = (M, L)corresponding to the metric N -linear connection DΓ(N) = (Li

jk, Cijk) with the

coefficients given by (6.64) have the following form:

Rij − 12(R + S)gij = kTij ,

Sij − 12(R + S)gij = kT(i)(j),

1P i

j = kT(i)j ,2P i

j = −kTi(j),

(6.124)

where Tij, T(i)j, Ti(j), T(i)(j) are d-tensor fields, the components of the energy-momentum tensor T in the basis Xα.

Proof. Making use of the above mentioned formulae, one shows that theequations (6.124) are equivalent with (6.107). q.e.d.

In vacuum, which corresponds to the case Tαβ = 0, if we multiply withGαβ the equations (6.107) written in the form,

Rαβ − 12Sc(D)Gαβ = 0(6.125)

we get Sc(D) − nSc(D) = 0, hence Sc(D) = 0 for n > 1. Thus (6.107) takesthe form Rαβ = 0 and we have the following result.

Proposition 6.12.1 For the vacuum state, the Einstein equations of the La-grange space Ln = (M, L) corresponding to the metric DΓ(N) whose coeffi-cients are given by (6.64) are as follows:

Rij = 0, Sij = 0,1P i

j = 0,2P i

j = 0.(6.126)

Theorem 6.12.2 The law of conservation, with respect to the metric connec-tion (6.64), for the Lagrange space Ln = (M, L) is given by

[Ri

j − 12(R + S)δi

j

]|i

+1P i

j |i = 0,

[Si

j − 12(R + S)δi

j

]|i+

2P i

j|i = 0.

(6.127)

Proof. If we look at (6.125), the conservation law reads as follows:

Dα

(Rα

β −12Sc(D)δα

β

)= 0.


This is equivalent with (6.127) when it is decomposed in the frame (Xα) =(δ/δxi, ∂/∂yj). We notice that since D is metric, DαGβγ = 0, it results

Rαβγδ + Rβαγδ = 0, Rααγδ = 0(6.128)

as for any pseudo-Riemannian space. But the equation

Dα

(Rα

β −12Sc(D)δα

β

)= 0g

does not reduce to an identity since D has torsion. Thus the conservation lawis not identically verified. There are several explanations of this fact from aphysical point of view, for one of this we refer to [92]. q.e.d.

We will give now a new form of the Einstein equations (6.124), following[145].

Theorem 6.12.3 Einstein equations (6.124) are equivalent with the followingequations

Rij − 12Rgij = k

hT ij , Sij − 1

2Sgij = k

vT ij ,

1P ij= kT(i)j ,

2P ij= −kTi(j).

(6.129)

Proof. Multiplying by gij the first two equations (6.124) and putting1T= gijTij ,

4T= gijT(i)(j) one obtains a system of two linear equations in R and S, whichcan be solved to give

R =k

2(n− 1)

[(n− 1)

1T −n

4T

],

S =k

2(n− 1)

[−n

1T +(n− 2)

4T

].

Now introducing

hT ij= Tij +

14(n− 1)

[−n

1T +(n− 2)

4T

]gij ,

vT ij= T(i)(j) +

14(n− 1)

[(n− 2)

1T −n

4T

]gij ,

one obtains the system (6.129).Conversely, multiplying the first two equations of (6.129) by gij , one ob-

tains

R =2k

2− n

hT , S =

2k

2− n

vT ,


wherehT= gij

hT ij ,

vT= gij

vT ij .

Using these again in (6.129), one easily deduces (6.124). q.e.d.

It has been shown in [145] that the conservation laws written for the equa-tions (6.129) are more easy to be used and handled.

Chapter 7

Generalized Lagrange spaces

In the first part of this book we studied the geometry of the tangent bundleof a smooth, real, n-dimensional manifold M , in the presence of a semisprayS, a nonlinear connection N , and an N -linear connection D. In the previoustwo chapters we have seen that for a Finsler or a Lagrange space there is acanonical metric structure on TM that uniquely determines such semisprays,nonlinear connections and N -linear connections. Consequently we can studythe geometry of a Finsler or Lagrange space by using the geometry of thesegeometric objects.

In this chapter we study the geometry of a d-tensor metric, which is notnecessarily associated to a Finsler or a Lagrange space. The geometry of suchmetric structures has been studied by R. Miron [119], [130] and it is knownas the geometry of generalized Lagrange spaces (or a GL-space for short).Examples of such spaces that are useful in geometric models for RelativisticOptics are given. Unlike for a Lagrange space a GL-space does not have, ingeneral, a canonical nonlinear connection. However, we study the geometryof pairs (S, N) that are compatible with the metric structure, where S is asemispray and N a nonlinear connection. A special attention is paid to theregular cases, when it is possible to associate a canonical semispray and anonlinear connection to a GL-metric. The geometry of generalized Lagrangespaces is developed with the same devices as the geometry of Lagrange spaces.That is, if we associate to a generalized Lagrange space a nonlinear connectionthen, a canonical metric connection is found. Using it, the main geometricobjects of a generalized Lagrange space are investigated. The structure ofgeneralized Lagrange space is lifted to an almost Hermitian structure on TMand its canonical metric N -linear connection appears as an almost Hermitianconnection on TM . Some classes of GL-metrics used by physicists R.G. Beiland P.R. Holland for a unified field theory are presented in the last part ofthis chapter. For this GL-metrics we determine corresponding nonholonomic

174 Chapter 7. Generalized Lagrange spaces

frames that unifies gravitational and electromagnetic fields.

7.1 Metric classes on TM

The most natural generalization of the notion of Lagrange space is that intro-duced and studied by R. Miron in [130], with the name of generalized Lagrangespace. See also [119], [5], [6] and [9] for this topic.

The notion was suggested by the fact that for many properties of the La-grange space, only the metric tensor field gij(x, y) is involved. In other words,many properties of a Lagrange space do not depend on the Lagrangian Litself but on the metric tensor gij(x, y) only. Thus, a generalized Lagrangespace is defined as a pair, which consists of a smooth n-dimensional manifoldand a d-tensor field gij(x, y) of rank n, symmetric and of constant signa-ture. Then Finsler and Lagrange spaces are examples of generalized Lagrangespaces. But there exist generalized Lagrange spaces that are not reducible toLagrange spaces, which means that their metric tensors are not provided byregular Lagrangians. Certain generalized Lagrange spaces are encountered inGeneral Relativity, Relativistic Optics, Biology. These will be treated withmore details.

Let gij(x, y) be a set of n2 smooth, real functions, defined on each coor-dinate neighborhood on TM such that when two such neighborhoods overlapwe have

gij(x, y) =∂xk

∂xi(x)

∂xh

∂xj(x)gkh(x, y).(7.1)

In the terminology used above (gij(x, y)) is a (0,2)-type d-tensor field. Fromthe coordinate transformation formula (7.1) we observe that the symmetrygij(x, y) = gji(x, y) condition and the condition det(gij(x, y)) 6= 0 are pre-served by a change of coordinates on TM .

Thus, we can introduce the following definition.

Definition 7.1.1 A (0, 2)-type d-tensor field gij(x, y) on TM is said to be ageneralized Lagrange metric, shortly a GL-metric, if

1) gij(x, y) = gji(x, y), i.e. it is symmetric,2) det(gij(x, y)) 6= 0, i.e. it is regular,3) The quadratic form gij(x, y)ξiξj , (ξi) ∈ Rn has constant signature. In

particular, if this quadratic form is positive definite, gij(x, y) is called a positivedefinite GL-metric.

For a GL-metric gij(x, y) on TM , one can consider g = gij(x, y)dxi ⊗ dxj ,which is a degenerate metric structure on TM , rank(g) = n < dim(TM) = 2n.However, gij is a metric structure on the pull-back bundle π∗(TM) or on thevertical subbundle V TM . For the geometry of metric structures on the verticalsubbundle V TM , we refer to [39].

7.1. Metric classes on TM 175

Definition 7.1.2 The pair GLn = (M, gij(x, y)) is called a generalized La-grange space, shortly a GL-space.

A first example of generalized Lagrange space is provided bellow. Let aij(x)be the local components of a Riemannian metric on M , which always exists ifM is paracompact. Then

gij(x, y) = eφ(x,y)aij(x),(7.2)

is a positive definite GL-metric, where φ is a smooth function on TM . As anexample, the function φ can be the absolute energy of the Riemannian metricaij(x), which is given by φ(x, y) = aij(x)yiyj . If aij is a Lorentz metric, theGL-metric (7.2) is called the Miron-Tavakol metric and it satisfies the Ehlers-Pirani-Schield axioms of relativity [143].

Definition 7.1.3 A GL-metric gij(x, y) is said to be reducible to a Lagrangianmetric, shortly an L-metric if there exists a smooth function L : TM → R suchthat

gij(x, y) =12

∂2L

∂yi∂yj(x, y).(7.3)

If (7.3) holds, then L is a regular Lagrangian and the pair Ln = (M,L) is aLagrange space. Notice that L in (7.3) is not uniquely determined. It may bereplaced with and only with L′(x, y) = L(x, y) + Ai(x)yi + U(x), where Ai isa d-covector field and U is a smooth function on M .

The first example of GL-space, given by expression (7.2), is not reducibleto a Lagrange space provided that ∂φ/∂yk 6= 0. With φ(x, y) = aij(x)yiyj ,this assures the existence of essential GL-metrics on paracompact manifolds.

It is possible to identify the class L-metrics among GL-metrics as follows.One associates with any GL-metric the Cartan tensor, which is a (0,3)-typed-tensor field defined as follows

Cijk =12

(∂gik

∂yj+

∂gij

∂yk− ∂gjk

∂yi

).(7.4)

This is symmetric in the subscripts j, k. Denote by (gjk) the inverse of thematrix (gij). Then functions gjk(x, y) are components of a d-tensor field oftype (2, 0). We shall use (gij) and (gij) for raising and lowering the indices ofa d-tensor field. We put

Cijk := gihChjk, A

ijk := gihCjkh = gihCjhk.(7.5)

Notice that Cijk is symmetric in the subscript j, k while Ai

jk it is not.For an L-metric, Cijk reduces to

Cijk =14

∂3L

∂yi∂yj∂yk


and it is totally symmetric, we refer to it also as the Cartan tensor. For thisparticular case we have

Cijk = Ai

jk =12gih ∂gjh

∂yk=

12gih ∂gkh

∂yj.

Proposition 7.1.1 A GL-metric gij(x, y) is reducible to an L-metric if andonly if Cijk is totally symmetric or, equivalently Ai

jk = Aikj, providing that

Cijk 6= 0.

Proof. The condition that Cijk is totally symmetric is equivalent with thefollowing condition

∂gij

∂yk=

∂gik

∂yj.(7.6)

Then, condition (7.6) is the integrability condition of the PDE (7.3) for theunknown function L. Thus, locally there exists a solution LU of (7.3) definedon π−1(U), where (U,ϕ) is a local chart on M .

Let (Uα)α∈I be an open covering of M with coordinate neighborhoods.Choose a smooth partition of unity (aα)α∈I subordinated to (Uα). Let LUα

be the solution of (7.3) on Uα. Then L =∑

α∈I

aαLUα is a regular Lagrangian

whose metric tensor is gij(x, y). q.e.d.

One can use this criteria to check when the GL-metric (7.2) is reducibleto an L-metric.

Proposition 7.1.2 For n > 1, the GL-metric (7.2) is an L-metric if and onlyif ∂φ/∂yk = 0, k = 1, 2, ..., n.

Proof. The symmetry condition (7.6) applied to the GL-metric (7.2) reads asfollows:

gij∂φ

∂yk= gik

∂φ

∂yj⇐⇒ δi

j

∂φ

∂yk= δi

k

∂φ

∂yj.

Last equation is true if and only if for all k ∈ 1, 2, ..., n, ∂φ/∂yk = 0, whichmeans that φ is a basic function. q.e.d.

Definition 7.1.4 A GL-metric gij(x, y) on TM is reducible to a Finslerianmetric, shortly an F-metric, if there exists a function F : TM → R+, ofC∞-class on TM and only continuous on the null section, which is positivelyhomogeneous of order 1 in yi and satisfies

gij(x, y) =12

∂2F 2

∂yi∂yj(x, y).(7.7)

7.1. Metric classes on TM 177

When (7.7) holds true, the function F is a fundamental Finsler functionand Fn = (M,F ) is a Finsler space. Notice that if F does exist, it is uniquelydetermined by (7.7) because of its p-homogeneity. Here p-homogeneity standsfor positively homogeneity with respect to y = (yi). By (7.7), a necessarycondition for a GL-metric on TM to be an F-metric is to be p-homogeneousof order zero that is gij(x, λy) = gij(x, y), ∀λ ∈ R+. However, the zero ho-mogeneity is not sufficient for a GL-metric to be an L-metric. For example ifφ(x, y) depends on y and is p-homogeneous of order zero, then the GL-metric(7.2) p-homogeneous of order zero and it is not reducible to a Finsler metric.

Definition 7.1.5 A Lagrange metric gij(x, y) on TM is called an AlmostFinsler Lagrange metric, shortly an AFL-metric, if one of its Lagrangians hasthe form

L(x, y) = F 2(x, y) + Ai(x)yi + U(x),(7.8)

where F is a fundamental Finsler function, Ai is a d-covector field, and U asmooth function on M . The pair (M, L) with L given by (7.8) is called anAFL-space.

Proposition 7.1.3 A GL-metric on TM that is p-homogeneous of order zerois an L-metric if and only if it is an AFL-metric.

Proof. An AFL-metric is clearly a zero homogeneous GL-metric. Conversely,suppose that we have (7.3) with gij(x, y) p-homogeneous of order zero. Letus consider F 2(x, y) = gij(x, y)yiyj . Then F 2(x, y) is homogeneous of secondorder in y and using Euler theorem on homogeneous functions we have

∂2F 2

∂yk∂yh= 2gkh, hence

∂2(L− F 2)∂yk∂yh

= 0,

and L has the form (7.8). q.e.d.

Proposition 7.1.4 A GL-metric gij on TM is an AFL-metric if and only if

∂gij

∂yk(x, y)yj = Cijky

j = 0(7.9)

Proof. If gij(x, y) is an AFL-metric, then by (7.4) and the Euler theoremon homogeneous functions, we obtain that expression (7.9) holds good. Con-versely, let gij(x, y) be a GL-metric that satisfies expression (7.9). We con-sider the energy function E(x, y) = gij(x, y)yiyj . Using (7.9) twice we find∂2E(x, y)/∂yk∂yh = 2gkh. Therefore, gij(x, y) is an L-metric. It satisfies (7.6)that allows us to write (7.9) into the form yj(∂gik/∂yj) = 0, which says thatgij(x, y) is p-homogeneous of order zero. Then, Proposition 7.1.3 completesthe proof. q.e.d.

Riemannian metrics enter also in the above scheme.


Definition 7.1.6 A GL-metric gij(x, y) on TM is said to be reducible to aRiemannian metric if it does not depend on y (an R-metric for short).

Proposition 7.1.5 A GL-metric gij(x, y) is an R-metric if and only if theCartan tensor Cijk vanishes identically.

One can summarize now the reducibility of a GL-metric to Lagrange,Finsler and Riemann metrics using the Cartan tensor Cijk as follows:

1) A GL-metric gij(x, y) is reducible to an L-metric if and only if the Cartantensor Cijk given by (7.4) is totally symmetric.

2) A GL-metric gij(x, y) is reducible to an F-metric if and only if the Cartantensor Cijk given by (7.4) satisfies Cijky

j = 0.

3) A GL-metric gij(x, y) is reducible to an R-metric if and only if the Cartantensor Cijk given by (7.4) vanishes identically.

The dependence of a GL-metric gij(x, y) on y only, can also be taken intoconsideration but with some precautions.

Definition 7.1.7 A GL-metric gij(x, y) is called a locally Minkowski metricif there exists a system of coordinates on TM in which the components gij ofthe metric depend on y only.

An example of a Minkowski metric is provided now. Let F : TM → R+ begiven by

F (y) = ((y1)m + ... + (yn)m)1/m,m ≥ 3,m ∈ N.(7.10)

Then

γij(y) =12

∂2F 2

∂yi∂yj

is a locally Minkowski F-metric and gij(x, y) = eφ(y)γij(y), with φ a smoothfunction on TM , is a locally Minkowski GL-metric.

Another interesting example of a GL-metric is given by

gij(x, y) = eφ(x,y)γij(y),(7.11)

with φ a smooth function on TM .This is an L-metric if and only if φ does not depend on y. If this happens

it becomes an F-metric that is conformal to a locally Minkowski F-metric.For φ = aix

i (ai constants, usually positive), (7.11) reduces to the Antonellimetric, which is very useful in Biology as was extensively discussed in the book[23], chapter 5.

7.2. Metric nonlinear connections and semisprays 179

The GL-metric

gij(x, y) = aij(x) +(

1− 1n2(x, y)

)yiyj ,(7.12)

where aij(x) is a Lorentz metric, yi = aik(x)yk and n(x, y) ≥ 1 is a smoothfunction on TM — called the refractive index — was used by R. Miron and T.Kawaguchi in Relativistic Optics [141]. The symmetry of gij(x, y) is obvious.We have det(gij(x, y)) 6= 0 since the metric gij(x, y) is invertible. As it is easyto check, its inverse is given by

gjk(x, y) = ajk(x)− 1a(x, y)

(1− 1

n2(x, y)

)yjyk,

where

a(x, y) = 1 +(

1− 1n2(x, y)

)‖y‖2, ‖y‖2 = aij(x)yiyj .

One can see that GL-metric (7.12) is obtained by a deformation of the Rieman-nian metric aij through a process that will be generalized in Section 7.5. Weremark here also that this GL-metric is not reducible neither to an L-metricnor to an F-metric.

7.2 Metric nonlinear connections and semisprays

Nonlinear connections and metric structures are important tools in the geom-etry of tangent bundles. There are situations, as in the geometry of a gen-eralized Lagrange spaces, [119], where both structures are considered, but nocondition of compatibility is required for these. Using the covariant derivativeone can associate to a semispray and a nonlinear connection, we introduce acompatibility condition between these geometric structures and a metric struc-ture. This compatibility is a natural generalization of the well known metriccompatibility of a linear connection in a Riemannian space, [59]. As the metriccompatibility is not enough to determine the Levi-Civita connection of a Rie-mannian space, similarly the metric compatibility does not determine uniquelya nonlinear connection. A whole family of metric nonlinear connections is de-termined when a metric tensor and a semispray are fixed. The problem ofcompatibility between a system of second order differential equations and ametric structure has been studied by numerous authors, [107], [67], [106], [180],[168]. In this case, the nonlinear connection is derived from a system of secondorder differential equations, and the metric condition we talk about is the firstHelmholtz condition from the inverse problem of Mechanics, [67]. We havealready seen in the previous chapter that for a Lagrange space the canonicalnonlinear connection is metric.


Let N be a nonlinear connection with δ/δxi = ∂/∂xi −N ji ∂/∂yj , ∂/∂yi

the adapted basis to the decomposition (2.14). Consider S a semispray. Thedynamical covariant derivative that corresponds to the pair (S, N) is definedby ∇ : χv(TM) −→ χv(TM) through the following formula:

∇(

Xi ∂

∂yi

)=

(S(Xi) + XjN i

j

) ∂

∂yi.(7.13)

One can extend ∇ to act on horizontal vector fields and then on arbitraryvector fields on TM by

∇(hX) = Θ∇(JX), ∇X = ∇(hX) +∇(vX), ∀X ∈ χ(TM).(7.14)

Here Θ is the adjoint structure, while h and v are the vertical and horizontalprojectors. Dynamical covariant derivative ∇ has the following properties:

1) ∇(X + Y ) = ∇X +∇Y,∀X, Y ∈ χv(TM),

2) ∇(fX) = S(f)X + f∇X, ∀X ∈ χv(TM),∀f ∈ F(TM).

According to the defining formula (7.13) and (7.14) we have that the dynam-ical covariant derivative ∇ preserves the horizontal and vertical distribution.Consequently, the almost product structure P, the adjoint structure Θ andthe tangent structure J are parallel with respect to ∇ that is the followingformulae are true:

∇P = ∇Θ = ∇J = 0.

One can consider that ∇ acts on functions according to the formula ∇f =S(f). Then, one can consider also the action of covariant derivative ∇ on1-forms given by

(∇ω)(X) = S(ω(X))− ω(∇X), ∀X ∈ χ(TM) and ω ∈ Λ1(TM).(7.15)

It is easy then to extend the action of∇ to the algebra of tensor fields by requir-ing for ∇ to preserve the tensor product. Since ∇ preserves both horizontaland vertical distribution, then ∇ will preserve also the algebra of d-tensorfields. We shall use this action, especially for d-tensor fields. For a GL-metric,which is a (0,2)-type d-tensor field, its dynamical covariant derivative is givenby

(∇g)(X, Y ) = S(g(X,Y ))− g(∇X,Y )− g(X,∇Y ), ∀X, Y.(7.16)

Definition 7.2.1 Let S be a semispray, N a nonlinear connection and ∇the associated covariant derivative. The nonlinear connection N is metric orcompatible with respect to the metric tensor g if ∇g = 0, which is equivalentto

S(g(X, Y )) = g(∇X,Y ) + g(X,∇Y ), ∀X, Y.

7.2. Metric nonlinear connections and semisprays 181

For a semispray S with local coefficients Gi one can associate a nonlinearconnection with local coefficients N i

j = ∂Gi/∂yj . If the GL-metric is reducibleto an L-metric then the pair (S, N) is metric, but in general this property isnot true. However, for a fixed semispray, we shall determine first a nonlinearconnection that is metric with respect to g and then we shall determine thewhole family of nonlinear connections with this property.

Let us consider the following Obata operators one can associate to a GL-metric gij :

Oijkl =

12(δi


kl =12(δi


Theorem 7.2.1 Let S be a semispray with local coefficients Gi and N theassociated nonlinear connection with local coefficients N i

j = ∂Gi/∂yj. Thereis a metric nonlinear connection N c, whose coefficients N ci

j are given by

N cij =

12gikS(gkj) + Oik

sjNsk .(7.18)

Proof. One can write coefficients N cij from (7.18) into the following equivalent

formN ci

j =12gikgkj| + N i

j .(7.19)

Here the covariant derivative gkj| is taken with respect to the pair (S, N). AsN i

j are local coefficients of a nonlinear connection and gikgkj| are componentsof a d-tensor field of (1,1)-type we have that N ci

j are also the local coefficientsof a nonlinear connection. Consider now the covariant derivative ∇ one canassociate to the pair (S, N c). It is a straightforward calculation to check that

S(gij)− gimN cmj − gmjN

cmi = 0,

which means that the nonlinear connection N c is metric. q.e.d.

Proposition 7.2.11) Let S be a semispray with local coefficients Gi, N the associated non-

linear connection with local coefficients N ij = ∂Gi/∂yj, and the nonlinear con-

nection determined above. The nonlinear connection N is metric if and onlyif N = N c.

2) If the GL-metric is reducible to an L-metric then N = N c and thecanonical nonlinear connection N is metric.

Now, we can determine the whole family of metric nonlinear connections thatcan be derived from a semispray S.


Theorem 7.2.2 Consider S a semispray with local coefficients Gi and N c themetric nonlinear connection with local coefficients given by expression (7.18).The family of all nonlinear connections that are metric with respect to themetric tensor gij of a GL-space is given by

N ij = N ci

j + OkijmXm

k ,(7.20)

where Xmk is an arbitrary (1, 1)-type d-tensor field.

Proof. The condition that both nonlinear connections N cij and N i

j are metricwith respect to the metric tensor g, can be written as S(gij) = gmjN

cmi +

gimN cmj and S(gij) = gmjN

mi +gimNm

j . If we subtract these two equations weobtain O∗is

jm(Nmi − N cm

i ) = 0. Using the fact that OijklO

∗kmpj = 0, the solution

of this tensorial equation is given by expression (7.20). q.e.d.

7.3 Metric N-linear connections

Let N be a nonlinear connection on TM with local coefficients N ij . We know

that such a connection always exists if M is a paracompact manifold. Aswe have seen in Section 6.6, an L-metric determines a canonical nonlinearconnection. This is no longer true for an arbitrary GL-metric. However,a compatibility condition of a nonlinear connection with the GL-metric hasbeen defined and studied in the previous section. Though there are someclasses of GL-metrics determining nonlinear connections, to be discussed later,we develop first a general theory taking an arbitrary nonlinear connection Nhaving no weak torsion, that is

tijk :=∂N i

j

∂yk− ∂N i

k

∂yj= 0.(7.21)

These condition are satisfied if the nonlinear connection is induced by a semis-pray S. In such a case the nonlinear connection is symmetric.

We denote as usual by DΓ(N) = (Lijk, C

ijk) an N -linear connection on TM

and by “|” and “|” the h- and v-covariant derivatives induced by it.

Definition 7.3.1 An N -linear connection DΓ(N) = (Lijk, C

ijk) on TM is

compatible with the GL-metric gij(x, y) or it is called metric if

gij|k = 0, gij |k = 0.(7.22)

A geometric meaning of the conditions (7.22) is given bellow.Let Xi be a d-vector field on TM . We can think to this d-vector field

either as to a horizontal vector field X = Xi(δ/δxi) or as to a vertical vectorfield Xi = Xi(∂/∂yi).


Consider c a parameterized curve on TM , c : t ∈ I → c(t) = (xi(t), yi(t))such that its image is contained into a coordinate neighborhood, that is Imc ⊂π−1(U) ⊂ TM .

We can define the covariant differential along the curve c, with respect toan N -linear connection, of a vector field. More generally, one can define thecovariant differential of d-tensor fields. For a d-vector field, this is given by

Dc(Xi) =DXi

dt= Xi

|kdxk

dt+ Xi|k δyk

dt.

For a metric tensor gij we have

Dc(gij) =Dgij

dt= gij|k

dxk

dt+ gij |k δyk

dt.(7.23)

Let ‖X‖ be the norm of the d-vector field Xi given by

‖X‖2 = gij(x, y)XiXj .(7.24)

Theorem 7.3.1 Let c be an arbitrary curve on TM of local equations xi =xi(t), yi = yi(t), t ∈ I ⊆ R. Then a d-vector field Xi with DXi/dt = 0 hasconstant length ‖X‖2 along the curve c if and only if (7.22) holds good that isthe N -linear connection is metric.

Proof. On the curve c we have

d‖X‖2

dt=

Dgij

dtXiXj + 2gij

DXi

dtXj

and within our hypothesis, d(‖X‖2)/dt = XiXjDgij/dt. Therefore, ‖X‖2 isconstant along c if and only if Dgij/dt = 0. By (7.24) and having in mindthat c is an arbitrary curve on TM , it follows that ‖X‖2 is constant along cif and only if (7.22) holds good. q.e.d.

There exists a general procedure to solve the tensorial equations of the type(7.22) that may be called the generalized Christoffel process. For instance, oneexpands gij|k = 0 in the form

δgij

δxk= Lh

ikghj + Lhjkgih,

one cyclically permutes the subscript i, j, k and one adds the first two andsubtract the last one. If the difference Lh

ij − Lhji is a priori given as T h

ij , thenLh

ij is completely determined.


Theorem 7.3.2 Let GLn = (M, gij(x, y)) be a generalized Lagrange space.i) There exists an unique N -linear connection CΓ(N) that is metric with

respect to (gij(x, y)) and has the torsions T ijk = 0 and Si

jk = 0.ii) The coefficients (Li

jk, Cijk) of the N -linear connection CΓ(N) are given

by the generalized Christoffel symbols

Lijk =

12gih

(δgjh

δxk+

δghk

δxj− δgjk

δxh

),

Cijk =

12gih

(∂gjh

∂yk+

∂ghk

∂yj− ∂gjk

∂yh

).

(7.25)

Proof. If we apply the Christoffel process to the tensorial equation gij|k =0, taking Li

jk − Likj := T i

jk = 0 we obtain Lijk from (7.25). Similarly, if

we use the Christoffel process for the tensorial equations gij |k = 0, takingCi

jk − Cikj := Si

jk = 0 we obtain Cijk from (7.25). The uniqueness of CΓ in i)

follows immediately. q.e.d.

If we take the torsion T ijk and Si

jk as a priory given tensors skew symmetrictensors in j and k, the tensorial equations (7.22) can also be solved and thesolution is that from (6.64) with appropriate notations.

The metric N -linear connection depends only on metric structure gij andon the nonlinear connection N i

j . We call it the canonical metric N -linearconnection of the pair (gij , N

ij). For this nonlinear connection we consider the

torsion and its curvature. With respect to the adapted basis there are threecomponents of torsion and three components of curvature as we did in Section2.6. These components of torsion and curvature of CΓ(N) appear in the Ricciidentities or the commutation formulae

Xi|k|h −Xi

|h|k = XrRirkh −Xi|rRr

kh,

Xi|k|h −Xi|h|k = XrP i

rkh −Xi|rC

rkh −Xi|rP r

kh,

Xi|r|h −Xi|h|k = XrSirkh,

(7.26)

where Xi(x, y) are the components of a d-vector field.

Theorem 7.3.3 The metric N -linear connection CΓ(N) has the followingproperties:

1) Rijkh + Rijhk = 0, Sijkh + Sijhk = 0,

2) Rijkh + Rjikh = 0, Pijkh + Pjikh = 0, Sijkh + Sjikh = 0,

3) Riikh = P i

ikh = Siikh = 0.

Proof. The first item 1) follows directly from the expressions of Rijkh and Si

jkh.The next two items 2)-3) are consequences of the commutation formulae of h-and v-covariant derivatives written for gij(x, y) and making use of (6.10). Thecalculations are similar with those from the Riemannian geometry. q.e.d.

7.4. Regular metrics 185

7.4 Regular metrics

Since a nonlinear connection plays an important role in the metric geometryof the tangent bundle, it is important to determine those GL-metrics that de-termine a canonical nonlinear connection. If this is the case, for a GL-metricgij(x, y), then the canonical N -linear connection of the corresponding GL-space depends only on gij(x, y) and consequently its geometry does dependonly on gij(x, y). We refer to such a metric as a regular metric. The regularityconditions for a GL-metric has been introduced and studied by R. Miron in[119]. These regular metrics were studied also by J. Szilasi, where a regularmetric is called Miron regular [179]. Here we discuss GL-metrics that deter-mine nonlinear connection following some ideas from paper [119] but usingdifferent notations and giving different proofs.

So far the most important GL-metrics determining nonlinear connectionsare the L-metrics, which are called also variational in [119]. These variationalmetrics were characterized in Theorem 7.1.1. Cartan d-tensor fields Cijk andAi

jk will continue to play an important role here.For any GL-metrics gij(x, y) on TM one may consider the 1-form

θg = yidxi, yi = gik(x, y)yk

as well as the 2-form ωg := dθg. We associate to any GL-metric gij(x, y) asmooth function on TM called the absolute energy of it, given by

E(x, y) = gij(x, y)yiyj = yiyi.(7.27)

Using the energy E , we can consider the d-tensor field g∗ with the components

g∗ij(x, y) =12

∂2E(x, y)∂yi∂yj

.

Lemma 7.4.1 The following formulae are true:

12

∂E∂yi

= Cikhykyh + gikyk = Ci00 + yi,

g∗ij = gij + 2Cjikyk + 2Cijky

k +∂Cikh

∂yjykyh.

Definition 7.4.1 A GL-space, GLn = (M, gij(x, y)), is said to be with weaklyregular metric or the GL-metric gij(x, y) is called weakly regular if the absoluteenergy E is a regular Lagrangian or (M, E) is a Lagrange space.

If the GL-metric gij(x, y) is weakly regular, then the Lagrange space (M, E)determines a nonlinear connection. One can check that the functions N i

j(x, y)defined bellow

N ij(x, y) =

∂Gi

∂yj, Gi(x, y) =

14g∗ij

(∂2E

∂yj∂xkyk − ∂E

∂xj

)(7.28)


are the local coefficients of a nonlinear connection on TM that depends ongij(x, y) only.

The functions Gi(x, y) from expression (7.28), are the local coefficients ofa semispray on TM . These functions can be written as follows

Gi(x, y) =14g∗ijgjrγ

r00 + yk ∂Cj00

∂xk.(7.29)

Here g∗ij denotes the inverse of the matrix g∗ij . This inverse exists since Eis, by hypothesis, a regular Lagrangian. We did use also, γi

jk the Christoffelsymbols of second kind of the metric gij . We may associate to E the Cartan1-form

θE =12

∂E∂yi

dxi = J∗(

12dE

)

and the Cartan 2-form ωE = dθE . We have then that E is a regular Lagrangianif and only if ωE is non-degenerate. For this reason the weakly regular GL-metrics are called also E-regular in [119].

Let us consider the d-tensor field K of (1,1)-type with the componentsgiven by

Kij(x, y) = gihg∗hj(x, y).(7.30)

Then we have that a GL-metric is weakly regular if and only if K is non-degenerate that is det(Ki

j(x, y)) 6= 0.Let us go back now to the Cartan 1-form θg and 2-form ωg one can associate

to a GL-metric gij(x, y). We have

Proposition 7.4.1 If g is an L-metric, which means that

gij(x, y) =12

∂2L

∂yi∂yj,

then, θg = θELand therefore ωg = ωEL

, where EL = yi(∂L/∂yi) − L is theenergy of the Lagrangian L.

Proof. Indeed, we have

12

∂EL

∂yi=

12

∂

∂yi

(yj ∂L

∂yj

)− 1

2∂L

∂yi= yjgij = yi

since the other two terms cancel. If we calculate ωg, we find

ωg = d(gikykdxi) =

12

(∂gik

∂xj− ∂gjk

∂xi

)dxj ∧ dxi +

(gij +

∂gik

∂yjyk

)dyj ∧ dxi.

7.4. Regular metrics 187

The factor of the second term can be put in the form gih

(δhj + ghs ∂gsk

∂yjyk

)

and so the following d-tensor field comes into the play

Bhj = δh

j + 2Ahjky

k.(7.31)

Then, Bhj are the components of a (1,1)-type d-tensor field. q.e.d.

Proposition 7.4.2 The 2-form ωg of the GL-metric g is non-degenerate ifand only if the d-tensor field (Bh

j ) given by expression (7.31) is non-degenerate,which means that det(Bh

j ) 6= 0.

Proof. The absence of the terms dyj ∧ dyk in the local expression of ωg showsthat the 2-form ωg is non-degenerate if and only if det(gihBh

j ) 6= 0 or, equiva-lently, det(Bh

j ) 6= 0 since det(gih) 6= 0. q.e.d.

Definition 7.4.2 A GL-metric (gij(x, y)) is called regular in Miron’s sense(briefly Miron regular) if det(Bh

j ) 6= 0.

Corollary 7.4.1 For an L-metric, the condition of Miron regularity is equiv-alent with the condition that the energy EL of L is a regular Lagrangian.

Proof. According to proposition 7.4.2 the condition of Miron regularity isequivalent with the condition that ωg is non-degenerate. Using Proposition7.4.1 we have that the Cartan 2-forms ωg and ωEL

coincide and hence EL is aregular Lagrangian. q.e.d.

Thus we have a case in which ωg = ωELwith EL a regular Lagrangian.

We may wonder if there exist situations in which θg = θLg for Lg a smoothfunction on TM . Let this be the case. In such a situation, we have

∂2Lg

∂yi∂yj= 2gisB

sj

and so the Miron regularity condition is equivalent with the condition that Lg

is a regular Lagrangian. We have

Proposition 7.4.3 For a GL-metric gij(x, y) there exists a Lagrangian Lg

such that θg = θLg if and only if Cij0 = Cji0.

Proof. If there exists Lg such that θg = θLg , we have 2gikyk = ∂Lg/∂yi. This

implies

4Cji0 + 2gij =∂2Lg

∂yi∂yj.


A change of j and i gives Cji0 = Cij0. Conversely, the system of PDE,∂Lg/∂yi = 2giky

k in the unknown Lg can be solved if the integrability condi-tions ∂yi/∂yj = ∂yj/∂yi, yi = giky

k, are satisfied. An easily calculation showsthat these conditions are equivalent with Cij0 = Cji0. Of course, Lg is first lo-cally found. Then using a smooth partition of unity on M a global LagrangianLg can be constructed. q.e.d.

Proposition 7.4.3 suggests that a new class of GL-metric can be introduced.

Definition 7.4.3 A GL-metric g = (gij(x, y)) is called weakly variational ifCij0 = Cji0.

An explanation for the term weakly variational is as follows. For an L-metricg, called also variational metric, we have ωg = ωEL

, where EL is a Lagrangian.The condition θg = θLg implies ωg = ωLg . From Proposition 7.4.1 we see thatany L-metric or any variational metric is weakly variational with θg = θEL

.We recall that θg = θLg implies

12

∂2Lg

∂yi∂yj= gisB

sj .(7.32)

This fact suggests to define a d-tensor field γg of type (0, 2) with localcomponents

γij(x, y) = gis(x, y)Bsj (x, y).(7.33)

A first question one can ask now is the following. When does γg become ametric? Here is the answer.

Proposition 7.4.4 For a GL-metric g the following conditions are equivalent:1) The d-tensor field γg with components (7.33) is a metric.2) The metric tensor g is Miron regular and weakly variational.3) γg is the Hessian of a regular Lagrangian Lg that is

γij =12

∂2Lg

∂yi∂yj.

Proof. If first statement is true, then the symmetry gisBsj = gjsB

si implies

Cij0 = Cji0 that is g is weakly variational. As det(γij) 6= 0 implies det(Bsj ) 6= 0,

g is also Miron regular. Thus condition 1) implies condition 2). If secondcondition holds true, then the condition of weakly regularity implies (7.32).Hence γg is the Hessian of the Lagrangian Lg. The condition Miron regularimplies, in view of (7.33), that det(γij) 6= 0, hence Lg is a regular Lagrangian.

Concluding, condition 2) implies condition 3). The implication from con-dition 3) to condition 2) is clear and therefore the proposition is completelyproved. q.e.d.

7.5. Variational problem for regular GL-metrics 189

A subclass of weakly variational GL-metrics is provided by those GL-metrics for which θg = θE . These metrics are called weakly normal, the termnormal being reserved for those GL-metrics for which Ai

j0 = 0. We have

Proposition 7.4.5 A GL-metric is weakly normal if and only if Ci00 = 0,which is equivalent to Ai

00 = 0.

Proof. According to Lemma 7.4.1, the equality 2gij(x, y)yj = ∂E/∂yi holds ifand only if Ci00 = 0 or equivalently Ai

00 = 0. q.e.d.

From Proposition 7.4.5 we have that any normal GL-metric is also weaklynormal. We notice that a normal GL-metric is also Miron regular since wehave the Bi

j = δij .

Proposition 7.4.6 If the GL-metric gij(x, y) is weakly normal then γg = g∗

and the absolute energy E is positively homogeneous of second order.

Proof. In our hypothesis the second formula in Lemma 7.4.1 reduces to g∗ij =gij + 2Cji0 and from (7.33) it follows g∗ij = γij . The equality 2gij(x, y)yj =∂E/∂yi yields 2E = yi∂E/∂yi and by Euler theorem of homogeneous functions,the absolute energy E is positively homogeneous of order two. q.e.d.

7.5 Variational problem for regular GL-metrics

In [131], R. Miron has considered under the name of regular GL-metrics thosemetrics that are weakly regular and weakly normal. We have

Proposition 7.5.1 If a GL-metric gij(x, y) is regular then the d-tensor fieldgij(x, y) is positively homogeneous of order zero in yi.

Proof. For a regular GL-metric we have g∗ij = gij + yk(∂gik/∂yj). It resultsg∗ijy

iyj = gijyiyj = E(x, y). From the second equality we deduce yi(∂E/∂yi) =

2E(x, y). Thus, the energy E is positively homogeneous of second order. Hence,gij(x, y) is positively homogeneous of order zero. q.e.d.

A geometric meaning of the regularity conditions was established by M.Hashiguchi, [84], as follows.

Let us assume that the GL-metric gij(x, y) is regular and zero homoge-neous. Then F (x, y) =

√E(x, y) is the fundamental function of a Finsler

space and the arc length s of a curve c : xi = xi(t), t ∈ [0, 1] can be consid-ered. It does not depend on the parameterization of c because F is positivelyhomogeneous of order 1 in y.

We take a variation cε : xi = xi(t) + εV i(t) of c with fixed end pointscε(0) = c(0) and cε(1) = c(1), ∀ε > 0. This implies that V i(0) = V i(1) = 0.Then we consider the integral of action I(cε) of function F along curves cε.


For this we find the following necessary condition for I(c) to be an extremalvalue for I(cε):

dI(cε)dε

∣∣∣∣ε=0

= 0.

The above condition implies the following equations:∫ 1

0

[∂F

∂xi− d

dt

(∂F

∂yi

)]V idt + V i ∂F

∂yi

∣∣∣∣1

0

= 0, yi =dxi

dt.(7.34)

We call a geodesic on M a curve c that satisfies the Euler-Lagrange equations

∂F

∂xi− d

dt

(∂F

∂yi

)= 0, yi =

dxi

dt.(7.35)

We say that the variational problem for F we presented above is regularif det

(∂2F 2/∂yi∂yj

) 6= 0. Thus the condition weakly regular for the GL-metric gij(x, y) is equivalent with the regularity of the variational problem for√E(x, y).If a curve c is geodesic, then the vector field V i that satisfies V i(∂F/∂yi) =

0 is called transversal to the curve c. The vector field V i with the propertygijy

iV j = 0 is called orthogonal to the curve c. Recall that yi = dxi/dt.The curves that satisfy equations (7.35) and have the end points in the

transversal directions, satisfy also the equations (7.34). These will be calledgeodesics.

In general, the transversality condition does not coincide with the orthog-onality. But we have

Theorem 7.5.1 If a GL-space GLn = (M, gij(x, y)) has regular zero homo-geneous metric, then

i) The variational problem for the Lagrangian√E(x, y) is regular;

ii) Transversality condition does coincide with the orthogonality condition.

Proof. First part of the theorem has been proved above. For the second partwe have

2F∂F

∂yi= 2gijy

j + yjyh ∂gjh

∂yi.

According to this formula we have F (∂F/∂yi) = gijyj . Thus FV i(∂F/∂yi) =

gijViyj . Hence the transversality coincides with the orthogonality. q.e.d.

7.6 Deformations of Finsler metrics

In this section we shall see that a large class of generalized Lagrange metrics arederived from Finsler metrics by a deformation process, which was developed

7.6. Deformations of Finsler metrics 191

by M. Anastasiei in [5], [6]. Let Fn = (M, F ) be a Finsler space with M

a smooth manifold and F : (x, y) ∈ TM 7→ F (x, y) ∈ R its fundamentalfunction. Assume that TM is endowed with a distinguished 1-form βi(x, y)dxi

and consider the function β(x, y) = βi(x, y)yi. Then one can define ∗F =L(F, β) such that in some conditions for L, ∗F is the fundamental function ofa new Finsler space. It is said that ∗Fn is obtained from Fn by a β-change[145], [98].

Classic examples for Finsler functions obtained by a β-change process aregiven by Randers and Kropina spaces, which are obtained from a Riemannianspace by particular β-changes. That is for a Randers space, L(F, β) = F + β,while for a Kropina space L(F, β) = F/β.

Let gij(x, y) be the Finsler metric tensor of Fn. If we want to construct anew Finsler metric ∗gij , which depends on gij(x, y), then because of the linearstructure of the set of Finsler tensor fields of a given type, the most generalchoice is

∗gij(x, y) = ρ(x, y)gij(x, y) + σ(x, y)Bij(x, y),(7.36)

for ρ and σ two scalar functions on TM and Bij(x, y) a symmetric d-tensorfield of type (0, 2). We may say that ∗gij is obtained from gij by a β-change.

It is clear that ∗gij is no longer a Finsler metric except if some strongconditions on ρ, σ and Bij are imposed. Metrics similar to (7.36) appear in[30] and [33] from physical considerations.

In order to relax such conditions we do not ask for ∗gij to be a Finsler metricbut a generalized Lagrange metric, which is a GL-metric. Consequently, werequire for the metric tensor ∗gij to satisfy the following conditions:

a) det(∗gij) 6= 0 andb) the quadratic from ∗gij(x, y)ξiξj , (ξi) ∈ Rn, has constant signature if

(x, y) varies on TM .Even this minimal requirements are not easy to be fulfilled except for some

particular σ, ρ and Bij .By our best knowledge the following two particular forms of the GL-metric

(7.36) were studied so far. One is conformal to a GL-metric, and it is givenby

∗gij(x, y) = e2α(x,y)gij(x, y).(7.37)

This class of GL-metrics contains the Miron-Tavakol metrics used by them inGeneral Relativity [144] and the Antonelli metrics that were introduced byP.L. Antonelli for some studies in Biology and Ecology [16]. For details see[130], ch.XI, and reference therein. Another important class of GL-metricsone can obtain from an a priory given GL-metric gij is given by

∗gij(x, y) = gij(x, y) + σ(x, y)yiyj , yi = gij(x, y)yj .(7.38)


Particular forms of GL-metrics given by above formula (7.38) were used by R.Miron and T. Kawaguchi in Relativistic Geometric Optics. See also [130], ch.XII for the geometry of such generalized Lagrange metrics.

Some particular forms of the GL-metric∗gij(x, y) = gij(x, y) + σ(x, y)Bi(x, y)Bj(x, y),(7.39)

with Bi(x, y) = gij(x, y)Bj(x, y) for Bj(x, y) a given d-vector field were intro-duced by R.G. Beil in order to develop an interesting unified field theory, see[35], [36] and the references therein. These were called Beil metrics. As suchwe refer to ∗gij in (7.39) as to the Beil metric.

This section is mainly devoted to the geometry of the GL-metrics given byexpression (7.39). First we assert that the functions ∗gij from (7.39) define forσ > 0 a positive definite GL-metric.

Indeed, it is clear that ∗gij are the components of a symmetric d-tensorfield. We look for the inverse of the matrix ∗gij in the form

∗gjk = gjk − ∗σBjBk,(7.40)

where ∗σ has to be determined. As ∗gij∗gjk = δk

i it follows that ∗σ =σ/(1 + σB2), with B2 = BiB

i = gijBiBj (the length of B with respect to

gij). Thus we have∗gjk = gjk − σ

1 + σB2BjBk.(7.41)

Consequently, we have det(gij) 6= 0.The quadratic form Φ(ξ) = ∗gijξ

iξj = gijξiξj +σ(Bkξ

k)2 is clearly positivedefinite in our hypothesis.

The GL-metric (7.39) has been used by R.G. Beil in [35] and [36] for Fn apseudo–Riemannian space or a Minkowski space.

We notice that for Bi = yi in (7.39) one obtains a general version ofthe Synge metric, which was used by R. Miron for a geometric theory ofRelativistic Optics (see [130], ch.XI).

In the following we shall assume Bi 6= yi and use some ideas and techniquesfrom [130], ch. XI.

As we know, a necessary and sufficient condition for ∗gij to be reducibleto an L-metric is the symmetry in all indices of the Cartan tensor field

∗Cijk =12

∂(∗gij)∂yk

i.e.∂(∗gij)

∂yk=

∂(∗gkj)∂yi

.(7.42)

Using (7.39) this condition becomes

∂σ

∂ykBiBj − ∂σ

∂yiBkBj + σ

(∂Bi

∂yk·Bj − ∂Bk

∂yi·Bj

)

+σ

(Bi · ∂Bj

∂yk−Bk · ∂Bj

∂yi

)= 0.

(7.43)

7.6. Deformations of Finsler metrics 193

Multiplying it by Bj one gets

B2

(∂σ

∂ykBi − ∂σ

∂yiBk

)+ σB2

(∂Bi

∂yk− ∂Bk

∂yi

)+

+σ

(Bi · ∂Bj

∂yk·Bj −Bk

∂Bj

∂yi·Bj

)= 0.

(7.44)

If (7.43) is an identity, then (7.44) should be an identity for any σ and Bi. Butfor Bi = Bi(x) and σ = F 2, (7.44) reduces to ykBi−yiBk = 0, which is not anidentity for any Bi. Thus in general ∗gij(x, y) is not reducible to an L-metric.

We have a case when ∗gij(x, y) is an L-metric as follows.

Proposition 7.6.1 Assume Bi = Bi(x). If σ(x, y) = f(Bi(x)yi) for a smoothfunction f : R→ R, then ∗gij is an L-metric.

Indeed, it is easy to check that, within these hypotheses, expression (7.43)is identically satisfied. Notice that we do not know which function L satisfies

∗gij =∂2L

∂yi∂yj.

We recall that ∗gij(x, y) is called weakly regular if its absolute energy

E(x, y) := ∗gij(x, y)yiyj = F 2(x, y) + σ(x, y)(Biyi)2(7.45)

is a regular Lagrangian, which means that the matrix with the entries

akh(x, y) =12

∂2E∂yh∂yk

(x, y)(7.46)

has maximal rank n.A direct calculation yields the following expression for the metric tensor

akh:

akh = gkh +12

∂2σ

∂yk∂yhβ2 + β

(∂σ

∂yk

∂β

∂yh+

∂σ

∂yh

∂β

∂yk

)

+σ∂β

∂yk

∂β

∂yh+ σβ

∂2β

∂yk∂yh.

(7.47)

It is hopeless to decide if akh given by expression (7.47) is invertible or not.However, we have some interesting particular cases.

Proposition 7.6.2a) If B is orthogonal to the Liouville vector field C = yi(∂/∂yi), then ∗gij

is weakly regular and akh(x, y) = gkh(x, y).b) If Bi = Bi(x) and σ(x, y) = f(β) for some smooth function f : R→ R,

then ∗gij is weakly regular if and only if 1 + ϕ(β)B2 6= 0, where 2ϕ(β) =β2f ′′ + 4βf ′ + 2f, f ′ = df/dβ, f ′′ = d2f/dβ2 and we have

akh(x, y) = gkh(x, y) + ϕ(x, y)Bk(x)Bh(x).(7.48)


Proof.a) The condition B is orthogonal to C is equivalent to β = 0. Thus

E(x, y) = F 2(x, y) and so akh = gkh.b) By a direct calculation we can derive expression (7.48). Therefore (akh)

has the same form as ∗gkh with σ replaced by ϕ, and the conclusion of ourproposition follows immediately. q.e.d.

We keep the hypothesis Bi = Bi(x) and σ = f(β), β 6= 0. From (7.48)we see that we have again akh = gkh when ϕ = 0. The differential equationβ2f ′′ + 4βf ′ + 2f = 0 takes the form (β2f ′ + 2βf)′ = 0 and so its generalsolution is f(β) = a/β + b/β2, a, b ∈ R. The metric ∗gij becomes

∗gij = gij +(

a

Bi(x)yi+

b

(Bs(x)ys)2

)Bi(x)Bj(x).(7.49)

Notice that although ∗gij is an L-metric, we do not know yet the LagrangianL.

The absolute energy of ∗gij is now E = F 2+a(Fi(x)yi)+b and the Lagrangespace Ln = (M, E) is an Almost Finsler Lagrange space.

We may put expression (7.48) into the following equivalent form

akh(x, y) = ∗gkh +(

12β2f ′′ + 2βf ′

)BkBh.(7.50)

Thus, we see that akh = ∗gkh if and only if f is a solution of the differentialequation

12f ′′β2 + 2βf ′ = 0 i.e. f(β) = c− d

β3, c, d ∈ R.

Within the previous hypothesis, we have that ∗gkh is an L-metric. The condi-tion akh = ∗gkh gives L in the form L(x, y) = E(x, y) + Ai(x)yi + ψ(x), whereAi is a covector and ψ a scalar. Inserting here E we get

L(x, y) = F 2(x, y) + c(Bi(x, y)yi)2 − d

Bi(x)yi+ Ai(x)yi + ψ(x),(7.51)

where c, d ∈ R. Therefore we found a case when ∗gij is an L-metric with L ofexplicit form (7.51).

From condition a) of Proposition 7.6.2, ∗gij is not necessarily an L-metric.If σ(x, y) and Bi(x, y) are positively homogeneous of order 0, then ∗gij(x, y)is so and (M,∗ gij) is a generalized Finsler space in Izumi’s sense (see [94]).

The condition that B is orthogonal to C is equivalent to the condition thatB is tangent to the indicatrix bundle I(M) ⊂ TM.

We remark here that conditions β = 0 and Bi = Bi(x) are incompatiblesince they lead to B = 0.

7.7. Connections for a deformed Finsler metric 195

If in expression (7.51) we take d = 0, Ai = 0, ψ = 0, and c > 0, then∗F 2 := L(x, y) is positively homogeneous of order 2 and consequently ∗Fn =(M,∗ F ) becomes a Finsler space. Notice that ∗F is obtained from F by aβ-change and in this case ∗gij reduces to a Finsler metric.

An interesting Beil metric can be associated to a Finsler space Fn with an(α, β)-metric. Here α2 = aij(x)yiyj and β = bi(x)yi, where aij is a Riemannianmetric an bi a covector field on M . One may consider

∗gij(x, y) = aij(x) + σ(x, y)bi(x)bj(x),(7.52)

where σ is a scalar function on TM such that 1+σb2 6= 0 for b2 = aijbibj . ThisGL-metric is not reducible to an L-metric or a Finsler metric. The previousdiscussion applies, too.

7.7 Connections for a deformed Finsler metric

In Finsler geometry as well as in their generalizations, the nonlinear connec-tions play an important role. For instance these connections allow us to workwith d-objects and we can keep and check easily the geometric meaning of thecalculation in local coordinates.

A nonlinear connection always exists if M is paracompact. But the nonlin-ear connections derived from or associated in a way to a GL-metric are muchmore useful. So far it is not possible to derive a nonlinear connection directlyfrom an arbitrary given GL-metric. However, as we have already shown inSection 7.4, there are some classes of GL-metrics for which we can associate ina canonical way a nonlinear connection to a GL-metric. This section is basedon M. Anastasiei and H. Shimada papers [5] and [14].

Given the GL-metric (7.39), we cannot derive a nonlinear connection fromit. But since it is constructed with gij(x, y), we may take into consideration theCartan nonlinear connection N i

j of the Finsler space Fn and then all possiblenonlinear connections have the form N i

j = N ij −Ai

j with Aij(x, y) an arbitrary

d-tensor field of (1,1)-type.We also consider the horizontal and the vertical components of a symmet-

ric, metric, N -linear connection:

F ijk =

12gih

(δghk

δxj+

δgjh

δxk− δgjk

δxh

),

Cijk =

12gih

(∂ghk

∂yj+

∂gjh

∂yk− ∂gjk

∂yh

),

(7.53)

where δ/δxj = ∂/∂xj − Nkj ∂/∂yk.


The Cartan connection has the local coefficients CΓ(N) = (N ij , F

ijk, C

ijk).

This connection is h- and v-metric and its h(h)- and v(v)-torsions vanish.Now we replace in the right hand side of (7.53) the metric gij by ∗gij and

the operator δ/δxj by sδ/δxj = ∂/∂xj − Nkj ∂/∂yk + Ak

j ∂/∂yk and denote theresults in the left hand side by sF i

jk and sCijk, respectively. Thus, we get an

N -linear connection sCΓ(N) = (N ij ,

s F ijk,

s Cijk), which we call standard metric

connection of the GL-metric ∗gij .

This connection is metric, which means that ∗gij

s

|k= 0, ∗gij

s

| k = 0 and

its h(h)-torsion and v(v)-torsion vanish. It is clear that it depends on Aij but

if Aij is a priori given it is the unique N -linear connection with the above

properties. For Aij = 0 we set ∗F := sF and ∗C := sC. Thus we have

sF ijk = ∗F i

jk +12∗gih

(As

j

∂(∗ghk)∂ys

+ Ask

∂(∗ghj)∂ys

−Ash

∂(∗gjk)∂ys

)

sCijk = ∗Ci

jk.

(7.54)

The first equation in (7.54) takes also the form

sFjik = ∗Fjik + ∗CkisAsj + ∗CjisA

sk −Al

i∗Cjkl.(7.55)

If ∗gij reduces to an L-metric or to a Finsler metric, (7.54) becomes

sF ijk = ∗F i

jk + CiksA

sj ,

sCijk = ∗Ci

jk.(7.56)

We notice the following possible choices of Aij : λ(x, y)δi

j , yiyj , Biyj , yiBj ,

BiBj .From expression (7.39) we find, [14],

∗F ijk = Bi

sFsjk +

σ

2∗g ih

[δ(BhBk)

δxj+

δ(BhBj)δxk

− δ(BjBk)δxh

]

+12∗gih(σjBhBk + σkBhBj − σhBjBk),

∗Cijk = Bi

sCsjk +

σ

2∗g ih

[∂(BhBk)

∂yj+

∂(BhBj)∂yk

− ∂(BjBk)∂yh

]

+12∗gih(σjBhBk + σkBhBj − σhBjBk),

(7.57)

where

Bis = δi

s − ∗σBiBs, σk :=δσ

δxk, σk :=

∂σ

∂yk, ∗σ =

σ

1 + σB2.(7.58)

7.7. Connections for a deformed Finsler metric 197

Now, sF ijk and sCi

jk can be easily deduced from (7.54).We remark here that matrix Bi

s is invertible. Its inverse is (B−1)sk =

δsk + σBsBk. As such from (7.57) we can find F and C as depending on ∗F

and ∗C.In order to evaluate the torsions and curvatures of ∗CΓ(cN) it is more

convenient to put (7.57) into the form

∗F ijk = F i

jk + Λijk,

∗Cijk = Ci

jk + Λijk, where

(7.59)

Λijk =

12∗gih

[δ(σBjBh)

δxk+

δ(σBhBk)δxj

− δ(σBjBk)δxh

]

−∗σBiBhFjhk and

Λijk =

12∗gih

[∂(σBjBh)

∂yk+

∂(σBhBk)∂yj

− ∂(σBjBk)∂yh

]

−∗σBiBhCjhk.

(7.60)

The components of the torsion of ∗CΓ(cN) are as follows:

∗T ijk = 0, ∗Ri

jk = Rijk,

∗Sijk = 0,

∗P ijk = P i

jk − Λikj ,

(7.61)

and ∗Cijk from (7.57). For the components of the curvatures we have

∗Sijkh = Si

jkh + Λijkh +A[kh](Cs

jkΛish + Λs

jkCish)

Λijkh = A[kh]

(∂Λi

jk

∂yh+ Λs

jkΛish

),

where A[kh] denotes a substraction after the interchange of the indices k andh.

∗P ijkh = P i

jkh +∂Λi

jk

∂yh− Λi

jh|k − Cijh|k − Λi

jh|k

+∂Ci

jh

∂yk+

∂Λijh

∂yk− Ci

jsΛshk + Λi

jsPshk − Λi

jsΛskh,

(7.62)

where |k denotes the covariant derivative constructed with Λijk.

∗Rijkh = Ri

jkh + Λijkh +A[kh](F

sjkΛ

ish + Λs

jkFish) + Λs

jsRskh,(7.63)

where

Λijkh = A[kh]

(δΛi

jk

δxh+ Λs

jkΛish

).(7.64)


Thus we have a sample of calculation for the geometric objects associated tothe GL-metric of the form (7.39). For more details and for physical consider-ations we refer to [13], [14] and [36].

7.8 New metric classes

In [87] and [88], P.R. Holland studies a unified formalism that uses a nonholo-nomic frame on space-time arising from consideration of a charged particlemoving in an external electromagnetic field. In fact, R.S. Ingarden in [93]was first to point out that the Lorentz force law can be written in this caseas geodesic equation on a Finsler space called Randers space, [159]. In [35],[36] a gauge transformation is viewed as a nonholonomic frame on the tangentbundle of a four dimensional base manifold. The geometry that follows fromthese considerations gives a unified approach to gravitation and gauge sym-metries. In the above mentioned papers, the common Finsler idea used by thephysicists R.G. Beil and P.R. Holland is the existence of a nonholonomic frameon the vertical subbundle V TM of the tangent bundle of a base manifold M .This nonholonomic frame relates a semi-Riemannian metric (the Minkovski orthe Lorentz metric) with an induced Finsler metric. In [18] and [19] such anonholonomic frame has been determined for two important classes of Finslerspaces that are dual in the sense of [89]: Randers and Kropina spaces. AsRanders and Kropina spaces are members of a bigger class of Finsler spaces,namely the Finsler spaces with (α, β)-metric, it appears a natural question:does a Finsler space with (α, β)-metric have such a nonholonomic frame? Theanswer is yes for Finsler space with (α, β)-metric, for Lagrange space with(α, β)-metric and for a class of generalized Lagrange metrics we studied inSection 7.6.

Consider aij(x) the components of a Riemannian metric on the base man-ifold M , a(x, y) > 0 and b(x, y) ≥ 0 two functions on TM and B(x, y) =Bi(x, y)dxi a vertical 1-form on TM . Then

gij(x, y) = a(x, y)aij(x) + b(x, y)Bi(x, y)Bj(x, y)(7.65)

is a generalized Lagrange metric, called the Beil metric . We say also that themetric tensor gij is a Beil deformation of the Riemannian metric aij . It hasbeen studied and applied by R. Miron and R.K. Tavakol in General Relativityfor a(x, y) = exp(2σ(x, y)) and b = 0. The case a(x, y) = 1 with variouschoices of b and Bi was introduced and studied by R.G. Beil for constructinga new unified field theory in [36].

In Finsler geometry an important class of Finsler metrics is provided by theclass of Finsler spaces with (α, β)-metrics, [116]. The first Finsler spaces with(α, β)-metric were introduced in 1940’s by the physicist G. Randers and hence

7.8. New metric classes 199

such spaces are called Randers spaces, [159]. Recently, R.G. Beil suggested toconsider a more general case, the class of Lagrange spaces with (α, β)-metric,which was discussed in [51].

Definition 7.8.1 A Finsler space Fn = (M, F (x, y)) is called with (α, β)-metric if there exists a 2-homogeneous function L(α, β) of two variables suchthat the Finsler metric F : TM → R is given by

F 2(x, y) = L(α(x, y), β(x, y)).(7.66)

Here α2(x, y) = aij(x)yiyj, where aij(x) is a Riemannian metric on M andβ(x, y) = bi(x)yi, where bi(x)dxi is a 1-form on M .

If we do not ask for the function L to be homogeneous of order two with respectto the (α, β) variables, then we have a Lagrange space with (α, β)-metric. Nextwe look for some different Finsler or Lagrange spaces with (α, β)-metric.

1) If L(α, β) = (α + β)2, then the Finsler space with Finsler metricF (x, y) =

√aij(x)yiyj + bi(x)yi is called a Randers space.

2) If L(α, β) = α4/β2, then the Finsler space with Finsler metric F (x, y) =(aij(x)yiyj)/|bi(x)yi| is called a Kropina space. These two classes of Finslerspaces play an important role in Finsler geometry and they are dual via theLegendre transformation, [89].

3) If L(α, β) = αnβm, then we have a Lagrange space with (α, β)-metric,where the Lagrange metric is F (x, y) = n/2

√aij(x)yiyj(bi(x)yi)m. This La-

grange spaces reduces to a Finsler spaces with (α, β)-metric if and only ifn + m = 2.

Throughout this section we shall rise and lower indices only with the Rie-mannian metric aij(x) that is yi = aijy

j , bi = aijbj , and so on.For a Lagrange space with (α, β)-metric F 2(x, y) = L(α(x, y), β(x, y)) we

consider the following four invariants as they are defined for a Finsler spacewith (α, β)-metric, [114]:

ρ =12α

∂L

∂α; ρ0 =

12

∂2L

∂β2;

ρ−1 =12α

∂2L

∂α∂β; ρ−2 =

12α2

(∂2L

∂α2− 1

α

∂L

∂α

).

(7.67)

For a Finsler space with (α, β)-metric that is if L is homogeneous of degreetwo with respect to α and β we have

ρ−1β + ρ−2α2 = 0.(7.68)

With respect to these notations we have that the metric tensor gij of a La-grange space with (α, β)-metric is given by

gij = ρaij + ρ0bibj + ρ−1(biyj + bjyi) + ρ−2yiyj .(7.69)


We may remark here that the formula (7.69) appears in [114] for Finsler spaceswith (α, β)-metric but it works more generally for Lagrange spaces with (α, β)-metric. The metric tensor gij of a Lagrange space with (α, β)-metric can bearranged into the form:

gij = ρaij +1

ρ−2(ρ−1bi + ρ−2yi)(ρ−1bj + ρ−2yj)

+1

ρ−2(ρ0ρ−2 − ρ2

−1)bibj .

(7.70)

If the bibj coefficient vanishes we have:

Proposition 7.8.1 If for a Lagrange space with (α, β)-metric the condition:

ρ2−1 = ρ0ρ−2(7.71)

holds true, then the metric tensor gij can be written in the equivalent form:

gij(x, y) = ρ(x, y)aij(x) +1

ρ−2Bi(x, y)Bj(x, y), where

Bi(x, y) = ρ−1(x, y)bi(x) + ρ−2(x, y)yi.(7.72)

If we compare (7.72) and (7.65) we have that if for a Lagrange space with(α, β)-metric the condition (7.71) holds true, then its fundamental metric ten-sor is a Beil metric.

R.G. Beil suggested to consider a Lagrange space with (α, β)-metric withthe fundamental function given by L(α, β) = αnβm. Then condition (7.71) istrue if and only if m2n2 = mn(m− 1)(n− 2). An example of Lagrange spacewith (α, β)-metric that satisfies the condition (7.71) has the Lagrange metricL(α, β) = α4/β.

7.9 Nonholonomic Finsler frames

In this section we find a nonholonomic frame for a class of generalized Lagrangemetrics (7.65). In [13], the metric tensor of such a generalized Lagrange spacehas been called the Beil metric. The Beil metric can be viewed also as a defor-mation of a Riemannian metric. In this work we consider the most general caseof Beil’s metric and we find a nonholonomic frame for it. This frame reducesin a particular case to that considered by R.G. Beil in [35] and [36]. Thenwe can use this ideas to find a nonholonomic frame for the class of Lagrangespaces with (α, β)-metric. We prove that the fundamental metric tensor of aFinsler space with (α, β)-metric can be derived from a Riemannian metric us-ing two deformations. Using these ideas we can find a nonholonomic frame for

7.9. Nonholonomic Finsler frames 201

a Finsler space with (α, β)-metric. As Randers and Kropina spaces are Finslerspaces with (α, β)-metric we may use these techniques to find nonholonomicFinsler frames for these Finsler spaces. More detailed information about thiscan be found in [51].

R. Miron studied such nonholonomic frames and the induced N -linearconnection in [139] for the so-called strongly non-Riemannian Finsler spaces.M. Matsumoto studied these nonholonomic frames also, in [114], where hecalled such frames the Miron frames of a strongly non-Riemannian Finslerspace. The Miron frame is a natural generalization of the Berwald framefor a two dimensional Finsler space or the Moor frame for a Finsler space ofdimension three, we studied in Sections 5.8 and 5.9.

In this section, we determine first a nonholonomic frame for a Beil metric(7.65). In the particular case when a(x, y) = 1 and b(x, y) is a constant k weget the frame used by R.G. Beil in [35]. In the previous section, we foundconditions in which the fundamental metric of a Lagrange space with (α, β)-metric is a Beil metric. Then we can determine a nonholonomic frame for aLagrange space with (α, β)-metric from the nonholonomic Finsler frame of aBeil metric. From expression (7.70) we can see that the fundamental metrictensor of a Finsler space with (α, β)-metric can be derived from a Riemannianmetric aij using the Beil deformation (7.65) in two steps. Using this idea wecan determine a nonholonomic frame for a Finsler space with (α, β)-metric asa product of two nonholonomic frames, each of these being determined by adeformation of the metric.

Let U be an open set of TM and

Vi : u ∈ U 7→ Vi(u) ∈ VuTM, i ∈ 1, ..., n

be a vertical frame over U . If Vi(u) = V ji (u)(∂/∂yj), then V j

i (u) are theentries of a invertible matrix for all u ∈ U . Denote by V j

k (u) the inverse ofthis matrix. This means that

V ij V j

k = δik, V i

j V jk = δi

k.

We call V ij a nonholonomic frame.

Theorem 7.9.1 Consider a GL-space with the GL-metric given by expression(7.65)

gij(x, y) = a(x, y)aij(x) + b(x, y)Bi(x, y)Bj(x, y)

and denote by B2(x, y) = aij(x)Bi(x, y)Bj(x, y). Then

V ij =

√aδi

j −1

B2

(√a±

√a + bB2

)BiBj(7.73)


is a nonholonomic frame. The metric GL-metric (7.65) and the Riemannianmetric aij(x) are related by

gij(x, y) = V ki (x, y)V l

j (x, y)akl(x).(7.74)

Proof. Consider also

V jk =

1√aδjk −

1B2

(1√a± 1√

a + bB2

)BjBk.(7.75)

It is a direct calculation to check that V jk is the inverse of V i

j that is V ij is a

nonholonomic frame. Next we have that V ki V l

j akl = aaij + bBiBj = gij so theformula (7.74) holds true. q.e.d.

Proposition 7.9.1 The GL-metric given by expression (7.65) is positive def-inite on TM .

Proof. As the functions a(x, y) and b(x, y) that define the metric (7.65) arepositive and the metric aij is positive definite from (7.75) we can see that V i

k

is well defined on TM . Then V ij from (7.73) is a nonholonomic frame on TM .

From (7.74) we have that gij and aij have the same signature, so gij is positivedefinite on TM . q.e.d.

We remark here that for a(x, y) = 1 and b(x, y) = k, the nonholonomicFinsler frame (7.73) is the frame that appears in formula (5.1) in [33].

Theorem 7.9.2 Let F 2(x, y) = L(α(x, y), β(x, y)) be the metric function ofa Lagrange space with (α, β)-metric for which the condition ρ2

−1 = ρ0ρ−2 istrue. Then

V ij =

√ρδi

j −1

B2

(√

ρ±√

ρ +B2

ρ−2

)σiσj(7.76)

is a nonholonomic Finsler frame, where σi = ρ−1bi + ρ−2y

i, B2 = σiσi =ρ2−1b

2 + ρ2−2α

2 + 2βρ−1ρ−2, ρ, ρ0, ρ−1 and ρ−2 are the invariants of the La-grange space with (α, β)-metric defined in (7.67).

For a Lagrange space with (α, β)-metric L = α4/β we have:

ρ =2α2

β, ρ0 =

α4

β3, ρ−1 =

−2α2

β2, ρ−2 =

4β

.

We have then that the condition (7.71) is true and B2 = 4α4b2/β4. Conse-quently, a nonholonomic frame for the given Lagrange space with (α, β)-metricis given by:

V ij = α

√2β

δij −

1α3b2

(√2β±

√2β

+α2b2

β3

)(2βyi − α2bi)(2βyj − α2bj).

7.9. Nonholonomic Finsler frames 203

Consider now a Finsler space with (α, β)-metric. From (7.70) we can seethat gij is the result of two deformations:

aij 7→ hij = ρaij +1

ρ−2(ρ−1bi + ρ−2yi)(ρ−1bj + ρ−2yj) and

hij 7→ gij = hij +1

ρ−2(ρ0ρ−2 − ρ2

−1)bibj .

(7.77)

The nonholonomic frame that corresponds to the first deformation (7.77) is,according to Theorem 7.9.1, given by

Xij =

√ρδi

j −1

B2

(√

ρ±√

ρ +B2

ρ−2

)σiσj ,(7.78)

where σi = ρ−1bi + ρ−2y

i, B2 = aij(ρ−1bi + ρ−2y

i)(ρ−1bj + ρ−2y

j) = ρ2−1b

2 +βρ−1ρ−2. The metric tensors aij and hij are related by

hij = Xki X l

jakl.(7.79)

According to Theorem 7.9.1, the nonholonomic frame that corresponds to thesecond deformation (7.77) is given by

Y ij = δi

j −1

C2

(1±

√1 +

ρ−2C2

ρ0ρ−2 − ρ2−1

)bibj ,(7.80)

where

C2 = hijbibj = ρb2 +

1ρ−2

(ρ−1b2 + ρ−2β)2.

The metric tensors hij and gij are related by the formula

gmn = Y imY j

n hij .(7.81)

From (7.79) and (7.81) we have that V km = Xk

i Y im, with Xk

i given by (7.78)and Y i

m given by (7.80), is a nonholonomic frame of the Finsler space with(α, β)-metric.

For a Randers space with the fundamental function L = (α + β)2 = F 2,the Finsler invariants (7.67) are given by

ρ =α + β

α=

F

α, ρ0 = 1, ρ−1 =

1α

, ρ−2 =−β

α3,

B2 =b2α2 − β2

α4.


We have that condition (7.71) is not satisfied. If we use the previous idea,then V k

m = Xki Y i

m is a nonholonomic frame of a Randers space, where:

Xij =

√α + β

αδij −

α2

α2b2 − β2

√α + β

α±

√αβ + 2β2 − b2α2

αβ

·(

bi − βyi

α2

)(bj − βyj

α2

),

Y ij = δi

j −1

C2

1±

√1 +

βC2

α + β

bibj , and

C2 =(α + β)b2

α− α

β

(b2 − β2

α2

)2

.

In a similar way we may find a nonholonomic frame for a Kropina space withthe fundamental function L = α4/β2 = F 2. In this case, the Finsler invariantsare given by

ρ =2α2

β2, ρ0 = 3

α4

β4, ρ−1 =

−4α2

β3, ρ−2 =

4β2

,

B2 = 16α2

β4

(α2b2

β2− 1

).

One can use also the two steps deformations (7.77) to determine the con-travariant tensor (gij) of a Finsler space with (α, β)-metric.

Part III

Dynamical systems

Chapter 8

Dynamical Systems.Lagrangian Geometries

The geometric study of dynamical systems is an important chapter of contem-porary mathematics due to its applications in Mechanics, Theoretical Physics,Control Systems, Economy and Biology. If M is a differentiable manifold thatcorresponds to the configuration space, a dynamical system can be locallygiven by a system of ordinary differential equations of the form xi = f i(t, x),which are called equations of evolution. Globally, a dynamical system is givenby a vector field X on the manifold M × R whose integral curves, c(t) aregiven by the equations of evolution, X c(t) = c(t). The theory of dynamicalsystems deals with the integration of such systems, determining the generalsolutions and particular solutions that correspond to some initial conditions,a qualitative study of these solutions and their stability.

The geometry of nonconservative mechanical systems, where the externalforce field depends on both position and velocity, was rigorously investigatedby Klein [100] and Godbillon [79]. The dynamical system of a nonconserva-tive mechanical systems is a second order vector field, or a semispray, and ithas been uniquely determined by Godbillon [79] using the symplectic struc-ture and the energy of the Lagrange space and the external force field. Usingthe external force field of the nonconservative mechanical system, Klein [100]introduces a force tensor, which is a second rank skew symmetric tensor. As-pects regarding first integrals for nonconservative mechanical systems wereinvestigated by Djukic and Vujanovic [75] and Cantrijn [60]. For the particu-lar case when the external force field is the vertical differential of a dissipationfunction, such systems were studied by Bloch [46].

In this chapter we extend the geometric investigation of nonconservativemechanical systems, using the associated evolution nonlinear connection. Weshow that the evolution nonlinear connection is uniquely determined by two

208 Chapter 8. Dynamical Systems. Lagrangian Geometries

compatibility conditions with the metric structure and the symplectic struc-ture of the Lagrange space, [55]. The covariant derivative of the Lagrangemetric tensor with respect to the evolution nonlinear connection is a secondrank symmetric tensor, which uniquely determines the symmetric part of theconnection. The difference between the symplectic structure of the Lagrangespace and the almost-symplectic structure of the nonconservative mechanicalsystem is the force tensor introduced by Klein [100], and used recently byMiron [128]. The force tensor, which is the vertical differential of the externalforce, uniquely determines the skew-symmetric part of the evolution nonlinearconnection. The force tensor vanishes in the work of Bloch [46] and thereforethe symplectic geometry of the nonconservative mechanical system coincideswith the symplectic geometry of the underlying Lagrange space as it has beendeveloped by Abraham and Marsden [2].

One can determine the equations of evolution either from a Lagrangianfunction by writing these equations as the Euler-Lagrange equations or byusing a Legendre transformation, determining a Hamiltonian function andconsidering the Hamilton equations. One can use then Lagrange or Hamiltongeometries for a geometric theory of the evolution problem. A geometricalapproach of this problem on the phase space for the Riemannian case hasbeen proposed by Munoz-Lecanda and Yaniz-Fernandez, [147]. This theoryhas been developed recently, for the case of Finsler and Lagrange spaces in[55] and [140]. For a mathematical model of the geomagnetic field, which hasaperiodic reversals, Yajima and Nagahama proposed recently in [193] a math-ematical model that corresponds to a nonlinear dynamical system (Rikitakesystem). This way the chaotic behavior of the system is expressed with theabove mentioned geometric and topologic invariants.

For a system of second order differential equations these geometric prob-lems where studied in Chapter 4, based on the geometry of the tangent bundleTM , to which we shall refer to as the phase space.

In this chapter we study dynamical systems on the phase space that aredefined by systems of second order differential equations that result from thetheory of scleronomic, holonomic mechanical systems given by Lagrange equa-tions when the external forces are a priori given.

The main idea is to determine a semispray S, whose integral curves give theevolution curves. We shall determine the evolution semispray of a mechanicalsystem by using the symplectic structure of the associated Lagrangian functionand the external force field. The geometry of the semispray will determine thegeometry of the associated dynamical system on the phase space.

We will study these problems first for a Riemann space Rn = (M, g) andthen for a Finsler space Fn = (M, F ), a Lagrange space Ln = (M, L), andfinally for a generalized Lagrange space GLn = (M,L), [140].

If the Lagrangian function is not homogeneous of second degree with re-

8.1. Riemannian mechanical systems 209

spect to the velocity-coordinates, which is the case in the Riemannian andFinslerian context, the energy of the system is different from the Lagrangianfunction and the evolution curves (solution of the Euler-Lagrange equations)are different from the horizontal curves of the system. Therefore, we shallstudy the variation of both energy and Lagrangian function along the evolu-tion curves and horizontal curves. Using the first geometric invariant fromthe KCC-theory [20] of an SODE we express the variation of energy along thehorizontal curves of the evolution nonlinear connection. As it happens for theRiemannian case, we prove that the energy is decreasing along the evolutioncurves of the system if and only if the external force field is dissipative.

Canonic nonlinear connection of a Lagrange manifold is the unique non-linear connection that is metric and symplectic. Conditions by which theevolution nonlinear connection is either metric or symplectic are determinedin terms of the symmetric or skew-symmetric part of a (1,1)-type tensor fieldassociated with the external force field. We provide examples of dissipativemechanical systems.

8.1 Riemannian mechanical systems

We start by considering gij(x) a Riemannian metric on the configuration spaceM , which is an n-dimensional, real, and smooth manifold. For the Riemannianspace Rn = (M, g) we consider ∇ the Levi-Civita connection and γi

jk(x) itslocal coefficients. A Riemannian mechanical system is defined by a tripleΣ = (M, g, F ), where F = F i(∂/∂xi) is a vector field on M , called the field ofexternal forces. The external force field F is dissipative if gij(x)yiF j(x) ≤ 0.Using the Riemannian metric gij one can define the one-form σ = Fi(x)dxi,where Fi(x) = gij(x)F j(x).

The kinetic energy of the system is given by

Ec(x, y) = gij(x)yiyj , yi =dxi

dt,(8.1)

while the Lagrange equations are given by

ddt

(∂Ec

∂yi

)− ∂Ec

∂xi= Fi(x), yi =

dxi

dt.(8.2)

One can write an equivalent form of Lagrange equations (8.2) as a system ofsecond order differential equations, given by

d2xi

dt2+ γi

jk(x)dxj

dt

dxk

dt=

12F i(x), F i(x) = gij(x)Fj(x).(8.3)


Equations (8.2) are called equations of evolution of the Riemannian mechanicalsystem Σ. Solution curves c(t) = (xi(t)) of Lagrange equations (8.2) or (8.3)are called evolution curves.

We remark here that equations (8.2) with Fi(x) = 0 are Euler-Lagrangeequations of kinetic energy. If we parameterize the solution curves of equations(8.2) or (8.3) by the arclength of the Riemannian space Rn, we have thefollowing results:

1) If the external forces are absent, then the evolution curves of system Σare the geodesics of the Riemannian space Rn.

2) If the external forces are conservative that is Fi(x) = ∂ϕ/∂xi, then thetotal energy Ec − ϕ is conserved along the evolution curves of the system Σ.

3) Using the Proposition 5.3.1 for the Riemannian case and Lagrange equa-tions (8.2) we obtain that along the evolution curves of the system Σ thefollowing formula is satisfied

dEc

dt=

dxi

dtFi(x).

In other words we have that the kinetic energy of the system is decreasing ifand only if the external force field is dissipative.

4) The kinetic energy Ec of the system is conserved along the evolutioncurves of the system Σ if and only if the external forces Fi are orthogonal tothese curves.

We assume now that the external forces of system Σ are given by a globallydefined vector field on the configuration space M . This implies that exists aglobally defined vector field S on TM whose integral curves are given by theequations of evolution (8.2), [147].

Theorem 8.1.1 There is a globally defined vector field S determined by themechanical system Σ, whose integral curves are given by the equations of evo-lution (8.2). The vector field S is a semispray on the phase space.

Proof. On every domain of induced local charts on TM , we consider thefunctions

2Gi(x, y) = γijk(x)yjyk − 1

2F i(x).(8.4)

One can check by direct calculation that, under a change of induced coordi-nates (1.1) on TM , the functions Gi change according to transformation law(4.2). Consequently, Gi are local coefficients of a semispray

S = yi ∂

∂xi− 2Gi(x, y)

∂

∂yi.(8.5)

The vector field S is globally defined if and only if the vector field F hasthis property. Moreover, the semispray S is determined by the Riemannian

8.2. Finslerian mechanical systems 211

mechanical system Σ, only. We call this semispray the evolution semispray ofthe mechanical system Σ.

Integral curves of the evolution semispray S are given by:

dxi

dt= yi,

dyi

dt= −γi

jk(x)yjyk +12F i(x).

The above equations coincide with the evolution equations of the mechanicalsystem Σ. q.e.d.

If the field F of external forces is a vector field on the configuration spaceM , we have that S is not a spray as functions Gi are not homogeneous ofsecond order with respect to y. If F is a d-vector field on the phase spaceTM , then S is a spray if and only if F i are homogeneous functions of secondorder with respect to yi.

Now, we can associate to the mechanical system Σ, using the evolutionsemispray S, some geometric objects, according to the theory we developed inChapter 4.

1) The nonlinear connection N that corresponds to Σ has the local coeffi-cients

N ij(x, y) =

∂Gi

∂yj= γi

jk(x)yk.(8.6)

2) The canonical N -linear connection CΓ(N) = (Lijk, C

ijk) has the follow-

ing coefficientsLi

jk = γijk, Ci

jk = 0.(8.7)

One can develop the geometry of the mechanical system Σ based on thegeometric objects: g, S, N , and CΓ(N). If one consider the Lagrange spaceLn = (M, Ec), we have that Ln is reducible to the Riemannian space Rn.As the nonlinear connection, given by (8.6), does not depend on the externalforce field F it results that g is the gravitational tensor field and h- and v-electromagnetic fields are trivial. We obtain then that Einstein equations ofthe space Ln, if g is the Lorentz metric, are the classical ones.

8.2 Finslerian mechanical systems

For a manifold M , that is the configuration space of a Finslerian dynamicalsystem, we consider the tangent bundle TM to which we shall refer to as thephase space. Suppose that there is a Finsler function F on TM and Fi(x, y)dxi

is a globally defined d-covector field on the phase space.A Finslerian mechanical system — which is a natural extension of the

Riemannian one presented in the previous section — is defined by a triple


Σ = (M,EF 2 , Fi). Here EF 2 is the energy of the Finsler space Fn = (M, F ),that is

EF 2 = yi ∂F 2

∂yi− F 2.

Since the fundamental function F of the Finsler space is homogeneous of orderone with respect to y, we have that F 2 is homogeneous of order two, while themetric tensor gij is zero homogeneous. Consequently, we have that the energyEF 2 coincides with F 2, that is

EF 2(x, y) = F 2(x, y) = gij(x, y)yiyj .(8.8)

Exactly, as for the Riemannian case, the Lagrange equations of the Finslerianmechanical system Σ are given by

ddt

(∂F 2

∂yi

)− ∂F 2


dxi

dt.(8.9)

Using expression (8.8), and the theory from Section 5.3 one can write anequivalent form of the Lagrange equations (8.9) as a system of second orderdifferential equations, given by

d2xi

dt2+ γi

jk(x, y)dxj

dt

dxk

dt=

12F i(x, y), yi =

dxi

dt,(8.10)

whereF i(x, y) = gij(x, y)Fj(x, y).(8.11)

Here γijk(x, y) are the Christoffel symbols of the metric tensor gij(x, y), given

by expression (5.28). Equations (8.9) are called the equations of evolution ofthe Finslerian mechanical system Σ. Solution curves c(t) = (xi(t)) of Lagrangeequations (8.9) or (8.10) are called evolution curves.

The system of equations (8.10) locally determine a dynamical system onthe phase space TM . If the external force field Fi(x, y) is globally definedon TM , we shall prove that there exists a globally defined vector field S onTM , whose integral curves are given by the equations of evolution (8.10) ofthe dynamical system. In order to do this, we consider the following functionsdefined on domains of induced local charts of TM :

2Gi(x, y) = γijk(x, y)yjyk − 1

2F i(x, y).(8.12)

Under a change of coordinates (1.1) on TM , functions Gi transform accordingto (4.2), which means that they are local coefficients of a semispray:

S = yi ∂

∂xi− 2Gi(x, y)

∂

∂yi.(8.13)

8.2. Finslerian mechanical systems 213

The equations of evolution (8.10) are the equations for the integral curves ofthe semispray S. As the semispray S is determined by the mechanical systemΣ only, we shall refer to S as the evolution semispray.

The geometry of the Finslerian mechanical system Σ is determined bythe geometry of the Lagrange space Ln = (M,F 2(x, y)) endowed with theevolution semispray S.

One can prove now using the theory developed by M.R. Santilli in [166]the following result: If the external forces of the mechanical system Σ dependon velocity yi = xi, then the equations on evolution (8.10) are not self-adjoint.

Next, we present another property of the kinetic energy EF 2 = F 2.

Theorem 8.2.1 The variation of the kinetic energy EF 2 = F 2 along the evo-lution curves (8.10) of the mechanical system Σ is given by

dEF 2

dt=

dxi

dtFi.(8.14)

Proof. Using expression (5.3.1) and the Lagrange equations (8.9) we obtain

dF 2

dt=

∂F 2

∂xi

dxi

dt

∂F 2

∂yi

dyi

dt=

[ddt

(∂F 2

∂yi

)− Fi

]dxi

dt+

∂F 2

∂yi

dyi

dt

=ddt

(∂F 2

∂yiyi

)− Fi

dxi

dt= 2

dF 2

dt− Fi

dxi

dt,

which shows that formula (8.14) is true. q.e.d.

Theorem 8.2.2 The kinetic energy of the Finsler space Fn is conserved alongthe evolution curves of the mechanical system Σ if the external forces F i areorthogonal to the evolution curves.

Next, we give some examples of Finslerian mechanical systems.1) Consider a Randers space Fn = (M, α+β), where α2(x, y) = aij(x)yiyj

is the kinetic energy of a Riemannian metric aij(x), while β(x, y) = bi(x)yi.For this space we consider the mechanical system Σ = (M, (α + β)2, Fi(x, y)),where the external force field has the covariant components

Fi(x, y) = β∂α2

∂yi= bj(x)yjaik(x)yk.

The equations of evolution for this mechanical system are given by (8.10),where the contravariant components of the external force field are given byF i(x, y) = β(x, y)yi = bj(x)yjyi. Local coefficients (8.12) of the evolutionsemispray S are then homogeneous of second order with respect to yi andconsequently S is a spray.


2) For an Ingarden space In = (M,α + β) we consider the mechanicalsystem Σ as in the previous example. The difference is that we do not workhere with the Cartan nonlinear connection of the Finsler space but with theLorentz nonlinear connection introduced in Section 5.11.

3) One can consider a Riemannian mechanical system Σ = (M, Ec, Fi,where Ec is the kinetic energy of a Riemannian metric gij(x) and the covariantcomponents of the external force field are given by Fi(x, y) = gij(x)yj . Weremark here that Riemannian geometry is not enough to study mechanicalsystem Σ, we have to use Finslerian techniques for this.

8.3 Nonlinear connection of a Finslerian mechanicalsystem

We have seen in the previous section that a Finslerian mechanical system Σdetermines a semispray S given by (8.13) to which we refer to as the evolutionsemispray. Consider now the nonlinear connection one can associate to thissemispray. This nonlinear connection is called the evolution nonlinear connec-tion of the mechanical system Σ = (M,F 2(x, y), Fi(x, y)). Local coefficientsof this nonlinear connection are given by

N ij = N i

j −14

∂F i

∂yj.(8.15)

Here N ji are local coefficients of Cartan nonlinear connection of the Finsler

space. We remark here that ∂F i/∂yj are local components of a d-tensor fieldof (1,1)-type. The evolution nonlinear connection N determines a direct de-composition of the tangent space TuTM into horizontal and vertical subspaces

TuTM = HuTM ⊕ VuTM, ∀u ∈ TM.(8.16)

A basis, adapted to this decomposition is given by (δ/δxi|u, ∂/∂yi|u), where

δ

δxi

∣∣∣∣u

=∂

∂xi

∣∣∣∣u

−N ji (u)

∂

∂yj

∣∣∣∣u

=δ

δxi

∣∣∣∣∣u

+14

∂F j

∂yi

∂

∂yj

∣∣∣∣u

.(8.17)

The dual basis adapted to the decomposition (8.16) is given by (dxi|u, δyi|u),where

δyi|u = dyi|u + N ij(u)dxj |u = δyi|u − 1

4∂F i

∂yjdxj |u,(8.18)

where (δ/δxi, ∂/∂yi) and (dxi, δyi) are adapted bases to the Cartan nonlinearconnection. As the nonlinear connection is symmetric, then its weak torsion

8.4. Metric N -linear connection of a Finslerian mechanical system 215

tijk vanishes, that is

tijk =∂N i

j

∂yk− ∂N i

k

∂yj= 0.

The curvature tensor Rijk of the evolution nonlinear connection is given by

Rijk =

δN ij

δxk− δN i

k

δxj.(8.19)

We have that the nonlinear connection N is integrable if and only if the cur-vature tensor Ri

jk vanishes.Another geometric object one can associate to the mechanical system Σ

is the Berwald connection BΓ(N) = (Bijk, 0) = (∂N i

j/∂yk, 0), where the hori-zontal coefficients are given by

Bijk = Bi

jk −14

∂2F i

∂yj∂yk.(8.20)

Using the theory developed in Section 2.8 about autoparallel curves of a non-linear connection, we obtain the following result:

Theorem 8.3.1 Autoparallel curves of the evolution nonlinear connection ofthe mechanical system Σ are given by the following system of second orderdifferential equations:

yi =dxi

dt,

δyi

dt=

δyi

dt− 1

4∂F i

∂yj

dxj

dt= 0.(8.21)

If the external force field vanishes then nonlinear connection N coincides withCartan nonlinear connection N and the autoparallel curves of N are the evo-lution curves of the mechanical system.

8.4 Metric N-linear connection of a Finslerian me-chanical system

In the previous sections we have seen that in the presence of an external forcefield Fi(x, y) on a Finsler space the geodesic spray and evolution semispray aredifferent. Consequently, Cartan nonlinear connection of the Finsler space andcanonical nonlinear connection N of the mechanical system Σ are differenttoo. In this section, we determine the canonical metric N -linear connection ofthe mechanical system Σ, which is different from the Cartan linear connectionof the Finsler space.

For a Finslerian mechanical system Σ = (M,F 2(x, y), Fi(x, y)), the metricN -linear connection D with local coefficients (F i

jk, Cijk) is uniquely determined

by the following system of axioms:


A1 : The nonlinear connection N is the canonical nonlinear connection(8.15) of the mechanical system Σ;

A2 : gij|k = 0, i.e. D is h-metric;A3 : T i

jk = F ijk − F i

kj = 0, i.e. D is h-symmetric;A4 : gij |k = 0, i.e. D is v-metric;A5 : Si

jk = Cijk − Ci

kj = 0, i.e. D is v-symmetric.According to Theorem 7.3.2, we obtain that axioms the A1–A5, uniquely

determine the local coefficients (F ijk, C

ijk) of the canonical metric N -linear

connection D. These coefficients are given by

F ijk =

12gih

(δgjh

δxk+

δghk

δxj− δgjk

δxh

),

Cijk =

12gih

(∂gjh

∂yk+

∂ghk

∂yj− ∂gjk

∂yh

).

(8.22)

Due to the fact that the N -linear connection D is determined by the mechan-ical system Σ only, we will refer to it as the canonical N -linear connectionof Σ. Next theorem will give us the relations between the canonical N -linearconnection of Σ and Cartan linear connection of the Finsler space Fn.

Theorem 8.4.1 Local coefficients of the canonical N -linear connection of theFinslerian mechanical system Σ have the following expression:

F ijk = F i

jk + Cijk, Ci

jk = Cijk,(8.23)

where Cijk is the d-tensor field given by

Cijk =

14gis

(Cskh

∂F h

∂yj+ Cjsh

∂F h

∂yk− Cjkh

∂F h

∂ys

)(8.24)

and CΓ(N) = (F ijk, C

ijk) is the Cartan connection of the Finsler space Fn.

Proof. If we use the Legrange equations (8.10) and F ijk given by expression

(8.22), we obtain expression (8.23). The equality Cijk = Ci

jk follows from astraightforward calculation. q.e.d.

We observe that the contraction yjCijk has a simpler form,

yjCijk =

14Ci

khys ∂F h

∂ys.(8.25)

Proposition 8.4.1 Consider a Finslerian mechanical system Σ.

8.5. Electromagnetic tensors of a Finslerian mechanical system 217

1) A horizontal curve c(t) = (xi(t), yi(t)) is a geodesic for the canonicalN -linear connection D of Σ if it satisfies the following system of second orderdifferential equations:

d2xi

dt2+ F i

jk

dxj

dt

dxk

dt+

14Ci

hk

∂F h

∂yj

dxj

dt

dxk

dt= 0.(8.26)

2) A vertical curve c(t) = (xi0, y

i(t)) is a geodesic for the canonical N -linearconnection D of Σ if it satisfies the following system of differential equations:

d2yi

dt2+ Ci

jk

dyj

dt

dyk

dt= 0.(8.27)

Proof. For an N -linear connection D the geodesic curves are solutions of thesystem (3.33). If a curve c(t) = (xi(t), yi(t)) is horizontal then

δyi

dt=

dyi

dt+ N i

j

(x,

dx

dt

)dxj

dt= 0.

If we substitute this into (3.33) and use the form (8.23) of F ijk then we obtain

equations (8.26). By a similar argument we obtain equations (8.27). q.e.d.

8.5 Electromagnetic tensors of a Finslerian mecha-nical system

If we consider the evolution nonlinear connection of a mechanical system, thenthe h-deflection tensor field has the following expression:

Dij = yi

|j = Dij +

12Ci

jhys ∂F h

∂ys+

14

∂F i

∂yj,(8.28)

where the horizontal deflection tensor field of the Finsler space Fn vanishes,which means that Di

j = 0. Then, according to Definition 6.11.1, the h-electromagnetic tensor field Fij of the canonical N -linear connection D hasthe following expression:

Fij =12(Dij −Dji) =

18

(gis

∂F s

∂yj− gjs

∂F s

∂yi

).(8.29)

We denote by

Fij =12

(∂Fj

∂yi− ∂Fi

∂yj

)(8.30)

the helicoidal tensor of the mechanical system Σ.


Theorem 8.5.11) The h-electromagnetic tensor Fij of the canonical N -linear connection

and the helicoidal tensor Fij of the mechanical system Σ are related by thefollowing formula:

Fij = −14Fij .(8.31)

2) The v-electromagnetic tensor fij of the canonical N -linear connectionvanishes.

Proof. It follows immediately from (8.29) and (8.30). q.e.d.

If we use the Theorem 6.11.2 for the h-electromagnetic tensor Fij we obtain:

Theorem 8.5.2 The helicoidal tensor Fij of a Finslerian mechanical systemΣ satisfies the following generalized Maxwell equations:

Fij|k + Fjk|i + Fki|j

=12(Rsijk + Rsjki + Rskij)ys − 1

2(Rijk + Rjki + Rkij),

Fij |k + Fjk|i + Fki|j = 0.

(8.32)

Proof. If we apply Ricci identities of the N -linear connection CΓ(N) =(F i

jk, Cijk) to the Liouville vector field yi, we obtain:

Dij|k −Di

k|j = ysRisjk −Ri

jk.

As the metric tensor is covariant constant, the previous identities are equiva-lent with the following identities:

Dij|k −Dik|j = ysRsijk −Rijk.

If we permute cyclically the indices i, j and k and add the correspondingidentities we obtain first equations (8.32). Second equation (8.32) followsimmediately using (8.31). q.e.d.

Remarks1) One can evaluate the curvature tensors Rijkh and Rijk using the follow-

ing expressions for the coefficients N ij of the nonlinear connection and for the

horizontal components F ijk of the N -linear connection:

N ij = N i

j −14

∂F i

∂yj,

F ijk = F i

jk + Cijk.

8.6. Almost Hermitian model of a Finslerian mechanical system 219

2) The left hand side of first identity (8.32) can be expressed also as follows:

Fij|k + Fjk|i + Fki|j = −14Σ(ijk)

(F

ij|k +14

∂Fm

∂yk

∂Fij

∂ym

),

where Σ(ijk) means the sum of all cyclic permutations of i, j and k.3) In particular if the external force field Fi of the Finslerian mechanical

system does not depend on directional variables yi then the electromagnetictensor Fij vanishes.

8.6 Almost Hermitian model of a Finslerian me-chanical system

For a Finslerian mechanical system Σ, we consider gij the metric tensor andthe nonlinear connection with local coefficients N i

j given by (8.15). Havingthese we can consider the N-lift G of the metric tensor gij given by

G = gijdxi ⊗ dxj + gijδyi ⊗ δyj .(8.33)

The metric N-lift G has the following properties:1) G depends on the Finslerian mechanical system only;2) G is a pseudo-Riemannian structure on the manifold TM ;3) The horizontal and vertical distributions are orthogonal with respect to

G.We can express the horizontal 1-forms δyi as

δyi = δyi − 14

∂F i

∂ysdxs.

Using this formula we can determine a relation between the N-lift G and N -liftG of the fundamental tensor gij of the Finsler space Fn.

The evolution nonlinear connection N is characterized by an almost com-plex structure F given by the following tensorial structure on TM :

F = − ∂

∂yi⊗ dxi +

δ

δxi⊗ δyi.(8.34)

Almost complex structure F has the following properties:1) F depends on the Finslerian mechanical system only;2) F is a tensor field of (1,1)-type on the manifold TM and F F = −Id;3) F is a complex structure if and only if the curvature tensor Ri

jk of thenonlinear connection vanishes.


Theorem 8.6.11) The pair (G,F) is an almost Hermitian structure on the differentiable

manifold TM .2) The associated 2-form ω is given by:

ω = gijδyi ∧ dxj .(8.35)

3) The 2-form ω determines an almost symplectic structure on TM .

Now, we pay a special attention to the case when ω is a symplectic form.This is the reason we shall express the 2-form ω using the symplectic structureω = gij δy

i ∧ dxj of the Finsler space Fn. The relation between the 2-forms ωand ω is as follows:

ω = ω − 14Fijdxi ∧ dxj .(8.36)

Theorem 8.6.2 The following three statements are equivalent:1) ω is a symplectic structure.2) The triple (TM,G,F) is an almost Kahlerian space.3) Σ(ijk)F(ij|k) = 0, Fij |k = 0.

Proof. First two statements are equivalent. We have that ω = dθ, whereθ = pidxi. Using formula (8.36) we have that ω is a symplectic structure ifand only if

dFij ∧ dxi ∧ dxj = 0,

which proves that 1) and 3) are equivalent. q.e.d.

We want to emphasize here that almost hermitian model (TM,G,F) givesa good geometric description of the Finslerian mechanical system Σ.

8.7 Lagrangian mechanical systems

We consider now a Lagrange space Ln = (M,L(x, y)) and a d-covector fieldwith components Fi(x, y) that lives on the phase space TM . We refer to thed-covector field Fi as to the external force field, while M is the configura-tion space. Exactly as we did for Finslerian mechanical systems, we define aLagrangian mechanical system as a triple Σ = (M, L(x, y), Fi(x, y)).

For the Lagrange space Ln, we consider the fundamental metric tensor:

gij(x, y) =12

∂2L

∂yi∂yj, ∀(x, y) ∈ TM.(8.37)

We assume that the evolution equations of the mechanical system Σ are givenby the Lagrange equations:

ddt

(∂L

∂yi

)− ∂L


dxi

dt.(8.38)

8.7. Lagrangian mechanical systems 221

These equations are not the Euler-Lagrange equations of a Lagrangian func-tion L′. This means that the system (8.38) is not self-adjoint except for theparticular case when Fi is the gradient of a function ϕ(x). As L is a regularLagrangian, then the system (8.38) is a system of second order differentialequations of the form:

d2xi

dt2+ 2Gi

(x,

dx

dt

)=

12F i

(x,

dx

dt

),(8.39)

where

2Gi =12gis

(∂2L

∂ys∂xjyj − ∂L

∂xs

)and F i = gisFs.(8.40)

Now, we determine a globally defined vector field on the phase space TM ,whose integral curves are solutions of the equations of evolution (8.39), [79].

Theorem 8.7.11) The following operator

S = yi ∂

∂xi− 2

(Gi(x, y)− 1

2F i

)∂

∂yi,(8.41)

is a globally defined vector field on TM . The vector field S is a semispraythat depends on the mechanical system Σ only. We call this semispray theevolution semispray of the mechanical system Σ.

2) The evolution semispray S is the unique vector field on TM solution ofthe equation

iSω = −dEL + σ.(8.42)

3) The integral curves of S are the evolution curves of the Lagrangianmechanical system Σ.

Proof. 1) Since Gi are local coefficients of the canonical semispray of theLagrange space Ln and F i are components of a d-vector field it follows thatGi = Gi− (1/2)F i are also local coefficients of a semispray. Consequently, thevector field (8.40) is globally defined on TM . One can see that JS = C so Sis a semispray on TM .

2) Due to the fact that ω is a symplectic structure on TM , equation (8.42)uniquely determine a vector field S on TM . The vector field F i(∂/∂yi) is theunique vector field that satisfies iF i(∂/∂yi)ω = σidxi. Using the linearity ofequations (6.25) and (8.42) we can see that S = S + (1/2)F i(∂/∂yi) is theunique solution of equation(8.42).

3) Its integral curves satisfy the system (8.39), so they are evolution curvesof the mechanical system Σ. q.e.d.


The geometry of the Lagrangian mechanical system Σ is the geometry ofthe Lagrange space Ln with the evolution semispray S.

Next, we investigate the variation of the energy EL along the evolutioncurves of the mechanical system Σ.

Corollary 8.7.11) The energy of the Lagrange space Ln is decreasing along the evolution

curves of the mechanical system if and only if the external force field is dissi-pative.

2) The energy is conserved along the evolution curves of the mechanicalsystem Σ if and only if the external forces Fi are orthogonal to the evolutioncurves.

Proof. Using the fact that the evolution semispray S is solution of equation(8.42) and the skew-symmetry of ω we obtain S(EL) = dEL(S) = σ(S) =Fiy

i. Along the evolution curves of the mechanical system, one can write thisexpression as follows:

ddt

(EL) = Fi

(x,

dx

dt

)dxi

dt.(8.43)

Therefore, the energy is decreasing along the evolution curves if and only ifwe have Fiy

i ≤ 0. q.e.d.

One can also prove the above result using local coordinates as follows. Asthe energy EL is given by

EL = yi ∂L

∂yi− L,

by taking the derivative we have

dEL

dt=

dyi

dt

∂L

∂yi+ yi d

dt

(∂L

∂yi

)− ∂L

∂xiyi − ∂L

∂yi

dyi

dt

= yi

[ddt

(∂L

∂yi

)− ∂L

∂xi

]= yiFi(x, y), yi =

dxi

dt.

The local coefficients Gi of the evolution semispray S are given by

Gi(x, y) = Gi(x, y)− 12F i(x, y).(8.44)

Now, we can consider some geometric objects fields determined by the evo-lution semispray S and we will refer to these as the geometric objects of themechanical system Σ.


The nonlinear connection N of the system Σ has the local coefficients

N ij = N i

j −14

∂F i

∂yj,(8.45)

where N ij are the local coefficients of the canonical nonlinear connection of

the Lagrange space Ln = (M,L). Therefore the adapted basis (δ/δxi, ∂/∂yi)to the horizontal and the vertical distributions is given by:

δ

δxi=

δ

δxi+

14

∂F j

∂yi

∂

∂yj,

∂

∂yi(8.46)

while its dual basis is

dxi, δyi = δyi − 14

∂F i

∂yjdxj .(8.47)

We remark here that the nonlinear connection is symmetric and hence its weektorsion vanishes.

Theorem 8.7.2 The evolution nonlinear connection N is metric if and onlyif the (0, 2)-type d-tensor field ∂Fi/∂yj is skew-symmetric.

Proof. The evolution nonlinear connection is metric if and only if the dynami-cal covariant derivative of the metric tensor gij with respect to the pair (S, N)vanishes. This covariant derivative is given by

gij| = S(gij)− gikNkj − gkjN

ki

=(

S − 12F

)(gij)− gik

(Nk

j −14

∂F k

∂yj

)− gkj

(Nk

i −14

∂F k

∂yi

)

= S(gij)− gikNkj − gkjN

ki −

14

(2F (gij) + gik

∂F k

∂yj+ gkj

∂F k

∂yi

)

= −F k

4

(2∂gij

∂yk− ∂gik

∂yj− ∂gkj

∂yi

)− 1

4

(∂Fi

∂yj+

∂Fj

∂yi

)

= −14

(∂Fi

∂yj+

∂Fj

∂yi

).

(8.48)

In the above calculations we did use the fact that the canonical nonlinearconnection N is metric and the Cartan tensor Cijk is totally symmetric.

The dynamical covariant derivative of the metric tensor gij with respectto the pair (S, N) is given by

gij| = −14

(∂Fi

∂yj+

∂Fj

∂yi

).(8.49)


Consequently, the evolution nonlinear connection is metric if and only if the(0, 2)-type d-tensor field ∂Fi/∂yj is skew-symmetric. q.e.d.

Theorem 8.7.3 The evolution nonlinear connection is compatible with thesymplectic structure if and only if the (0, 2)-type d-tensor field ∂Fi/∂yj issymmetric.

Proof. The evolution nonlinear connection N is compatible with the symplecticstructure of the Lagrange space if and only if ω(hX, hX) = 0, ∀X ∈ χ(TM),where h is the corresponding horizontal projector. Let us consider the almostsymplectic structure:

ω = 2gijδyj ∧ dxi,(8.50)

with respect to which both horizontal and vertical subbundles are Lagrangiansubbundles. Using expression (8.47) the canonical symplectic structure canbe expressed as follows:

ω = 2gij δyj ∧ dxi = 2gij

(δyj +

14

∂F j

∂ykdxk

)∧ dxi

= 2gijδyj ∧ dxi +

14

(∂Fi

∂yj− ∂Fj

∂yi

)dxj ∧ dxi

= ω +12Fijdxj ∧ dxi.

(8.51)

Here Fij is the helicoidal tensor of the mechanical system ΣL introduced byR. Miron in [119]:

Fij =12

(∂Fi

∂yj− ∂Fj

∂yi

).(8.52)

Therefore, the evolution nonlinear connection is compatible with the sym-plectic structure if and only if ω = ω which is equivalent to the fact thatthe helicoidal tensor of the mechanical system ΣL, given by expression (8.52),vanishes. q.e.d.

The Berwald connection BΓ(N) = (Bijk, 0) one can associate to the me-

chanical system Σ has the horizontal coefficient given by

Bijk = Bi

jk −14

∂2F i

∂yj∂yk.(8.53)

The metric N -linear connection D of the mechanical system Σ has thecoefficients given by

Lijk =

12gih

(δgjh

δxk+

δghk

δxj− δgjk

δxh

),

Cijk =

12gih

(∂gjh

∂yk+

∂ghk

∂yj− ∂gjk

∂yh

).

(8.54)


The relation between these coefficients and the coefficients of the canonical N -linear connection of the Lagrange space Ln is given by the following theorem.

Theorem 8.7.4 The local coefficients of the canonical linear connection D ofthe mechanical system Σ have the following expression:

Lijk = Li

jk + Cijk, C

ijk = Ci

jk,(8.55)

where Cijk is given by

Cijk =

14gis

(Cskh

∂F h

∂yj+ Cjsh

∂F h

∂yk− Cjkh

∂F h

∂ys

),(8.56)

and (Lijk, C

ijk) is the canonical N -linear connection of the Lagrange space Ln.

Using the canonical N -linear connection of the mechanical system Σ we candefine and study the horizontal and vertical geodesics.

The electromagnetic tensors Fij and fij of the mechanical system Σ canbe defined using the deflection tensors as follows:

Fij =12(Dij −Dji), fij =

12(dij − dji).

Their expression is given by the following theorem and it is different from theone we obtained is Section 8.5.

Theorem 8.7.5 The following formulae hold true:

Fij = Fij +14Fij ,

fij = 0.

(8.57)

Proof. We have the following formula that relates the deflection tensors:

Dij = Di

j +14

∂F i

∂yj+ yhCi

hj .

The above formula is equivalent to

Dij = Dij +14gis

∂F s

∂yj+ yhCihj .

Using the defining formula (8.30) for the helicoidal tensor field Fij , we obtainthat the formulae (8.57) are true. q.e.d.

One can write Maxwell equations for the h-electromagnetic field using sim-ilar methods as we did use in Theorem 8.8.


8.8 Almost Hermitian model of a Lagrangian me-chanical system

For a Lagrangian mechanical system Σ, we consider the evolution nonlinearconnection N , the adapted basis (δ/δxi, ∂/∂yi) to the horizontal and verticaldistributions and the N-lift G of the fundamental tensor gij(x, y). The N-liftG is given by

G = gijdxi ⊗ dxj + gijδyi ⊗ δyj .

The evolution nonlinear connection N is perfectly determined by its associatedalmost complex structure F, which is given by

F =δ

δxi⊗ δyi − ∂

∂yi⊗ dxi.

Both structure G and F depend on our mechanical system, only.

Theorem 8.8.11) The pair (G,F) is an almost Hermitian structure on the phase manifold

TM .2) The almost symplectic structure

ω = gijδyi ∧ dxj

is related to the symplectic form ω of the Lagrange space Ln through the fol-lowing formula

ω = ω − 14Fijdxi ∧ dxj .(8.58)

Proof. First part of the theorem follows the proof we did use for Theorem8.10. For the second part we have the following:

ω = gijδyi ∧ dxj = gij

(δyi − 1

4∂F i

∂ysdxs

)∧ dxj = ω − 1

4Fijdxi ∧ dxj .

From above calculation we have that formula (8.58) holds true. q.e.d.

As ω is the symplectic structure of the Lagrange space Ln, we have thatdω = 0. By taking the exterior derivative of (8.58) we obtain:

dω = −14dFij ∧ dxi ∧ dxj .(8.59)

The 3-form dFij ∧dxi∧dxj from the right hand side of the above formula canbe expressed as follows:

dFij ∧ dxi ∧ dxj =13

(δFij

δxk+

δFjk

δxi+

δFki

δxj

)dxi ∧ dxj ∧ dxk

+∂Fij

∂ykδyk ∧ dxi ∧ dxj .

8.9. Generalized Lagrangian mechanical systems 227

Using the covariant derivative

Fij|k =δFij

δxk−FsjL

sik −FisL

sjk

one can rewrite it as:

dFij ∧ dxi ∧ dxj =13(Fij|k + Fjk|i + Fki|j)dxi ∧ dxj ∧ dxk

+∂Fij

∂ykδyk ∧ dxi ∧ dxj .

Using these we can write expression (8.59) as follows:

dω = − 112

(Fij|k + Fjk|i + Fki|j)dxi ∧ dxj ∧ dxk

−14Fij |kδyk ∧ dxi ∧ dxj .

(8.60)

If we use expression (8.60) and the above calculation then we can prove thefollowing theorem:

Theorem 8.8.2 The following statements are equivalent:1) The 2-form ω is a symplectic structure.2) The manifold (TM,G,F) is almost Kahlerian.3) Fij|k + Fjk|i + Fki|j = 0 and Fij |k = 0.

One can conclude that the geometry of the manifold (TM,G,F) is thegeometry of the mechanical system Σ on the phase space TM . (TM,G,F) isthe almost Hermitian model of the mechanical system Σ. Using this model onecan study the Einstein equations of Σ, with respect to the canonical N -linearconnection.

8.9 Generalized Lagrangian mechanical systems

In this section we consider gij a generalized Lagrange metric as we discussedin Chapter 7 and N i

j a nonlinear connection. If gij is weakly regular, thenN i

j is determined only by gij . For this section we assume that the nonlinearconnection N is canonically associated to gij . We can generalize the theoryof Lagrangian mechanical systems to a theory of generalized Lagrangian me-chanical systems (GL-mechanical systems for short) as follows.

Definition 8.9.1 A generalized Lagrangian mechanical system is defined by atriple Σ = (M, gij , Fi), where gij is a GL-metric and Fi are the given externalforces.


Let E(x, y) = gij(x, y)yiyj be the absolute energy of the GL-space GLn =(M, gij). We assume that the evolution equations of the GL-mechanical systemΣ are given by the following Lagrange equations:

ddt

(∂E∂yi

)− ∂E


dxi

dt.(8.61)

Since we assumed that our GL-metric is weakly regular, then the absoluteenergy E is a regular Lagrangian. Consider

∗gij the fundamental metric tensor

induced by E . Then Lagrange equations (8.61) are equivalent to the followingsystem of second order differential equations:

d2xi

dt2+ 2

∗G i

(x,

dx

dt

)=

12F i

(x,

dx

dt

), F i = gisFs, where

2∗G i =

12∗g is

(∂2E

∂ys∂xjyj − ∂E

∂xs

).

(8.62)

At this moment one can apply the theory from the previous sections to theLagrangian mechanical system Σ′ = (M, E(x, y), Fi(x, y)). According to thistheory, the following statements are true:

1) The vector field on TM

∗S= yi ∂

∂xi− 2

( ∗G i − 1

2F i

)∂

∂yi

is a semispray that depends on the mechanical system, only.

2) The integral curves of the evolution semispray∗S are the evolution curves

of the mechanical system Σ.3) The variation of the energy EE is given by

dEEdt

=dxi

dtFi

(x,

dx

dt

), where EE = yi ∂E

∂yi− E .

4) The nonlinear connection∗N of Σ has the local coefficients

∗N i

j =∂∗G i

∂yj− 1

4∂F i

∂yj.

5) The∗N -linear metric connection of the mechanical system Σ has the

coefficients:∗L i

jk =12gih

∗δ gjh

δxk+

∗δ ghk

δxj−

∗δ gjk

δxh

,

∗C i

jk =12gih

(∂gjh

∂yk+

∂ghk

∂yj− ∂gjk

∂yh

).

(8.63)

8.9. Generalized Lagrangian mechanical systems 229

A geometric theory of the GL-mechanical system can be developed using the

fundamental objects: gij , Fi,∗N and the

∗N -linear connection.

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Index

(α, β)-metric,Lagrange space with, 199

(α, β)-metric,Finsler space with, 199

B-change, 191N -linear connection, 47N -linear connection,

curvature, 52horizontal coefficients, 47nonholonomic coefficients, 55, 113symmetric, 51torsion, 50vertical coefficients, 47

β-change, 190ϕ-Lagrange space, 161ϕ-Lagrangian, 135, 161h-metric, 154, 156h-symmetric, 154q-form, 8v-metric, 154, 156v-symmetric, 154(h)h-torsion, 51, 101, 156(v)v-torsion, 51, 101, 156

absolute energy, 175, 185, 227adjoint structure, 28, 36, 46, 180AFL-metric, 177AFL-space, 134, 157, 177almost

complex structure, 30, 36, 150, 219,225

Hermitian structure, 151, 152, 219Kahlerian model, 152Kahlerian space, 151, 220product structure, 30, 36, 65, 180

symplectic structure, 219, 226Almost Finsler Lagrange metric, 177Almost Finsler Lagrange space, 194angular metric, 87, 108, 109, 114anisotropic medium, 125Antonelli metric, 178arclength, 88, 91, 119, 162autoparallel curves, 24, 42, 61, 77, 97,

106, 146, 159, 215

Beildeformation, 198metric, 191, 195, 198

Berwaldbasis, 27, 47, 56, 97connection, 48, 51, 68, 103, 121,

155, 215frame, 109, 112

Berwald connection,curvature, 53, 69flat, 69local coefficients, 68structure equations, 69torsion, 68

Bianchi identities, 169

Caratheodory transformations, 145Cartan

1-form, 85, 136, 141, 1862-form, 85, 99, 136, 151, 186linear connection, 102, 110, 195nonlinear connection, 96, 100, 110symmetry, 94, 142symmetry,

Index 247

exact, 95, 143tensor, 84, 136, 175vector, 109

Cartan’s structure equations, 56, 69,155

Chern connection, 104Chern-Rund connection, 104, 155Christoffel matrix, 124Christoffel symbols, 91, 212Christoffel symbols,

generalized, 154, 183closed 1-form, 95, 143complete lift of

a curve, 65a function, 13, 90, 141a vector field, 13, 37, 72, 73, 94,

142an 1-form, 13

complex structure, 150configuration space, 63, 210connection, 22connection 1-forms, 56, 68, 155connection map, 29conservation law, 72, 94, 142, 171constant of motion, 72, 94, 142contact of order 1, 4cotangent

bundle, 6space, 6structure, 12, 85, 136

covariant derivative, 22, 23Crystallographic connection, 55curvature

2-forms, 56, 69, 155components, 52, 169d-tensor, 33, 42, 74tensor, 33, 43, 52, 169

d-connection, 46d-covector field, 32d-tensor field, 31d-vector field, 32deflection tensors, 105, 157, 160

distinguished tensor field, 31dynamical

covariant derivative, 34, 70, 97,146, 179

symmetry, 38, 72, 94, 142

ecological metric, 133eigenvalue, 124eikonal equation, 125Einstein equations, 152, 170elasticity tensor, 124electromagnetic tensor, 119, 122, 145,

158, 161, 166energy function, 89, 90energy of a Lagrangian, 138energy-momentum tensor, 170Euclidean metric, 126Euler theorem, 16Euler-Lagrange

equations, 89, 118, 138, 189operator, 90, 138

evolutioncurves, 209, 212equations, 209, 212, 220nonlinear connection, 214semispray, 210, 212, 221

external forces, 209, 212, 220external forces,

dissipative, 209, 221

F-metric, 176fibre bundle, 5fifth invariant, 74Finsler

metric, 82, 191, 199space, 83

Finsler connection, 105, 106Finsler frame, 55first integral, 38, 94, 143first invariant, 67, 74flag curvature, 108flow, 7, 94, 143

248 Index

fourth invariant, 74, 77Frolicker-Nijenhuis bracket, 9, 12Frobenius theorem, 33

generalized Finsler space, 194generalized Lagrange metric, 174generalized Lagrange space, 174geodesic

equations, 77, 91spray, 92, 100, 162stability, 112, 115

geodesics, 24, 42, 59, 91, 97, 106, 121,189, 210

GL-metric, 174GL-space, 174

h-connection vector, 114h-covariant derivative, 49, 101Hamilton equations, 126Hamilton-Jacobi equation, 126Hamiltonian gradient, 128Hamiltonian metric, 127Hashiguchi connection, 105, 155helicoidal tensor, 217Helmholtz condition, 179Holland frame, 117holonomic frame, 58homogeneous

connection, 22, 41function, 16q-forms, 17vector field, 16

horizontalcodistribution, 28distribution, 25, 30, 41, 46lift, 26subbundle, 25

horizontal curve, 36, 59, 118, 216horizontal projector, 180

Ingarden space, 121integrable nonlinear connection, 34integral of action, 137

invariant transformation, 38invariant vector field, 94, 142isotropic medium, 125

Jacobiendomorphism, 71, 108equations, 37, 71, 107, 115vector field, 37, 71, 111, 115

kinetic energy, 209Kropina space, 190, 199

L-metric, 175, 178Lagrange

equations, 209, 212, 220, 227Lagrange space, 132Lagrangian subbundle, 100, 147, 148Lagrangian submanifold, 152Lagrangian,

differentiable, 132regular, 132

Liealgebra, 9bracket, 7derivative, 37subalgebra, 10, 33symmetry, 38, 72, 94, 108, 142

linear connection, 22Liouville vector field, 10, 16Lorentz

equations, 119, 146, 159nonlinear connection, 120

Maxwell equations, 161Maxwell equations,

generalized, 166, 218mechanical system,

Finslerian, 211generalized Lagrangian, 227Lagrangian, 220Riemannian, 209

metricconnection, 153

Index 249

nonlinear connection, 35, 99, 149,180

metric tensor, 83Minkovski norm, 82Minkowski metric, 178Miron

frame, 112regular, 185, 187

Miron-Tavakol metric, 175Moor frame, 112

Nijenhuis tensor, 9, 12, 33, 150Noether theorem, 95, 143nonholonomic frame, 54, 201nonlinear connection, 22nonlinear connection,

canonical, 145local coefficients, 66symmetric, 33, 68

normalized supporting element, 87, 108

Obata operators, 99, 149, 181one-parameter group, 7, 40, 144orthogonal vector, 190

parallel transport, 20, 45phase-slowness covector, 125phase-velocity function, 125projector

horizontal, 26, 28, 66vertical, 26, 28, 66

pull-back bundle, 10

R-metric, 178Randers space, 116, 121, 190, 199ray-theory Hamiltonian, 125reducible to

Finsler space, 133Finslerian metric, 176Lagrangian metric, 175Riemannian metric, 83, 178

refractive index, 179regular GL-metric, 189

regular metric, 185Ricci identities, 54, 156, 184Ricci tensor field, 167, 170Rund connection, 104

S-differential, 141Sasaki lift, 151scalar-type curvature, 108, 114Schouten connection, 46second invariant, 71, 74semispray, 9, 66, 119semispray,

canonical, 139, 145local coefficients, 64

SODE, 63, 75SODE,

linearizable, 69spray, 67, 76strongly non-Riemannian space, 112structural

functions, 5group, 5

structure equations, 69, 155symplectic form, 100, 136symplectic structure, 85, 92, 100, 106,

136, 140, 151, 219, 226

tangentbundle, 5space, 4structure, 11, 28, 36, 85, 136, 180vector, 4

tensor field, 8third invariant, 74, 77torsion

2-forms, 56, 68, 155components, 51

transversal vector, 190trivialization chart, 4

v-covariant derivative, 49, 101variational equations, 37variational metric, 185

250 Index

variational problem, 88, 118, 137vertical

distribution, 10endomorphism, 11, 71lift, 13subbundle, 10

vertical lift ofa function, 141

vertical projector, 180Vranceanu identities, 168

wavefront, 125weak torsion, 33, 182, 214weakly normal, 188weakly regular metric, 185weakly variational, 187

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