STUDY ON SOMETRANSFORMATIONS OF

RIEMANN-FINSLER SPACES

THESIS SUBMITTED TO THEDST-CIMS

FACULTY OF SCIENCEIN PARTIAL FULFILMENT OF THE DEGREE OF

Doctor of Philosophyin

Mathematical Sciences

by

Manoj KumarEnrolment No. 341755

UNDER THE SUPERVISION OFDr. Bankteshwar Tiwari

Banaras Hindu UniversityOctober-2014

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Dedicated

To

My Beloved Parents

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COPYRIGHT c⃝, FACULTY OF SCIENCEBANARAS HINDU UNIVERSITY

VARANASI-221005, INDIA

2014.

ALL RIGHTS RESERVED.

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DST-CENTRE FOR INTERDISCIPLINARY MATHEMATICAL SCIENCES

FACULTY OF SCIENCE

BANARAS HINDU UNIVERSITY

VARANASI-221005, INDIA.

UNDERTAKING FROM THE CANDIDATE

I hereby declare that I have completed the research work for the full time pe-

riod described under the clause VIII.(1) of the Ph.D. ordinance of the Banaras

Hindu University, Varanasi and that the research work embodied in this thesis

entitled “STUDY ON SOME TRANSFORMATIONS OF RIEMANN-

FINSLER SPACES” is my own work.

Date: Manoj Kumar

Place: Varanasi (Research Scholar)

CANDIDATE’S DECLARATION

I, Manoj Kumar certify that the work embodied in this Ph.D. thesis as “Study

on some transformations of Riemann-Finsler spaces ” is my own bonafide work

carried out by me under the supervision of Dr. Bankteshwar Tiwari for a

period from September 2011 to October 2014 at DST-CIMS, Banaras Hindu

University, Varanasi. The matter embodied in this Ph.D. thesis has not been

submitted for the award of any other degree/diploma.

I declare that I have faithfully acknowledged, given credit to and referred to

the research workers wherever their works have been cited in the text and the

body of the thesis. I further certify that I have not willfully lifted up some other’s

work, para, text, data, results, etc. reported in the journals, books, magazines,

reports, dissertations, thesis, etc., or available at web-sites and included them in

this Ph.D. thesis and cited as my own work.

Date: Manoj Kumar

Place: Varanasi (Research Scholar)

CERTIFICATE FROM THE SUPERVISOR

This is to certify that the above statement made by the candidate is correct to

the best of my knowledge.

Prof. Umesh Singh Dr. B. Tiwari

(Co-ordinator) (Supervisor)

DST-CENTRE FOR INTERDISCIPLINARY MATHEMATICAL SCIENCES

FACULTY OF SCIENCE

BANARAS HINDU UNIVERSITY

VARANASI- 221005.

COURSE WORK ANDPRE-SUBMISSION SEMINAR COMPLETION

CERTIFICATE

This is to certify that Mr. Manoj Kumar, a bonafide research scholar of

this centre, has successfully completed the Ph.D. course work and pre-submission

seminar requirement which is a part of his Ph.D. programme, on his thesis en-

titled, “STUDY ON SOME TRANSFORMATIONS OF RIEMANN-

FINSLER SPACES”.

Date: Prof. Umesh Singh

Place: Varanasi (Co-ordinator)

COPYRIGHT TRANSFER CERTIFICATE

Title of the Thesis: STUDY ON SOME TRANSFORMATIONS OF RIEMANN-

FINSLER SPACES.

Candidate’s Name: Manoj Kumar

Copyright Transfer

The undersigned hereby assigns to the Banaras Hindu University all rights under

copyright that may exist in and for the above thesis submitted for the award of

the Ph.D. degree.

Copyright c⃝, Faculty of Science

Banaras Hindu UniversityVaranasi-221005, INDIA

2014.

All rights reserved.

Manoj Kumar

Note: However, the author may reproduce or authorize others to reproduce ma-

terial extracted verbatim from the thesis or derivative of the thesis for author’s

personal use provided that the source and the University’s copyright notice are

indicated.

Acknowledgements

I am indebted to many people for their assistance and inspiration in this work.

First of all, I would like to express my sincere and deep gratitude to my supervisor;

Dr. Bankteshwar Tiwari, Associate Professor, DST-CIMS; for introducing this

interesting topic; his patience and guidance. He has always supported my Ph.D.

program with trust and enthusiasm, believing in my capabilities, doing all that

he could to improve my knowledge and sharing his passion for academic research

and teaching.

I am thankful to various Professors of B. H. U., in particular, Prof. Umesh

Singh (Co-ordinator, DST-CIMS), Prof. A. K. Shrivastava (Dean, Faculty of

Science and Additional Co-ordinator, DST-CIMS), Prof. R. S. Pathak (Ex-Head

and Professor, Deptt. of Mathematics, BHU), Prof. M. M. Tripathi (Deptt. of

Mathematics, BHU), Dr. M. K. Singh (DST-CIMS), Dr. R. Chaubey (DST-

CIMS) and Dr. S. K. Upadhayay (Deptt. of Mathematical Sciences, IIT, BHU)

for their valuable time and guidance during the Ph.D. program.

I would be failing my duty if I do not express my sincere thanks to Dr. Ab-

hishek Singh (Post doctoral fellow, DST-CIMS) for assisting with library searches

and other help.

All my work is a generous, wise and understanding critic. Our tireless and

efficient office assistant Manish Srivastava suffered through many drafts and was

able to guide me through the complexities of organizing the manuscript. I have

benefited from the helpful staff and the resources of the DST-CIMS, Faculty of

Science, B. H. U.

I am grateful to the DST-Centre for Interdisciplinary Mathematical Sciences

(CIMS) for awarding Junior Research Fellowship in march 2012 and Senior Re-

search Fellowship in march 2014. I relied heavily on the excellent supportive work

of my fellow Anjani Kumar Shukla, Dhram Raj Singh and Pranav Waila for ad-

vanced understanding of MS office, Latex, Matlab and Mathematica.

I am also conscious to suggestions made from the helpful elders researchers

and the resources of the DST-CIMS. It can not be expressed in words the helping

and cooperating nature of Dr.R. Chaubey (DST-CIMS) and Dr.Sapna Devi

(Deptt.of Mathematics, University of Allahabad, Allahabad) who motivated

me to do researches of interdisciplinary nature, specially related to other areas

Cosmology and Modeling, respectively.

A special thanks to my friends Dinesh Kumar, Ghanashyam Kr. Prajapati,

Ranadip Gangopadhyay, Ashutosh Singh of DST-CIMS and my Senior Vishal

Singh, Krishna Kumar Singh of Deptt. of Mathematics (BHU Varanasi), who

were kind enough to read a very rough first draft of the manuscript and offer

comments and advice.

Thanks to my old friends in Delhi and the new ones, who met along the way,

in Varanasi as well as all over India, during my Ph.D. period for their love and

support given me in hard times.

My thanks are also due to those who either directly or indirectly helped and

encouraged me for this work.

Last but not least, I owe a great debt of gratitude to my father Shri Harish

Chandra Verma and my mother Smt. Kanya Wati for their endless patience, love

and support throughout my schooling, for their blessing and encouraging me to

excel in every aspect of life, which always motivate me to choose a carrier in

research in Mathematical Sciences.

—Manoj Kumar

Preface

The object of this thesis is to study some transformations of Riemann-Finsler

spaces. For instance, Randers space may be treated as a little deformation of

a Riemannian space. More generally Finsler spaces with (α, β)- metrics may be

treated as a deformation of Riemannian space. To avoid referring to previous

knowledge of Riemann-Finsler Geometry, we include chapter 1, which contains

those concepts and results on Riemann-Finsler Geometry which are used in an

essential way in the rest of the book.

With this approach in mind, the thesis has been divided into Seven chapters.

Chapter 1 contains some definitions and literature survey relevant to the proposed

problems. A brief introduction of Riemann-Finsler geometry with some historical

development is given.

Thereafter in chapter 2, a special (α, β)- metric, which is considered as a

generalization of the Randers metric as well as of the Z. Shen’s square metric, has

been studied and the conditions for a Finsler space with this special metric to be

a Berwald space, a Douglas space and Weakly-Berwald space respectively, have

also been found.

In the chapter 3, we find a condition under which a Finsler space with

Randers change of m-th root metric is projectively related to a m-th root metric

and also we find a condition under which this Randers transformed m-th root

Finsler metric is locally dually flat. Moreover, if transformed Finsler metric is

conformal to the m-th root Finsler metric, then we prove that both of them reduce

to Riemannian metrics.

In chapter 4, we study the conformal transformation of m-th root Finsler

metric. The spray coefficients, Riemann curvatures and Ricci curvature of confor-

mally transformed m-th root metrics are shown to be certain rational functions of

direction. Further under certain conditions it is shown that a conformally trans-

formed m-th root metric is locally dually flat if and only if the transformation is

a homothety. Moreover the conditions for the transformed metrics to be Einstein

and Isotropic mean Berwald curvature are also found.

Chapter 5 is devoted to study the properties of a modified Finsler space

obtained by transformation of a Finsler space with the help of two normalised

semi-parallel vector fields.

The last two chapters are devoted to applications of Finsler spaces in Mathe-

matical Modelling and Mathematical Cosmology which shows the interdisciplinary

nature of our work.

In the second last chapter 6, the dynamics of a predator-prey model is pro-

posed and analyzed. Three types of refuges: those that protect a constant number

of prey population, a constant proportion of prey population and a function of

predator-prey encounters using refuges are considered. Linear stability analysis

based on Lyapunov theory and Jacobi stability analysis based on KCC theory are

carried out. Comparisons of results obtained in both cases shows that, Jacobi

stability analysis of these models reflects the better ecological interpretation.

In chapter 7, Finsler-Randers cosmological models in modified gravity the-

ories have been investigated. The de Sitter, power law and general exponential

solutions are assumed for the scale factor in the corresponding cosmological mod-

els. For each scenario, we have discussed all energy conditions in detail. We have

also investigated the behaviour of FR cosmological models in modified theories of

gravity like Einstein theory, Hoyle-Narlikar Creation field theory, Lyra geometry

and General class of scalar-tensor theories.

In the last a list of number of books and research papers on the subject is

given in References.

ii

Contents

Preface i

1 Introduction 1

1.1 Historical Development of Finsler Geometry . . . . . . . . . . . . . 1

1.2 The Geometry of Finsler Spaces . . . . . . . . . . . . . . . . . . . . 3

1.3 Review of Literature about Finsler Geometry . . . . . . . . . . . . 6

1.4 Connections and Covariant Differentiations . . . . . . . . . . . . . . 11

1.5 Flag curvature and S-curvature in Finsler geometry . . . . . . . . . 17

1.6 Some special Finsler spaces . . . . . . . . . . . . . . . . . . . . . . 19

1.7 KCC theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.8 Mathematical Cosmology in Finslerian space-time . . . . . . . . . . 22

2 On Finsler space with a special (α, β)-metric 24

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 The condition to be a Berwald space . . . . . . . . . . . . . . . . . 27

2.3 The condition to be a Douglas space . . . . . . . . . . . . . . . . . 28

2.4 The condition to be a Weakly-Berwald space . . . . . . . . . . . . . 32

3 On Randers change of a Finsler space with m-th root metric 38

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Fundamental metric tensor of Randers transformed m-th root metric 41

3.4 Spray coefficients of Randers transformed m-th root metric . . . . 42

3.5 Conformally related Randers transformed m-th root metric . . . . 46

iii

3.6 Locally dually flatness of Randers transformed m-th root metric . . 49

4 On Conformal Transformation of m-th root Finsler metric 51

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Fundamental tensor and Spray coefficients of conformally trans-

formed m-th root metric . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Locally dually flat conformally transformed m-th root metric . . . . 55

4.5 Conformally transformed Einstein m-th root metric . . . . . . . . . 57

4.6 Conformally transformed m-th root metric with Isotropic E-curvature 58

5 Transformation of a Finsler Space by Normalised Semi-Parallel

Vector Fields 60

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Transformed Finsler space obtained by Normalised semi-parallel

vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Special Finsler spaces with semi-parallel vector fields . . . . . . . . 65

6 Predator-prey model with prey refuges: Jacobi stability vs Linear

stability 68

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Applications of geometric theory to second order system . . . . . . 72

6.4 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.5 Numerical simulations and discussion . . . . . . . . . . . . . . . . . 78

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Finsler-Randers Cosmological models in Modified Gravity Theo-

ries 87

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.2 Finsler-Randers Cosmological Model in Einstein Theory . . . . . . . 89

7.3 Finsler-Randers cosmological model in Lyra geometry . . . . . . . . 97

iv

7.4 Finsler-Randers Cosmological Model in General Class of Scalar-

tensor theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.5 Finsler-Randers Cosmological Model in C-field theory . . . . . . . . 109

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Bibliography 116

v

Chapter 1

Introduction

1.1 Historical Development of Finsler Geometry

The differential geometry of Non-Euclidean spaces was studied systematically by

German mathematician B. Riemann (1854). He proposed metrics which are more

general than Euclidean. One metric, which is a quadratic differential one form,

is called Riemannian metric. Einstein’s general theory of relativity (1915) was its

first important and fully successful application. In the renowned lecture of 1854 ‘

Uber die Hypothesen, welcheder Geometrie zu Grunde liegen’ (On the Hypotheses,

which lie at the Foundations of Geometry), Riemann had himself conjectured the

existence of a geometry which is more general than the one proposed by him.

More than sixty years after his lecture a systematic study of generalized geometry,

began after the publication of a thesis by P. Finsler (1894-1970) in 1918, called

Finsler geometry. Finsler geometry has started with Finsler’s famous dissertation

under the supervision of C. Caratheodory who intended to geometrize the calculus

of variations.

The calculus of variations with specific contact to other geometrical back-

ground, are closely related by Finsler geometry and which was introduced by

Caratheodory in connection with problems in parametric form. In fact, the noti-

fiable relationship between some contact of differential geometry and the calculus

of variations have been seen some years ago to the publication of Finsler’s thesis,

in particular by Bliss (1906), Landsberg (1908) and Blaschke (1923). Definitions

of angle in terms of invariants of a parametric problem in the calculus of varia-

tion was introduced by Bliss and Landsberg. Although the geometrical theories

1

given by Bliss and Landsberg were against an Euclidean background, which were

not supporting the true objectives of the generalization of Riemann’s proposal.

Finsler’s thesis must be regarded as the first step in this direction.

In the year 1900 in Paris, the problem of geometry of the calculus of variations

was the one of the problems formulated by Hilbert. This problem was posed by

Hilbert without any formulation of specific question or conjecture. By providing

an account to the invariant integral, as well as emphasizing on the importance of

the problem of multiple integrals, Hilbert praised Kneser’s book also.

In all modern treatments of the subject, the role of Hilbert invariant integral

is crucial one. The geometrical idea in Finsler geometry is of a smoothly varying

family of Minkowski norms, rather than a family of inner products and this family

of Minkowski norms is known as a Finsler structure. In contrast to Riemannian ge-

ometry, there is also an equivalence problem that states that ‘how transformation

induced from a coordinate change can affect two given Finsler structures’? It is

quite reasonable to expect that a connection and its curvature, will be involved in

the solution of the equivalence problem in permissible set up of Finsler geometry.

A Finsler manifold is a manifold where each tangent space is equipped with

a Minkowski norm, that is, a norm that is not necessarily induced by an inner

product. This norm also induces a canonical inner product. However, in sharp

contrast to the Riemannian case, these Finsler-inner product does not only depend

on where you are, but also in which direction you are looking. Despite this quite

large step away from Riemannian geometry, Finsler geometry contains analogues

for many of the natural objects in Riemannian geometry. For example, length,

geodesics, curvature, connections, covariant derivative, and structure equations.

The history of development of Finsler geometry can be divided into four periods.

The first period of the history of development of Finsler geometry began in

1924, when three geometers J. H. Taylor, J. L. Synge and L. Berwald simulta-

neously started work in this field. However according to Matsumoto the creator

of this geometry is L. Berwald in 1925. The name “Finsler geometry” was first

given by J. Taylor in 1927.

The second period began in 1934, when E. Cartan published his thesis on

2

Finsler geometry. He showed that is was indeed possible to define connection

coefficients and hence covariant derivatives such that the Ricci lemma is satisfied.

On this basis Cartan developed the theory of curvature and torsion. All subsequent

investigations considering the geometry of Finsler spaces were dominated by this

approach. Several Mathematicians such as E. T. Davies, S. Golab, O. Varga, V.

V. Wagner have studied Finsler geometry along Cartan’s approach.

The third period of the history of Finsler geometry began in 1951, when H.

Rund introduced a new process of parallelism from the stand-point of Minkowskian

geometry. Cartan had introduced parallelism from the stand-point of locally Eu-

clidean geometry. Later on E. T. Davies and A. Diecke showed that the two con-

cepts of parallelism were the same. Several Mathematicians such as W. Barthel,

A. Diecke, D. Laugwitz have studied Finsler spaces on Rund’s approach.

The fourth period of the history of developmental of Finsler geometry began

in 1963, where H. Akabar developed the modern theory of Finsler spaces based

on the geometry of connections of fibres bundles. The reason of modernization is

to establish a global definition of connections in Finsler spaces and to re-examine

Cartan’s system of axioms. Mathematicians and Physicists began to study special

Finsler spaces since Matsumoto organised Symposium on the models of Finsler

spaces in 1970.

1.2 The Geometry of Finsler Spaces

The goal which Riemann set for himself was the definition and discussion of the

most general finite-dimensional space in which every curve has a length derived

from an infinitesimal length or line element. In modern terminology Riemann’s

approach is this: Let a differentiable manifold M of a certain class be given. In

any local coordinate system (x1, ..., xn) = (x) a length F (x, dx) must be assigned

to a given line element (x, dx) = (x1, ..., xn; dx1, ..., dxn) with origin x. If x(t) is a

(smooth) curve in M then∫F (x, x)dt is its length.

In order to insure that the length of a curve is positive and independent of the

3

sense in which the curve is traversed, Riemann requires F (x, dx) > 0 for dx = 0

and F (x, dx) = F (x,−dx).

Next Riemann assumes that the length of the line element remains unchanged

except for terms of second order, if all points undergo the same infinitesimal

change. This amounts to the condition F (x, kdx) = kF (x, dx) for k > 0. Now a

days we rather justify this condition by requiring that a change of the parametriza-

tion of the curve does not change its length. Riemann then turns immediately to

the special case where F (x, dx) =√∑

gij(x)dxidxj, that is, to those spaces which

are now called Riemann spaces. The general case is passed over with the following

remarks: the next simplest case would comprise the manifolds, in which the line

element can be expressed as the fourth root of a bi quadratic differential form.

The investigation of these more general types would not require any essentially

different principles, but it would be time consuming and contribute comparatively

little new to the theory of space (verhaltnismassig auf die Lehre vom Raume wenig

neues Licht werfen), because the results cannot be interpreted geometrically.

Here is one of the few instances where Riemann’s feeling was wrong. Nev-

ertheless the passage had a great influence: the general case was for along time

entirely neglected, and when it was taken up the principles of Riemannian geom-

etry were applied. The results thus obtained are not different enough to enrich

geometry materially, moreover they frequently do not lend themselves to a naive

geometric interpretation.

Finsler was the first who studied the general spaces systematically in his thesis

written under Caratheodory’s guidance. His main idea is this: if a curve is given,

then a field which contains the curve defines a Riemann metric as above. Those

results which are independent of the choice of the field are the real geometric

properties of the curve. In this way Finsler succeeded in developing a theory of

curves, and also the foundations for a theory of surfaces. This is the reason why

the name Finsler space was generally accepted for the general class of spaces first

defined by Riemann.

Then a new line of thought developed in the geometric school at Prague with

Berwald, Funk, and Winternitz as principle representatives. A Finsler space is

4

not considered as a point space but primarily as a set of line elements in which a

Riemannian metric is associated with each line element. The main emphasis in this

theory lies on the definitions and properties of parallelism and similar questions.

The whole development culminated in Cartan’s monograph, which is considered

to have given the theory of Finsler spaces its final form.

Under the restriction to use nothing but Riemannian methods it may be true

that the theory has reached its limits. However, as soon as this restriction is

dropped it appears that the surface has hardly been scratched.

A first, but not the most interesting or decisive, step beyond the mentioned

results consists in extending the results of Riemannian geometry rather than its

methods.

Finsler geometry is an alternative approach to geometrization of fields, and

its fundamental idea can be traced back to a lecture of Riemann (1854). In this

lecture Riemann suggested that the positive fourth root of a fourth order differen-

tial form might serve as a metric function. This function has three properties that

it is convex and common with the Riemannian quadratic form it is positive and

homogeneous of degree one in the differentials. Therefore, it is a natural general-

ization of the notion of distance between two neighbouring points xi; xi + dxi to

consider as given by some function F (xi; dxi), where i = 0, ..., n, satisfying these

three properties.

A systematic study of these kind of manifolds was first considered by Funk

(1929), and in 1925, the method of tensor calculus were applied to the theory of

Berwald (1947) independently but almost simultaneously by Synge, Taylor and

Berwald. It was found that the second derivatives of 12F 2(xi, dxi) with respect

to differentials serves as components of a metric tensor in analogy with Riemann

geometry. By this process, parallel displacements and connection coefficients in

Finsler spaces are defined, but with these connections Ricci lemma was no longer

valid. Cartan (1934) showed that it was indeed possible to define connections

and a covariant derivative so that Ricci lemma is preserved. This development is

closely related to the present application of Finsler geometry in physics, namely,

to geometrize both Cosmology and gravity simultaneously.

5

Finsler geometry was first applied in gravitational theory, and this application

lead to corrections to observational results predicted by general relativity.

As mentioned before, the main application of Finsler geometry is the ge-

ometrization of electromagnetism and gravitation. A Finslerian approach to this

geometrization was first introduced by Randers (1941), but in his work Finsler

geometry was not mentioned, although it was used. Randers metric produces a

geodesic equation identical with Lorentz equation for a charged particle. But the

metric depends on qm

and defines a different space for each type of particle by

Ingarden (1976).

Some (α, β)- metrics are important for Cosmology, in application point of

view. Presently Finsler geometry has many developments which focus to the

researchers. Now days Finsler geometry has found an abundance of applications

in both physics and practical applications.

1.3 Review of Literature about Finsler Geometry

Finsler geometry has been studied and developed by geometers like J. H. Taylor,

L. Berwald, E. Cartan, E. T. Davis, O. Varga, A. Deicke, M. Matsumoto, H.

Shimada, S. S. Chern, D. Bao, Z. Shen, L. Tamassy and R. Miron etc. This

geometry has the crucial applications due to G. S. Asanov, P. L. Antonelli, R. G.

Beil, R. M. Santilli, and R. S. Ingarden, etc.

1.3.1 Differentiable Manifold

An n-dimensitional differentiable manifold is a set M together with a family of

injective maps fi : Ui ⊂ Rn → fi(Ui) ⊆M of open sets Ui in Rn into M such that

(i) ∪ifi(Ui) =M ,

(ii) For each pair i, j with fi(Ui) ∩ fj(Uj) = W = ϕ,

the sets f−1i (W ) and f−1

j (W ) are open sets in Rn and f−1i ofj, f−1

j ofi are differen-

tiable.

(iii) The family (Ui, fi) is maximal relative to (i) and (ii).

Examples.

6

1. Rn is an n-dimensitional differentiable manifold.

2. Let Sn be the standard unit sphere in Rn+1 defines as

Sn :=

ξ = (ξi) ∈ Rn+1 : |ξ| =

√n+1∑i=1

(ξi)2 = 1

is n-dimensitional differentiable

manifold.

1.3.2 Minkowski Norm

Definition 1.1 A Minkowski norm on a finite dimensional (n−dim) vector

space V is a function F : V → [0,∞) such that

(i) F (y) ≥ 0 for any y ∈ V and F (y) = 0 if and only if y = 0,

(ii) F is positively 1-homogeneous, i.e. F (λy) = λF (y), for every y ∈ V and

λ > 0,

(iii) F is C∞ on V \0 such that for any y ∈ V , the bilinear symmetric functional

gy on V , defined as gy(u, v) := 12

∂2

∂s∂t[F 2(y + su + tv)]s=t=0, is an inner product,

i.e. for a fixed basis bi of V and y = yibi = 0 the matrix (gij(y)) defined by

gij(y) := gy(bi, bj) =12

∂2

∂yi∂yj[F 2] (y) should be positive definite.

The inner product gy is called fundamental form in the y direction. The pair

(V, F ) is called a Minkowski space. A Minkowski norm is said to be reversible if

F (−y) = F (y) for all y ∈ V .

Examples

1. Let V = Rn and y = (yi) ∈ Rn, |y| :=√

n∑i=1

(yi)2 be standard Euclidean norm,

then F (y) = |y| is a Minkowski norm and (Rn, F ) is a Minkowski space.

2. Let V = R2 and for y = (y1, y2) ∈ R2, let F (y) = ((y1)4 + (y2)4)14 , called quar-

tic norm. This F is NOT a Minkowski norm on R2, because det(gij) = 3(y1)2(y2)2

(y1)4+(y2)4

vanishes on co-ordinate axes and hence positive definiteness of matrix (gij) is vi-

olated at some nonzero y.

3. Let V = R2 and for y = (y1, y2) ∈ R2, let F (y) =√√

(y1)4 + (y2)4 + λ [(y1)2 + (y2)2];

this may be viewed as a perturbation of the quartic norm. This F is a Minkowski

norm on R2 for positive λ.

7

1.3.3 Finsler space

Let M be an n-dimensional C∞-manifold. TxM denotes the tangent space of M

at x. The tangent bundle of M is the union of tangent spaces TM :=∪

x∈M TxM .

We denote the elements of TM by (x, y) where y ∈ TxM . Let TM0 = TM \ 0

is called slit tangent bundle.

Definition 1.2 A Finsler metric on M is a function F : TM → [0,∞) with

the following properties:

(i) F is C∞ on slit tangent bundle TM0,

(ii) F is positively 1-homogeneous on the fibers of tangent bundle TM ,

and (iii) the Hessian of F 2

2with element gij(x, y) = 1

2∂2F 2

∂yi∂yjis positive definite on

TM0.

The pair (M,F ) is then called a Finsler space. A Finsler metric F = F (x, y) on a

manifold is said to be reversible if F (x,−y) = F (x, y) for all y ∈ TxM . Normally,

one does not impose this reversibility condition on a Finsler metric, because it

excludes some interesting examples such as a Randers metric.

A Finsler metric F on M is said to be Riemannian, if the restriction of F , Fx(y) :=

F (x, y) on TxM is a Euclidean norm for any x ∈ M ; that is, Fx(y) =√⟨y, y⟩

x,

for any y ∈ TxM , where ⟨, ⟩x is an inner product on TxM . One usually denotes a

Riemannian metric by a family of inner products gx = ⟨y, y⟩x, on tangent spaces

TxM .

The Riemannian metrics, which are reversible Finsler metrics, are important ex-

amples of Finsler metrics.

Examples [Tiwari (2012a)]

1. Euclidean metric

Let |y| :=

√n∑

i=1

(yi)2, y = (yi) ∈ Rn be the standard Euclidean norm on Rn.

Considering the identification TxRn ∼= Rn, define α0 = α0(x, y) by α0 := |y|, for

y ∈ Rn, then α0 is a Finsler metric called the standard Euclidean metric.

2. Spherical metric

Let Sn be the standard unit sphere in Rn+1 defines as

8

Sn :=

ξ = (ξi) ∈ Rn+1 : |ξ| =

√n+1∑i=1

(ξi)2 = 1

.

Then spherical metric is given by

α+1 := α+1(y) =

√|y|2 + (|x|2|y|2 − ⟨x, y⟩2)

1 + |x|2,

where notation ⟨, ⟩ denotes usual Euclidean inner product on Rn.

3. Klein Hyperbolic metric

Let Hn+ be upper portion of hyperboloid of two sheets given by

Hn := ξ = (ξi) ∈ Rn+1 : −(ξ1)2 − (ξ2)2 − ...− (ξn)2 + (ξn+1)2 = 1.

Then Klein Hyperbolic metric is given by

α−1 := α−1(y) =

√|y|2 − (|x|2|y|2 − ⟨x, y⟩2)

1− |x|2.

The three metrics in Examples (1), (2) and (3) can be incorporated in a single

formula:

αµ =

√|y|2 + µ(|x|2|y|2 − ⟨x, y⟩2)

1 + µ|x|2, y ∈ TxDn(rµ) ∼= Rn,

where rµ := 1√−µ

if µ < 0 and rµ := +∞ if µ ≥ 0.

4. Poincare Hyperbolic metric

We can construct Poincare Hyperbolic metric through stereographic projection of

the upper portion of the hyperboloid

HnR := ξ = (ξi) ∈ Rn+1 : −(ξ1)2 − (ξ2)2 − ...− (ξn)2 + (ξn+1)2 = R2

onto the ball Dn2R := x ∈ Rn : |x| < 2R, as follows

F = 4R2 |y|4R2 − |x|2

.

5. Funk metric

Let

Θ :=

√|y|2−(|x|2|y|2−⟨x,y⟩2)+⟨x,y⟩

1−|x|2 , y ∈ TxDn ∼= Rn.

Then Θ = Θ(x, y) is a Finsler metric on Dn, called the funk metric on Dn.

6. Berwald metric

Let

B :=

(√|y|2−(|x|2|y|2−⟨x,y⟩2)+⟨x,y⟩

)2

(1−|x|2)2√

|y|2−(|x|2|y|2−⟨x,y⟩2),

where y ∈ TxDn ∼= Rn. Then B = B(x, y) is a Finsler metric on Dn, first con-

structed by L. Berwald.

9

7. Randers metric

This metric was first introduced by a physicist named Randers (1941), who was

concerned with the unified field theory of gravitation and electromagnetism. His

justification was: ‘Perhaps the most characteristic property of the physical world is

the uni direction of time-like intervals. Since there is no reason why this asymme-

try should disappear in the mathematical description, it is of interest to consider

the possibility of a metric with asymmetrical property’. Later on, Ingarden (1957)

also used this metric in the context of electron microscopes and called it the Ran-

ders metric, for the first time, taking the name of G. Randers. Let α =√aij(x)yiyj

is a Riemannian metric and β = bi(x)yi is a one-form. Then the metric F = α+β

is a Finsler metric, called Randers metric.

8. (α, β)-metrics

M. Matsumoto has introduced the concept of (α, β)- metric in 1972 as a general-

ization of Randers metric.

A Finsler metric F on a differentiable manifold M is called an (α, β)-metric, where

α is a Riemannian metric α =√aij(x)yiyj and β is a one-form β = bi(x)y

i, if

F is a positively homogeneous function of degree one in α and β. The study of

some important (α, β) metrics such as Randers metric F = α+β, Kropina metric

F = α2

β, Matsumoto metric F = α2

α−β, generalized Kropina metric F = αn+1

βn and Z.

Shen’s square metric F = (α+β)2

αhave greatly contributed to the growth of Finsler

geometry and its applications to the theory of relativity and Cosmology and other

areas.

An (α, β)-metric can be expressed in the form F = αϕ(s) where ϕ is C∞ posi-

tive function and s = βα. F = αϕ(s) is a Finsler metric for any α and β with

b = ||β||α < b0 if and only if

ϕ(s) > 0, ϕ(s)− sϕ′(s) + (b2 − s2)ϕ′′ (s) > 0 , (|s| ≤ b < b0).

9. m-th root metrics

The concept of m-th root metric was introduced by Shimada (1979) and is given

by F := ai1i2...im(x)yi1yi2 ...yim1m , where ai1...im is a symmetric covariant tensor

of order m. The m-th root metric has been successfully applied to ecology by An-

tonelli et al. (1993) and studied by several authors [Srivastava and Arora (2012),

10

Tayebi et al. (2012), Tayebi and Najafi (2011), Yu and You (2010) and Prasad and

Dwivedi (2002)]. It is regarded as a generalization of Riemannian metric in the

sense that the second root metric is a Riemannian metric. For m = 3, it is called

a cubic Finsler metric and for m = 4, it is quartic metric. In four-dimension, the

special fourth root metric in the form F = 4√y1y2y3y4 is called the Berwald-Moor

metric.

1.3.4 Geodesics and Spray coefficients

A smooth curve (C∞) in a Finsler space is a geodesic if it has constant speed and

is locally minimizing. Thus a geodesic in (M,F ) is a curve c : I = [a, b] → M

with F (c(t), c(t)) = constant and for any t0 ∈ I, there is a small number ϵ > 0

such that c is minimizing on [t0 − ϵ, t0 + ϵ] ∩ I.

Using the calculus of variations, one can show that geodesics in a Finsler space F n

are given by a system of second order ordinary differential equations. If a geodesic

is represented locally by the equations xi = xi(t), i = 1, 2, ..., n for an arbitrary

parameter t, then the equations of a geodesics are given by

d2xi

dt2+Gi

(x,dx

dt

)= 0,

where

Gi =1

4gil[F 2]xkyl

yk −[F 2]xl

are called the spray coefficients of F n.

1.4 Connections and Covariant Differentiations

1.4.1 Finsler connection

A Finsler connection FΓ on a manifold M is a triad (F ijk, N

ik, C

ijk) of h-connection

F ijk, a non-linear connectionN i

k and a vertical connection C ijk such that h-covariant

derivative Kij|k and v-covariant derivative Ki

j|k of a Finsler tensor field Kij is given

by

Kij|k =

δKij

δxk+Km

j Fimk −Ki

mFmjk

11

and

Kij|k =

∂Kij

∂yk+Km

j Cimk −Ki

mCmjk,

whereδ

δxk≡ ∂

∂xk−N j

k

∂

∂yj.

The components of five torsion tensors T ijk, R

ijk, C

ijk, P

ijk, S

ijk and three curva-

ture tensors in terms of coefficients FΓ = (F ijk, N

ij , C

ijk) of the Finsler connection

can be written as:

Torsion Tensors

(h) h-torsion:

T ijk = F i

jk − F ikj, (1.1)

(v) h-torsion:

Rijk =

δN ij

δxk− δN i

k

δxj, (1.2)

(h) hv-torsion: Cijk,

(v) hv-torsion:

P ijk = ∂kN

ij − F i

kj, (1.3)

(v) v-torsion:

Sijk = C i

jk − C ikj. (1.4)

Curvature Tensors

h-curvature:

Rihjk =

δF ihj

δxk− δF i

hk

δxj+ Fm

hjFimk − Fm

hkFimj + Ci

hmRmjk, (1.5)

hv-curvature:

P ihjk = ∂kF

ihj − Ci

hk|j + CihmP

mjk , (1.6)

v-curvature:

Sihjk = ∂kC

ihj − ∂jC

ihk + Cm

hjCimk − Cm

hkCimj. (1.7)

The deflection tensor field Dik of a Finsler connection FΓ is given by

Dik = yjF i

jk −N ik. (1.8)

12

The Ricci identities w.r.to FΓ can be written as:

Khi|j|k −Kh

i|k|j = KriR

hrjk −Kh

rRrijk −Kh

i|rTrjk −Kh

i|rRrjk,

Khi|j|k −Kh

i|k|j = Kri P

hrjk −Kh

r Prijk −Kh

i|rCrjk −Kh

i|rPrjk,

Khi |j|k −Kh

i |k|j = Kri S

hrjk −Kh

r Srijk −Kh

i|rSrjk.

In Riemannian geometry, there is a unique connection, called the Riemannian

connection, which was introduced by Levi-Civita using Christoffel symbols. It has

two remarkable properties.

(1) The connection is compatible with the metric. In other words, the covariant

derivative gij|k of the metric tensor vanishes. This means

gij|k =∂gij(x)

∂xk− grj(x)Γ

rik(x)− gir(x)Γ

rjk(x) = 0.

Here Γijk denote the Christoffel symbols and given by

Γijk =

1

2gri(x)

(∂grj(x)

∂xk+∂grk(x)

∂xj− ∂gjk(x)

∂xr

).

(2) Torsion is zero. This means that Γijk(x) = Γi

kj(x).

Although in Finsler geometry there is no such connections that satisfy above two

conditions. However there are several connections in Finsler geometry such as

the Berwald connection, Cartan connection and Chern connection. In 1926, L.

Berwald was the first to introduce the concept of connection in Finsler geom-

etry. Berwald started his theory from the equations of geodesics to define the

Berwald connection. Berwald’s work stems from the study of systems of differen-

tial equations and is very much rooted in the calculus of variations. The Berwald

connection and Chern connection are not compatible with the metric, but it is

torsion free. The Cartan connection is compatible with the metric, but its torsion

is non-zero.

1.4.2 Berwald connection

Definition 1.3 The Berwald connection BΓ(Gijk, G

ij, 0) of a Finsler space

F n with Finsler metric F is a Finsler connection which is uniquely determined by

13

the following five axioms:

B1) (h) h−torsion free : T ijk = 0,

B2) F is h−metrical : F|i = 0,

B3) (h) hv−torsion free : C ijk = 0,

B4) (v) hv−torsion free : P ijk = 0,

B5) Deflection free : Dik = 0.

The connection coefficients of BΓ are given by

Gij = ∂jG

i, Gijk = ∂jG

ik,

where Gi are spray coefficients of the Finsler metric F .

The h-curvature tensor of Berwald connection BΓ is H ihjk, given by

H ihjk = π(jk)

[δkG

ihj +Gr

hjGirk

],

where π(jk) stands for interchange of indices j, k and subtraction.

The surviving curvature and torsion tensors of BΓ are as follows:

The hv-curvature tensor of Berwald connection BΓ is denoted by Bihjk also known

as Berwald curvature, given by

Bihjk = ∂hG

ijk.

The (h) torsion tensor Rijk is given by

Rijk = H i

0jk = δkGij − δjG

ik.

1.4.3 Cartan connection

It is an amusing irony that although Finsler geometry starts with only a norm

in any given tangent space, it regains an entire family of inner products, one for

each direction in that tangent space. This is why one can still make sence of

metric-compatibility in the Finsler setting.

Cartan (1934) introduced a connection called the Cartan connection which

was metric-compatible but not torsion free. The Cartan connections remain, to

this day, immensely popular with the Matsumoto and the Miron schools of Finsler

14

geometry. Besides the curvature tensors of hh-and hv- type, there is a third

curvature tensor associated with the Cartan connection. It is of vv- type. In

Cartan’s theory of Finsler spaces we have three curvature tensors Rihjk, P i

hjk, Sihjk

and three torsion tensors Rijk(= yhRi

hjk), P ijk(= yhP i

hjk), Cijk.

Definition 1.4 The Cartan connection CΓ of a Finsler space F n with Finsler

metric F is a Finsler connection which is uniquely determined by a system of five

axioms:

C1) h−metrical, i.e., gij|k = 0,

C2) (h) h−torsion free, i.e., T ijk = 0,

C3) Deflection free, i.e., Dik = 0,

C4) v−metrical, i.e., gij|k = 0,

C5) (v) v−torsion free, i.e., Sijk = 0.

The coefficients (Γijk, N

ij , C

ijk) of CΓ are as follows:

1. Γijk = girΓjrk,

where

Γijk =1

2(δkgij + δigjk − δjgik)

and

δk = ∂k −N rk ∂r.

2. N ij = γi0j − Ci

jrγr00,

where

γijk =1

2(∂kgij + ∂igjk − ∂jgik)

and the subscript 0 stands for the transvection by y.

3. C ijk = girCjrk,

where

Cijk =1

2∂kgij.

15

The surviving torsion tensors of CΓ are given by:

(v) h-torsion

Rijk = Ri

0jk,

(h) hv-torsion

Cijk =1

2∂kgij

and (v) hv-torsion

Pijk = Cijk|0.

The curvature tensors of CΓ are as follows:

h-curvature Rihjk satisfy

Rhijk = −Rihjk = −Rhikj,

hv-curvature

Phijk = π(hi)[Cjik|h + Cr

hjPrik

]and v-curvature

Shijk = π(jk) [CrhkCrij] .

1.4.4 Chern connection

S. S. Chern introduced a connection called the Chern connection, Chern (1943).

S. S. Chern studied the equivalence problem for Finsler spaces using the Cartan’s

exterior differentiation method. He discovered a very simple connection. Later on,

H. Rund independently introduced this connection in a different setting. Thus,

Chern’s connection was also called the Rund connection in literatures.

Definition 1.5 There is a unique linear connection, called Chern connection

on a Finsler manifold satisfying the following axioms:

1. Torsion freeness, i.e.,

Γijk = Γi

kj.

16

2. Almost metric compatibility, i.e.,

dgij = gikwkj + gkjw

ki + 2Cijkδy

k,

i.e., gij|k = 0 and gij;k = 2FCijk.

In this case, the connection coefficients Γijk are same as that of the Cartan con-

nection, i.e.,

Γijk =

gis

2

(δgsjδxk

− δgjkδxs

+δgksδxj

).

The surviving curvature tensors, h−curvature tensor Rihjk and hv−curvature ten-

sor P ihjk are given by

Rihjk =

δΓihk

δxj−δΓi

hj

δxk+ Γi

ljΓlhk − Γi

lkΓlhj

and

P ihjk = −F

∂Γihj

∂yk,

respectively.

The surviving torsion tensors are

Rijk = lhRi

hjk

and

P ijk = lhP i

hjk.

1.5 Flag curvature and S-curvature in Finsler ge-

ometry

1.5.1 Flag curvature

First of all in this section, we discuss flag curvature, for Finsler metrics. L. Berwald

first successfully extended the notion of Riemann curvature to Finsler spaces. The

flag curvature in Finsler geometry is an extension of the sectional curvature in

17

Riemannian geometry. For a Finsler space, at each point on the manifold the flag

curvature is a function of a tangent plane and a vector in the plane called the pole

vector.

We say that a Finsler metric is of scalar curvature if the flag curvature is

independent of the tangent planes containing the pole vectors. So the scalar

curvature is a function on the tangent bundle. In dimension two, every Finsler

metric is of scalar curvature.

If the flag curvature is constant then the Finsler metric is said to be of constant flag

curvature. One of the important problems in Finsler geometry is to characterize

Finsler metrics of scalar curvature.

Definition 1.6 For a Finsler metric F , the Riemann curvature Ry : TxM →

TxM is defined by Ry(u) = Rik(x, y)u

k ∂∂xi , u = uk ∂

∂xk , where

Rik = 2

∂Gi

∂xk− yj

∂2Gi

∂xj∂yk+ 2Gj ∂2Gi

∂yj∂yk− ∂Gi

∂yj∂Gj

∂yk.

Definition 1.7 For a tangent plane P ⊂ TxM containing y, the flag curva-

ture K(x, y, P ) with pole vector y is defined by

K(x, y, P ) :=gy(Ry(u), u)

gy(y, y)gy(u, u)− gy(y, u)gy(y, u),

where u ∈ P such that P = span y, u.

If K(x, y, P ) = K(x, y), then the Finsler metric is said to be of scalar flag curva-

ture.

If K(x, y, P ) = K(x), then the Finsler metric is said to be of isotropic flag curva-

ture.

If K(x, y, P ) = 3θF+ c(x), where c = c(x) is a scalar functions on M and θ is an

exact form on M , then the Finsler metric F is said to of almost isotropic flag

curvature.

If K(x, y, P ) = constant, then the Finsler metric is said to be of constant flag

curvature.

18

1.5.2 S-curvature

For a vector y ∈ TxM\ 0, let c = c(t) be the geodesic with c(0) = x and c(0) = y.

Then the S-curvature of the Finsler metric F is defined by

S(x, y) :=d

dt[τF (c(t), c(t))] |t=0,

where τF is called distortion of the Finsler metric F and defined by

τF = ln

√det(gij)

σF

and

σF =V ol(Bn)

V ol (yi) ∈ Rn|F (yibi) < 1.

The S-curvature of Finsler metric F is rewritten as

S =∂Gm

∂ym(x, y)− ym

∂ (logσF (x))

∂xm.

A Finsler metric F is said to have isotropic S-curvature if S = (n + 1)c(x)F , for

some scalar function c(x) on M . Further, if c is a constant, then F is said to be

of constant S-curvature.

F is said to be of almost isotropic S-curvature if

S = (n+ 1) cF + η ,

where η = ηi(x)yi is a 1-form and c = c(x) is a scalar function on M .

1.6 Some special Finsler spaces

1.6.1 Locally Minkowksi space

If a Finsler space has a covering of coordinate neighborhoods in which fundamental

metric tensor gij do not depend on xi, then it is called Locally Minkowksi.

Remark A Finsler space is a Locally Minkowksi space, iff one of the following

two conditions are satisfied:

(1) H ihjk = Gi

hjk = 0 in BΓ,

19

(2) Rihjk = Ci

hj|k = 0 in CΓ.

Thus, a Finsler space is locally Minkowski, iff it is a Berwald space and the h-

curvature tensor of BΓ or CΓ vanishes.

1.6.2 Berwald space

A Finsler space is called Berwald space if the spray coefficients Gi = 12Γijk(x)y

jyk

are quadratic in y, i.e., the coefficients of Berwald connection Gijk is independent

of y. Riemannian metrics are special Berwald metrics. It is well known that S-

curvature of a Berwald metric vanishes identically. Thus a Finsler metric with

vanishing S-curvature may be regarded as generalised Berwald metric.

1.6.3 Landsberg space

The Landsberg tensor is defined as

Ljkl = −1

2FFyi

∂3Gi

∂yj∂yk∂yl.

Finsler spaces with Ljkl = 0 are called Landsberg spaces. Obviously if the spray

coefficients Gi are quadratic in y for any x, then Ljkl = 0. Thus every Berwald

metric is a Landsberg metric . Landsberg metric can also be regarded as an

generalized Berwald metric. It is long open existing problem whether or not any

Landsberg metric is a Berwald metric.

1.6.4 Douglas space

In projective geometry of Finsler manifolds there is an important quantity called

Douglas tensor and defined by

Dihjk =

∂3πi

∂yh∂yj∂yk,

where

πi = Gi − 1

n+ 1

∂Gm

∂ymyi.

In local coordinate system the following three conditions are equivalent

Dihjk = 0.

20

Gi =1

2Γijk(x)y

jyk + P (x, y)yi.

Dij = Giyj −Gjyi = Aijklm(x)y

kylym.

A Finsler metric is called a Douglas metric if Dihjk = 0. Douglas metric is another

generalization of Berwald metric.

1.6.5 Weakly-Berwald space

The E-curvature is defined by the trace of the Berwald curvature, i.e., Eij =12Bm

mij.

A Finsler space is said to be Weakly-Berwald space if, Eij =12Bm

mij = 0. Weakly-

Berwald space with an (α, β)-metric has been investigated by several authors [Lee

and Lee (2006) and Matsumoto (1991)].

1.7 KCC theory

The notion of the KCC (Kosambi-Cartan-Chern) theory was initiated by Kosambi

(1933), Cartan (1933) and Chern (1939), and the abbreviation KCC comes natu-

rally from the names of these three initiators, who profound the geometric theory of

a system of second order ordinary differential equation (SODE). The first attempt

to establish and to develop systematically the KCC theory is due to Antonelli and

Bucataru (2001). The KCC theory describes the evolution of a dynamical system

in geometric terms, by considering it as a geodesic in a Finsler space.

The most significant applications of the KCC theory have been developed for

second-order autonomous systems. For several such systems which provide Lotka-

Volterra models from biology, the Jacobi stability has been investigated. Recent

advances have been obtained in the Riemannian KCC frame-work.

1.7.1 Linear stability vs Jacobi stability

A second order differential equation can be investigated in geometric terms by

KCC-theory inspired by the geometry of a Finsler space. By associating a non-

linear connection and a Berwald type connection to the differential system, five

21

geometrical invariants are obtained with the second invariant giving the Jacobi

stability of the system.

The Jacobi stability gives a simple and powerful tool for constraining the

physical properties of different systems such as dynamical systems, which is given

by second order differential equations (SODE) [Bohmer et al. (2010)].

The study of Jacobi stability is complementary to the study of linear stability

(Lyapunov’s) and is based on the study of Lyapunov stability of whole trajectories

in a region, and hence the perturbation yields trajectories close to the reference

trajectory.

Similarly, in the case of Lyapunov stability, the perturbations of a stable

equilibrium point lead to trajectories which will be dumped out, provided that

these are small enough so as not to escape from a basin of attraction [Antonelli et

al. (1993)].

Linear stability analysis involves the linearisation of the dynamical system

via the Jacobian matrix of a non-linear system, while the KCC theory addresses

stability of a whole trajectory in a tubular region given by Bohmer et al. (2010).

The Jacobi stability gives a global stability than the linear stability. The

Jacobian matrix of the linearised system plays an important role in linear stability

analysis. In linear stability we consider the signs of the eigen values of Jacobian

matrix of corresponding linear system at equilibrium point, where as in Jacobi

stability we consider the signs of the eigenvalues of the deviation curvature tensor

P ij evaluated at the same point.

1.8 Mathematical Cosmology in Finslerian space-

time

One of the main aims of Cosmology is to express all known forces of nature in

one unified theory. Practically, all unification efforts nowadays proceed from the

assumption that quantum field theory is fundamental and gravitation must be

squeezed into a quantum context.

22

On the other hand, there exist other approaches to unification which uses

some geometrical theories. They assumed that a geometrical theory, Einstein’s

general relativity is a fundamental theory, thus electromagnetism and other fields

can be unified by means of a geometrical theory.

The most known geometrical approaches to unification are the theories of

Weyl and Kaluza-Klein, which aim to geometrize electromagnetism like gravita-

tion. These theories faced with serious problems such as, in Weyl’s theory, the

norms of vectors are not invariant under parallel transport, and in the approach

of Kaluza-Klein theories, electrodynamics is geometrized in a five dimensional

space-time. Also quantization of space-time is another existing problem when the

electromagnetic field is quantized.

Finsler geometry is the geometry of space and motion. In our universe we remark

that “there is no position without motion ”. A Finsler space can be considered as

a manifold of positions (coordinate systems xi) and of tangent vectors yi (veloci-

ties) along the curves (world lines of the moving particles) of the background. The

general spaces of paths are closely connected with the principle of equivalence.

In the four dimensional world of space-time the trajectory of a particle falling

freely in a gravitational field is a certain fixed curve. Its direction at any point

depends on the velocity of the particle. The principle of equivalence implies that

there is a preferred set of curves in space-time at any point, pick up any direction

and there is a unique curve in that direction that will be trajectory of any particle

starting with that velocity. These trajectories are thus the properties of space-time

itself.

This standpoint reveals a profound relation between the principle of equiva-

lence and the space-time of paths in Finsler spaces. In addition with a Finsler-

Randers type space-time, as we shall present in the following, the limits of the

equivalence principle of General Relativity can be extended since the presence of

the electromagnetic field does not affect the geodesic motion of a charged par-

ticle in the space. The electromagnetic field is intrinsically incorporated in the

geometry of the space.

23

Chapter 2On Finsler space with a special (α, β)-metric

The purpose of the present chapter is to study a special (α, β)-metric, which is

considered as a generalization of the Randers metric as well as of the Z. Shen’s

square metric and to find the conditions for a Finsler space with this special metric

to be a Berwald space, a Douglas space and Weakly-Berwald space respectively.

2.1 Introduction

The notion of (α, β)-metric in Finsler spaces was introduced by Matsumoto (1972)

as a generalization of Randers metric F = α + β, where α is a regular Rieman-

nian metric α =√aij(x)yiyj, i.e., det(aij) = 0 and β is a one-form β = bi(x)y

i

and studied by many authors [Matsumoto (1991)and Matsumoto (1992)]. Some

authors also assume α to be positive definite [Shen and Yildirim (2008) and Lee

and Lee (2006)].

A Finsler metric F on a differentiable manifold M is called an (α, β)-metric,

if F is a positively homogeneous function of degree one in α and β. Other than

Randers metric F = α + β, there are several important (α, β)-metrics, namely

Kropina metric F = α2

β, Matsumoto metric F = α2

α−β, generalized Kropina metric

F = αn+1

βn and Z. Shen’s square metric F = (α+β)2

α. Z. Shen’s square metric is also

interesting because the metric,

F (x, y) =

(√(1− |x|2)|y|2 + ⟨x, y⟩2 + ⟨x, y⟩

)2(1− |x|2)2

√(1− |x|2)|y|2 + ⟨x, y⟩2

, (x, y) ∈ TRn

24

constructed by L. Berwald in 1929, which is projectively flat on unit ball Bn with

constant flag curvature K = 0, can be written in form F = (α+β)2

αfor some suitable

α and β. Here | · | and ⟨, ⟩ denote the standard Euclidean norm and inner product

respectively on Rn. Shen and Yildirim (2008) also introduced a special (α, β)-

metric F = α+ ϵβ+ k β2

α, which may be considered as a generalization of Randers

metric (k = 0, ϵ = 1) and square (α, β)-metric (k = 1, ϵ = 2). If the Riemannian

metric α is positive definite then in view of Chern and Shen (2004), F is positive

definite with ||β|| < b0 iff 1+ ϵs+ks2 > 0 and 1+2kb2−3ks2 > 0 for any numbers

s and b with |s| ≤ b < b0 where s = βα. In particular Z. Shen square metric is a

positive definite Finsler metric if ||β||α < 1. Z. Shen and Yildirim classified this

special (α, β)-metric under the restriction of projectively flatness and constant flag

curvature. Recently Zhou (2010) proved that a square metric with constant flag

curvature must be projectively flat and hence classified all projectively flat square

(α, β)-metrics. For the rest of the chapter we assume k = 0, ϵ = 0.

Definition 2.1 A Finsler space is called Berwald space if the spray coeffi-

cients Gi = 12Γijk(x)y

jyk are quadratic in y, i.e., the coefficients of Berwald con-

nection Gijk is independent of y.

Riemannian metrics are special Berwald metrics. Weakly-Berwald space and Dou-

glas spaces are generalizations of Berwald spaces.

Definition 2.2 The tensor D := Dijkl∂i ⊗ dxj ⊗ dxk ⊗ dxl is called Douglas

curvature, introduced by Douglas (1927-28), where

Dijkl :=

∂3

∂yj∂yk∂yl

(Gi − 1

n+ 1

∂Gm

∂ymyi).

Douglas curvature, always vanishes for a Riemannian metrics. Finsler metrics

with vanishing Douglas curvature are called Douglas metric and the space is called

Douglas space.

Definition 2.3 The E-curvature is defined by the trace of the Berwald cur-

vature, i.e., Eij =12Bm

mij. A Finsler space is said to be Weakly-Berwald space if,

Eij =12Bm

mij = 0.

25

Weakly-Berwald space with an (α, β)-metric has been investigated by several au-

thors [Lee and Lee (2006) and Matsumoto (1991)]. A Finsler space with an (α, β)-

metric is a Weakly-Berwald space iff Bmm = ∂Bm

∂ymis a one form, i.e., Bm

m is a homo-

geneous polynomial in (yi) of degree one. Further Matsumoto (1991) investigated

that a Finsler space with an (α, β)-metric is a Weakly-Berwald space, iff Bm are

homogeneous polynomials in (yi) of degree 2.

The purpose of the present chapter is to find the conditions for a Finsler space

with special (α, β)-metric F = α + ϵβ + k β2

αto be Berwald space, Douglas space

and Weakly-Berwald space.

Let M be an n-dimensional C∞-manifold. TxM denotes the tangent space of M

at x. The tangent bundle of M is the union of tangent spaces TM :=∪

x∈M TxM .

We denote the elements of TM by (x, y) where y ∈ TxM . Let TM0 = TM \ 0.

Definition 2.4 A Finsler metric on M is a function F : TM → [0,∞) with

the following properties:

(i) F is C∞ on TM0,

(ii) F is positively 1-homogeneous on the fibers of tangent bundle TM ,

and (iii) the Hessian of F 2

2with element gij(x, y) = 1

2∂2F 2

∂yi∂yjis regular on TM0,

i.e., det(gij) = 0.

The pair (M,F ) is then called a Finsler space. F is called fundamental function

and gij is called fundamental tensor. Let Cijk =12

∂gij∂yk

be Cartan tensor. Consider

the Finsler space F n = (M,F ) equipped with an (α, β)-metric F (α, β). Let γijkdenote the Christoffel symbols in the Riemannian space (M,α). Denote by bi;j, the

covariant derivative of the vector field bi with respect to Riemannian connection

γijk, i.e., bi;j = ∂bi∂xj − bkγ

ijk.

Consider the following notations from Chern and Shen (2004)

rij =1

2bi;j + bj;i , rij = aihrhj, rj = bir

ij,

sij =1

2bi;j − bj;i , sij = aihshj, sj = bis

ij,

bi = aihbh, b2 = bibi.

26

2.2 The condition to be a Berwald space

In the present section, we find the condition that a Finsler space F n with a special

(α, β)-metric

F = α + ϵβ + kβ2

α(2.1)

where ϵ, k are non zero constants, to be a Berwald space.

A Finsler space is called Berwald space if the coefficients Gij of Berwald con-

nection BΓ are linear. If the spray coefficients Gi of a Finsler space with an

(α, β)-metric are given by 2Gi = γi00 + 2Bi, then we have Gij = γi0j + Bi

j and

Gijk = γijk + Bi

jk, where Bijk = ∂kB

ij and Bi

j = ∂jBi. Thus a Finsler space with

an (α, β)-metric is a Berwald space iff Gijk = Gi

jk(x), equivalently Bijk = Bi

jk(x).

Moreover on account of Matsumoto (1991) Bijk is determined by

FαBtjiy

jyt+αFβ(Btjibt− bj;i)y

j = 0 where yk = aikyi. (2.2)

For the special (α, β)-metric (2.1) we have,

Fα = 1− kβ2

α2, Fβ = ϵ+

2kβ

α, Fαα =

2kβ2

α3, Fββ =

2k

α. (2.3)

Substituting (2.3) in (2.2) equation, we have

(α2 − kβ2)Btjiy

jyt + α2(αϵ+ 2kβ)(Btjibt − bj;i)y

j = 0. (2.4)

Assume that F n is a Berwald space, i.e., Bijk = Bi

jk(x).

Separating (2.4) in rational and irrational terms of yi as

(α2 − kβ2)Btjiy

jyt + 2kα2β(Btjibt − bj;i)y

j + α3ϵ(Btjibt − bj;i)y

j = 0 (2.5)

which yields two equations

(α2−kβ2)Btjiy

jyt+2kα2β(Btjibt−bj;i)yj

(2.6)

and

(Btjibt − bj;i)y

j = 0, α = 0, ϵ = 0. (2.7)

27

Substituting (2.7) in (2.6), we have

(α2 − kβ2)Btjiy

jyt = 0. (2.8)

Case (i) If Btjiy

jyt = 0, we have

Btjiath +Bt

hiatj = 0 and Btjibt − bj;i = 0. (2.9)

Thus we obtain Btji = 0 by Christoffel process in the first equation of (2.9) and

from second of (2.9), we have bi;j = 0.

Case (ii) If (α2 − kβ2) = 0

⇒ α is a one form, which is a contradiction.

Conversely, if bi;j = 0, then Btji = 0 are uniquely determined from (2.4).

Hence, we conclude the following

Theorem 2.1 A Finsler space with a special (α, β)-metric F = α+ ϵβ+k β2

α

where ϵ, k are non zero constants, is a Berwald space iff bi;j = 0.

2.3 The condition to be a Douglas space

In this section, we find the condition for a Finsler space F n with a special (α, β)-

metric (2.1), to be of Douglas type.

Bacso and Matsumoto (1997) characterizes a Douglas space as a Finsler space for

which Bij = Biyj − Bjyi are homogeneous polynomials of degree 3, in short we

write Bij is hp(3).

In view of Matsumoto (1991), if β2Fα +αγ2Fαα = 0, then the function Gi(x, y) of

F n with an (α, β) -metric is written in the form

2Gi = γi00 + 2Bi,

where

Bi =αFβs

i0

Fα

+ C∗βFβy

i

αF− αFαα

Fα

(yi

α− αbi

β

),

28

C∗ =αβ(r00Fα − 2s0αFβ)

2(β2Fα + αγ2Lαα),

and

γ2 = b2α2 − β2.

The vector Bi(x, y) is called the difference vector. Hence Bij is written as

Bij =αFβ

Fα

(si0yj − sj0y

i) +α2Fαα

βFαC∗(biyj − bjyi).

Substituting (2.3) in above equation, we get

[α2(1 + 2kb2)− 3kβ2][(α2 − kβ2)Bij − α2(ϵα+ 2kβ)(si0yj − sj0y

i)]

−kα2[r00(α2 − kβ2)− 2s0α

2(ϵα + 2kβ)](biyj − bjyi) = 0. (2.10)

If F n is a Douglas space, that is, Bij are hp(3). Arranging the rational and

irrational terms, equation (2.10) can be re written as

[(α2 − kβ2)Bij − 2kβα2(si0yj − sj0y

i)][α2(1 + 2kb2)− 3kβ2]

−kα2[r00(α2 − kβ2)− 4ks0α

2β)](biyj − bjyi)

−α[α2(1 + 2kb2)− 3kβ2

α2ϵ(si0y

j − sj0yi)− 2s0kα

4ϵ(biyj − bjyi)] = 0. (2.11)

Separating rational and irrational terms of yi in (2.11) we have the following two

equations

[(α2 − kβ2)Bij − 2kβα2(si0yj − sj0y

i)][α2(1 + 2kb2)− 3kβ2]

−kα2[r00(α2 − kβ2)− 4ks0α

2β)](biyj − bjyi) = 0 (2.12)

and

α2(1 + 2kb2)− 3kβ2

ϵ(si0y

j − sj0yi)

−2s0kα2ϵ(biyj − bjyi) = 0. (2.13)

Substituting (2.13) in (2.12), we have

[α2(1 + 2kb2)− 3kβ2](α2 − kβ2)Bij

+α2r00(k2β2 − kα2)(biyj − bjyi) = 0. (2.14)

29

Only the term 3k2β4Bij of (2.14) does not contain α2.

Hence we must have hp(5), vij5 satisfying

3k2β4Bij = α2vij5 . (2.15)

Now, we study the following two cases:

Case (i) α2 0(modβ).

In this case, (2.15) is reduced to Bij = α2vij, where vij are hp(1).

Thus (2.14) gives

[α2(1 + 2kb2)− 3kβ2]vij − kr00(biyj − bjyi) = 0. (2.16)

Transvecting this by biyj, where yj = ajkyk, we have

α2[(1 + 2kb2)vijbiyj − kb2r00] = β2(−kr00 + 3kvijbiyj). (2.17)

Since α2 0(modβ), there exists a function h(x) satisfying

[(1 + 2kb2)vijbiyj − kb2r00] = h(x)β2,

(−kr00 + 3kvijbiyj) = h(x)α2.

Eliminating vijbiyj from the above two equations, we obtain

r00(−k + k2b2) = h(x)[(1 + 2kb2)α2 − 3kβ2]. (2.18)

From (2.18), we get

bi;j = l [(1 + 2kb2)aij − 3kbibj], (2.19)

where l = h(x)k(1−kb2)

provided b2 = 1k. Hence bi is a gradient vector.

Conversely, if (2.19) holds, then sij = 12(bi;j − bj;i) = 0 and we get (2.18).

Therefore, (2.12) is written as follows

Bij = −lk[α2(biyj − bjyi)]

which are hp(3), that is, F n is a Douglas space.

Case (ii) α2 ∼= 0(modβ).

30

Consider the following Lemma.

Lemma. [Lee and Park (2004)] If α2 ∼= 0(modβ), that is, aij(x)yiyj contains

biyi as a factor, then the dimension n is equal to 2 and b2 vanishes. In this case

we have 1-form δ = di(x)yi satisfying α2 = βδ and dib

i = 2. In this case the

equation (2.15) is reduced to Bij = δwij2 , where wij

2 are hp(2).

Hence the equation (2.13) leads to

−2s0kδϵ(biyj − bjyi) + ϵ[δ(−3kβ](si0y

j − sj0yi) = 0. (2.20)

Transvecting the above equation by yibj, we get −s0ϵβδ[−kβ + δ] = 0 but β and

δ are non zero and β = δk, we have ⇒ s0 = 0. Substituting s0 = 0 in equation

(2.20), we have sij = 0. Therefore, (2.16) reduces to

(δ − 3kβ)wij2 − kr00(b

iyj − bjyi) = 0.

Transvecting the above equation by biyj, we get (δ − 3kβ)wij2 biyj + kr00β

2 = 0,

which can be written as δwij2 biyj = −kβ(βr00 − 3wij

2 biyj). Therefore, there exists

an hp(2), λ = λijyiyj such that wij

2 biyj = −kβλ, βr00 − 3wij2 biyj = δλ.

Eliminating wij2 biyj from these equations, we get

βr00 = λ(δ − 3kβ), (2.21)

which implies there exists an hp(1), v0 = vi(x)yi such that

r00 = v0(δ − 3kβ), λ = v0β. (2.22)

In view of equation (2.22) and considering sij = 0, we have

bi;j = rij =1

2[vi(dj − 3kbj) + vj(di − 3kbi)]. (2.23)

Hence bi is a gradient vector.

Conversely, if (2.23) holds, then sij = 0, which implies r00 = v0(δ − 3kβ).

Therefore, (2.10) is written as Bij = v0kδ(biyj − bjyi), which are hp(3), i.e., F n is

a Douglas space.

Thus, we have

31

Theorem 2.2 A Finsler space with a special (α, β)-metric F = α+ ϵβ+k β2

α

where ϵ, k are non zero constants, is a Douglas space iff either

(1) α2 0(modβ), b2 = 1k: bi;j is writtten in the form (2.19),

or (2) α2 ∼= 0(modβ) : n = 2 and bi;j is writtten in the form (2.23),

where α2 = βδ, δ = di(x)yi, v0 = vi(x)y

i.

Remark: If α is a positive definite Riemannian metric then the set of non Rie-

mannian (α, β)-metric satisfying case (ii), i.e., α2 ∼= 0(modβ) is void, otherwise

b2 = 0 ⇒ bi = 0 that is β = 0. For positive definite case Li et al. (2009) have

found a condition for a general (α, β)- metric to be Douglas. Comparing the re-

sults of Theorem (1.1) [Li et al. (2009)], we have k1 = 2k, k2 = 0, k3 = −3k

and bi;j = 2τ (1 + 2kb2)aij − 3kbibj. Thus we have τ = l(x)2

, where l is given by

l = h(x)k(1−kb2)

provided b2 = 1k.

2.4 The condition to be a Weakly-Berwald space

In the present section, we consider the condition that the Finsler space with an

(α, β)- metric (2.1), to be a Weakly-Berwald space. Weakly-Berwald space is a

generalization of Berwald space, introduced by M. Matsumoto and studied by

several authors [Lee and Lee (2006) and Matsumoto (1992)].

The spray coefficients Gi(x, y) of F n with an (α, β)-metric, given by Lee and Lee

(2006), can be written as

2Gm = γm00 + 2Bm,

where

Bm =

(E∗

α

)ym +

(αFβ

Fα

)sm0 −

(αFαα

Fα

)C∗(

ym

α

)−(α

β

)bm, (2.24)

where

E∗ =

(βFβ

F

)C∗, C∗ =

αβ (r00Fα − 2αs0Fβ)2 (β2Fα + αγ2Fαα)

, γ2 = b2α2 − β2. (2.25)

Differentiating (2.24) w.r.t yn and summing over suffixes m and n, we have

Bmm =

1

2αF (βFα)2Ω

22Ω2AC∗ + 2αFΩ2Bs0 + α2FFαFαα (Cr00 +Ds0 + Er0)

,

(2.26)

32

where

A = (n+ 1) β2Fα (βFαFβ − αFFαα) + αγ2Fα(Fαα)

2 − 2FαFαα − αFαFααα

,

B = α2FFαα,

C = βγ2−β2(Fα)

2 + 2b2α3FαFαα − α2γ2(Fαα)2 + α2γ2FαFααα

,

(2.27)

D = 2αβ3(γ2 − β2

)FαFβ − α2β2γ2FαFαα

−2αβγ2(γ2 + 2β2

)FβFαα − α3γ4(Fαα)

2 − α2βγ4FβFααα

,

E = 2α2β2FαΩ

and

Ω =(β2Fα + αγ2Fαα

),

(2.28)

provided Ω = 0.

Substituting (2.3) in equation (2.24), (2.25), (2.27) and (2.28), we have

Bm = C∗[

β (αϵ+ 2kβ)

α(α2 + ϵαβ + kβ2)− 2kβ2

α (α2 − kβ2)

ym +

2kβα

(α2 − kβ2)

bm]

+

α2 (αϵ+ 2kβ)

(α2 − kβ2)

sm0 , (2.29)

where

C∗ =αβ r00 (α2 − kβ2)− 2α2s0 (αϵ+ 2kβ)2 β2 (α2 − kβ2) + 2kβ2 (b2α2 − β2)

.

From (2.3) and (2.27), we get

A =(n+ 1) β3 (α2 − kβ2)

α5

[α3ϵ− 3kϵαβ2 − 4k2β3

]+

2γ2kβ2

α5

×(α2 + ϵαβ + kβ2

) (α2 + kβ2

),

B =(α2 + ϵαβ + kβ2) (2kβ2)

α2,

C =β3γ2

α4

[−(α2 − kβ2

)2+ 4b2kα2

(α2 − kβ2

)− 2γ2

(3kα2 − k2β2

)],

(2.30)

33

D =2β3

α2

[(γ2 − β2

) (α2 − kβ2

)(αϵ+ 2kβ)− 2k

(α2 − kβ2

)βγ2]

−[4kγ2

(γ2 + 2β2

)(αϵ+ 2kβ)− 2γ4

(4k2β + 3kϵα

)],

Ω =β2

α2

[α2(1 + 2kb2

)− 3kβ2

],

E =2β4

α2

(α2 − kβ2

) [(1 + 2kb2

)α2 − 3kβ2

]and

C∗ =α r00 (α2 − kβ2)− 2α2s0 (αϵ+ 2kβ)

2β α2 (1 + 2kb2)− 3kβ2.

Substituting (2.30) into (2.26), we getα10β

(8b4k2 + 8b2k + 2

)+ α4β7

(8b4k5 + 32b2k4 − 4k3

)+ α6β5

(−8b4k4 + 16b2k3 + 28k2

)+ α8β3

(−8b4k3 − 32b2k2 − 14k

)+ 18k5β11 + α2β9

(−24b2k5 − 30k4

)+ α9β2

(8b4k2ϵ+ 8b2kϵ+ 2ϵ

)+ α7β4

(−16b4k3ϵ− 40b2k2ϵ− 16kϵ

)+ α5β6

(8b4k4ϵ+ 56b2k3ϵ+ 44k2ϵ

)+ αβ10

(18k4ϵ

)+ α3β8

(−48k3ϵ− 24b2k4ϵ

)Bm

m

+α6β4

(−20b2k3 − 16k2 − 8b2k3n− 4k2n

)+ α4β6

(4b2k4 + 32k3 + 16b2k4n+ 20k3n

)+ α8β2

(12k2b2 + 8b4k3

)+ α2β8

(−16k4 − 8b2k5n− 28k4n+ 12b2k5

)+ β10

(16k5 + 12k5n

)+α7β3

(2k2b2ϵ− 8kϵ− 8knϵ− 10b2k2nϵ

)+ α9β

(2b2kϵ+ 2b2knϵ+ nϵ+ ϵ

)+ α5β5

(−10b2k3ϵ+ 10k2ϵ+ 14b2k3nϵ+ 22k2nϵ

)+ αβ9

(−3k4ϵ+ 9k4nϵ

)+ α3β7

(6b2k4ϵ− 6b2k4nϵ− 24k3nϵ

)r00

+α10β

(−4b2ϵ2k − 4b2ϵ2kn− 2nϵ2 − 2ϵ2 + 8b2k2 + 4k

)+α8β3

(−16k2 + 14kϵ2 + 14knϵ2 − 8b2k2ϵ2 + 16b2k2ϵ2n− 56k3b2

)+α6β5

(24b2k4 + 72k3 + 32b2k4n+ 16k3n+ 20k3b2ϵ2 − 6k2ϵ2 − 12b2k3nϵ− 30k2ϵ2n

)+α2β9

(−12k5 + 48k5n

)+ α4β7

(24b2k5 − 48k4 − 32b2k5n− 64k4n− 6k3ϵ2 + 18k3ϵ2n

)+α7β4

(−16b2k3ϵ+ 48b2k3nϵ+ 36k2nϵ+ 40k2ϵ

)+α5β6

(−16k3ϵ− 40b2k4nϵ− 92k3nϵ+ 40b2k4ϵ

)+ α9β2

(−32k2b2ϵ− 8b2k2nϵ− 4knϵ

)+α3β8

(−16k4ϵ+ 60k4nϵ

)s0 +

α10β

(8b2k2 + 4k

)+ α8β3

(−8b2k3 − 16k2

)+α6β5

(−8b2k4 + 8k3

)+ α2β9

(−12k5

)+ α4β7

(8b2k5 + 16k4

)+ α9β2

(8b2k2ϵ+ 4kϵ

)+α5β6

(8b2k4ϵ+ 28k3ϵ

)+ α3β8

(−12k4ϵ

)+ α7β4

(−16b2k3ϵ− 20k2ϵ

)r0 = 0.

(2.31)

34

Suppose that F n is Weakly-Berwald space, i.e., Bmm is hp(1). Since α is irrational

in (yi), the equation (2.31) is divided into two equations as follows

F1Bmm + βG1r00 + α2H1s0 + α2I1r0 = 0 (2.32)

βF2Bmm +G2r00 + α2βH2s0 + α2βI2r0 = 0, (2.33)

where

F1 =α10(8b4k2 + 8b2k + 2

)+ α4β6

(8b4k5 + 32b2k4 − 4k3

)+ α6β4

(−8b4k4 + 16b2k3 + 28k2

)+ α8β2

(−8b4k3 − 32b2k2 − 14k

)+ 18k5β10 + α2β8

(−24b2k5 − 30k4

),

F2 =α8(8b4k2ϵ+ 8b2kϵ+ 2ϵ

)+ α6β2

(−16b4k3ϵ− 40b2k2ϵ− 16kϵ

)+ α4β4

(8b4k4ϵ+ 56b2k3ϵ+ 44k2ϵ

)+ 18k4ϵβ8 + α2β6

(−48k3ϵ− 24b2k4ϵ

),

G1 =α6β2(−20b2k3 − 8b2k3n− 16k2 − 4k2n

)+ α4β4

(4b2k4 + 32k3 + 16b2k4n+ 20k3n

)+ α8

(12k2b2 + 8b4k3

)+ α2β6

(−16k4 − 8b2k5n− 28b4n+ 12b2k5

)+ β8

(16k5 + 12k5n

),

G2 =α8(2b2kϵ+ 2b2kϵn+ nϵ+ ϵ

)+ α6β2

(2k2b2ϵ− 8kϵ− 8knϵ− 10k2b2nϵ

)+ α4β4

(−10b2k3ϵ+ 10k2ϵ+ 14b2k3nϵ+ 22k2nϵ

)+ β8

(−3k4ϵ+ 9k4nϵ

)+ α2β6

(6b2k4ϵ− 6b2k4nϵ− 24k3nϵ

),

H1 =α8(−4b2ϵ2k − 4b2ϵ2kn− 2nϵ2 − 2ϵ2 + 8b2k2 + 4k

)+ α6β2

(−16k2 + 14kϵ2 + 14knϵ2 − 8b2k2ϵ2 + 16b2k2ϵ2n− 56k3b2

)+ α4β4

(24b2k4 + 72k3 + 32b2k4n+ 16k3n+ 20k3b2ϵ2 − 6k2ϵ2 − 12b2k3nϵ− 30k2ϵ2n

)+ β8

(−12k5 + 48k5n

)+ α2β6

(24b2k5 − 48k4 − 32b2k5n− 64k4n− 6k3ϵ2 + 18k3ϵ2n

),

H2 =α4β2(−16b2k3ϵ+ 48b2k3nϵ+ 36k2nϵ+ 40k2ϵ

)+ α2β4

(−16k3ϵ− 40b2k4nϵ− 92k3nϵ+ 40b2k4ϵ

)+ α6

(−32k2b2ϵ− 8k2b2nϵ− 4knϵ

)+ β6

(−16k4ϵ+ 60k4nϵ

),

I1 =α8(8b2k2 + 4k

)+ α6β2

(−8b2k3 − 16k2

)+ α4β4

(−8b2k4 + 8k3

)+ β8

(−12k5

)+ α2β6

(8b2k5 + 16k4

)35

and

I2 =α6(8b2k2ϵ+ 4kϵ

)+ α2β4

(8b2k4ϵ+ 28k3ϵ

)+ β6

(−12k4ϵ

)+ α4β2

(−16b2k3ϵ− 20k2ϵ

).

Eliminating Bmm from these equations, we obtain

Rr00 + α2βSs0 + α2βTr0 = 0, (2.34)

where

R = F1G2 − β2G1F2, S = F1H2 −H1F2, T = F1I2 − F2I1.

Since only the term (8b4k2 + 8b2k + 2) (2b2kϵ+ 2b2knϵ+ nϵ+ ϵ)α18r00 of Rr00 in

(2.34) does not contain β, we must have, hp(19)V19 such that

α18r00 = βV19. (2.35)

First we are concerned with α2 0(modβ) and b2 = 0. Hence (2.35) shows the

existence of a function V 1 satisfying V19 = V 1α18, and we get r00 = V 1β.

Then (2.34) is reduced to

RV 1 + α2Ss0 + α2Tr0 = 0. (2.36)

Only the term 18k5 (−3k4ϵ+ 9k4nϵ)− 18k4ϵ (16k5 + 12k5n) β18V 1 of the above

does not contain α2, and hence we must have hp(17), V17 satisfying

18k5

(−3k4ϵ+ 9k4nϵ

)− 18k4ϵ

(16k5 + 12k5n

)β18V 1 = α2V17.

Since α2 0(modβ), we have V17 = 0, i.e., V 1 = 0. Hence we obtain r00 =

0; rij = 0 and r0 = 0; rj = 0. Substituting V 1 = 0, r0 = 0 in (2.36), we get

Ss0 = 0 ⇒ s0 = 0 since S = 0.

Conversely, substituting r00 = 0, s0 = 0 and r0 = 0 into (2.31), we have Bmm = 0.

That is, the Finsler space with (2.1) is a Weakly-Berwald space. On the other

hand, we suppose that the Finsler space with (2.1) be a Berwald space. Then we

have r00 = 0, s0 = 0 and r0 = 0, because the space is a Weakly-Berwald space

from the above discussion. Substituting the above into (2.29), we have Bm = 0

that is, the Finsler space with (2.1) is a Berwald space. Hence sij = 0 hold good.

36

Theorem 2.3 A Finsler space with the metric (2.1) is Weakly-Berwald space

iff rij = 0 and sj = 0.

Remark: In view of Theorem (2.1) a Finsler space with metric (2.1) is a Berwald

space iff rij = 0 and sij = 0.

37

Chapter 3On Randers change of a Finsler spacewith m-th root metric

The purpose of the present chapter is to find a condition under which a Finsler

space with Randers change of m-th root metric is projectively related to a m-th

root metric and also to find a condition under which this Randers transformed m-

th root Finsler metric is locally dually flat. Moreover, if the transformed Finsler

metric is conformal to the m-th root Finsler metric, then it is proved that both of

them reduce to Riemannian metrics.

3.1 Introduction

The concept of m-th root metric was introduced by Shimada (1979), applied to

ecology by Antonelli et al. (1993) and studied by several authors [Srivastava and

Arora (2012), Tayebi et al. (2012), Tayebi and Najafi (2011), Yu and You (2010)

and Prasad and Dwivedi (2002)]. It is regarded as a generalization of Riemannian

metric in the sense that the second root metric is a Riemannian metric. For m = 3,

it is called a cubic Finsler metric studied by Matsumoto and Numata (1982) and

for m = 4, it is quartic metric studied by Li and Shen (2012). In four-dimension,

the special fourth root metric in the form F = 4√y1y2y3y4 is called the Berwald-

Moor metric. This metric has been studied by Balan (2010) and considered by

physicists as an important subject for a possible model of space time. Recent

studies show that m-th root Finsler metrics play a very important role in physics,

space-time and general relativity as well as in unified gauge field theory [Pavlov

(2006), Lebedev (2006), Balan and Brinzei (2006), Balan and Brinzei (2005) and

38

Asanov (1984)]. Li and Shen (2012) have studied the geometric properties of

locally projectively flat fourth root metrics in the form F = 4√aijkl(x)yiyjykyl and

generalized fourth root metrics in the form F =√√

aijkl(x)yiyjykyl + bij(x)yiyj.

Tayebi and Najafi (2011) have characterized locally dually flat and Antonelli m-

th root metrics and Tayebi et al. (2012) have found a condition under which a

generalized m-th root metric is projectively related to m-th root metric. Brinzei

(2009) has investigated necessary and sufficient condition for a Finsler space with

m-th root metric to be projectively related.

Recently, Srivastava and Arora (2012) have introduced Randers change of m-

th root metric and studied relations between various tensors of the transformed

Finsler space and Finsler space with m-th root metric. In this chapter a condition

under which the transformed Finsler space is projectively related with given Finsler

space has been investigated. Also the condition under which the transformed

Finsler space is locally dually flat has been found. Further it is proved that if the

transformed Finsler space and original Finsler space with m-th root metric are

conformally related then both reduce to Riemannian metrics.

3.2 Preliminaries

Let M be an n-dimensional C∞-manifold, TxM denotes the tangent space of M

at x. The tangent bundle TM is the union of tangent spaces, TM :=∪

x∈M TxM .

We denote the elements of TM by (x, y), where x = (xi) is a point of M and

y ∈ TxM called supporting element. We denote TM0 = TM \ 0.

Definition 3.1 A Finsler metric on M is a function F : TM → [0,∞) with

the following properties:

(i) F is C∞ on TM0,

(ii) F is positively 1-homogeneous on the fibers of tangent bundle TM and

(iii) the Hessian of F 2

2with element gij = 1

2∂2F 2

∂yi∂yjis positive definite on TM0.

The pair (M,F ) = F n is called a Finsler space. F is called the fundamental

function and gij is called the fundamental tensor of the Finsler space F n.

39

The normalized supporting element li, angular metric tensor hij and metric tensor

gij of F n are defined respectively as:

li =∂F

∂yi, hij = F

∂2F

∂yi∂yjand gij =

1

2

∂2F 2

∂yi∂yj. (3.1)

Let F be a Finsler metric defined by F = m√A, where A is given by A :=

ai1i2...im(x)yi1yi2 ...yim , with ai1...im symmetric in all its indices, Shimada (1979).

Then F is called an m-th root Finsler metric. Clearly, A is homogeneous of degree

m in y.

Let

Ai = aii2...im(x)yi2 ...yim =

1

m

∂A

∂yi, (3.2)

Aij = aiji3...im(x)yi3 ...yim =

1

m(m− 1)

∂2A

∂yi∂yj, (3.3)

Aijk = aijki4...im(x)yi4 ...yim =

1

m(m− 1)(m− 2)

∂3A

∂yi∂yj∂yk. (3.4)

The normalized supporting element of F n is given by

li := Fyi =∂F

∂yi=∂ m√A

∂yi=

1

m

∂A∂yi

Am−1m

=Ai

Fm−1. (3.5)

Consider the transformation

F = F + β, (3.6)

where F = m√A is an m-th root metric and β(x, y) = bi(x)y

i is a one form

on the manifold M . Clearly F is also a Finsler metric on M . Throughout the

chapter we call the Finsler metric F as the Randers transformed m-th root metric

and (M,F ) = Fn as Randers transformed Finsler space. We restrict ourselves

for m > 2 throughout the chapter and also the quantities corresponding to the

Randers transformed Finsler space F n will be denoted by putting bar on the top

of that quantity.

40

3.3 Fundamental metric tensor of Randers trans-

formed m-th root metric

A Finsler metric F = α + β, where α =√aijyiyj is a Riemannian metric and

β = bi(x)yi is a differential one form, was introduced by physicist Randers (1941)

from view point of general theory of relativity. Further Antonelli et al. (1993)

and Matsumoto (1986) studied this metric as a Finsler metric and investigted

its properties. In 1971, M. Matsumoto introduced a Finsler metric F (x, y) =

F (x, y)+β(x, y), where F is a Finsler metric and β is a one form on the manifold

M . This metric is called Randers change of Finsler metric.

The differentiation of (3.6) with respect to yi yields the normalized supporting

element li given by

li = li + bi. (3.7)

In view of (3.5), we have

li =Ai

Fm−1+ bi. (3.8)

Again differentiation of (3.8) with respect to yj yields

hij = (m− 1)F

F

(Aij

Fm−2− AiAj

F 2(m−1)

). (3.9)

From (3.8) and (3.9), the fundamental metric tensor gij of Finsler space F n is

given by:

gij = hij + lilj

gij = (m− 1)F

F

(Aij

Fm−2− AiAj

F 2(m−1)

)+

(Ai

Fm−1+ bi

)(Aj

Fm−1+ bj

)gij =

(m− 1)τAij

Fm−2+Aibj + AjbiFm−1

+ bibj +(1−m− 1τ

) AiAj

F 2(m−1), (3.10)

where τ = FF.

The contravariant metric tensor gij of Finsler space F n is given by, Srivastava and

Arora (2012)

gij =1

τ(m− 1)

[Fm−2Aij − 1

1 + q

biyj + bjyi

F+b2 +m− 1τ − 1

(1 + q)2yiyj

F 2

].

41

Thus contravariant metric tensor gij is rewritten in the form

gij =1

τ(m− 1)

[Fm−2Aij − biyj + bjyi

F+b2 +m− 1τ − 1

F2 yiyj

], (3.11)

where q = βF, (1 + q) = F

F= τ , and matrix (Aij) denotes inverse of (Aij), Yu and

You (2010). Here we have used AijAj = Ai = yi.

Proposition 3.1 The covariant metric tensor gij and contravariant metric

tensor gij of Randers transformed m-th root Finsler space F n are given as :

gij =(m− 1)τAij

Fm−2+Aibj + AjbiFm−1

+ bibj + (1−m− 1τ)AiAj

F 2(m−1)

and

gij =1

τ(m− 1)

[Fm−2Aij − biyj + bjyi

F+b2 +m− 1τ − 1

F2 yiyj

].

3.4 Spray coefficients of Randers transformed m-

th root metric

The geodesics of a Finsler space F n are given by the following system of equations

d2xi

dt2+Gi

(x,dx

dt

)= 0,

where

Gi =1

4gil[F 2]xkyl

yk −[F 2]xl

(3.12)

are called the spray coefficients of F n.

Two Finsler metrics F and F on a manifold M are called projectively related if

there is a scalar function P (x, y) defined on TM0 such that Gi= Gi +Pyi, where

Gi and Gi are the geodesic spray coefficients of F n and F n respectively. In other

words two metrics F and F are called projectively related if any geodesic of the

first is also geodesic for the second and vice versa.

In view of equation (3.10), the metric tensor gij of F n can be rewritten as [Yu and

You (2010)]

gij = τgij +Aibj + AjbiFm−1

+ bibj + (1− τ)AiAj

F 2(m−1), (3.13)

42

where

gij = (m− 1)Aij

Fm−2− (m− 2)

AiAj

F 2(m−1). (3.14)

Further in view of equation (3.11), the contravariant metric tensor gij can be

rewritten as [Yu and You (2010)]

gij =1

τ

[gij − biyj + bjyi

(m− 1)F+

b2 +m− 1τ − 1

(m− 1)F2 − (m− 2)

(m− 1)F 2

yiyj

], (3.15)

where

gij =Fm−2

(m− 1)Aij +

(m− 2)

(m− 1)

yiyj

F 2. (3.16)

The Spray coefficients of Randers transformed Finsler space F n are given by

Gi=

1

4gil[F

2]xkyl

yk −[F

2]xl

.

It can also be written as

Gi=

1

4gil(

2∂gjl∂xk

−∂gjk∂xl

)yjyk

. (3.17)

From (3.13), (3.15) and (3.17), we get

Gi=

1

4τ

[gil + yi

(ϕyl − bl

(m− 1)F

)− biyl

(m− 1)F

]×[(

2∂

∂xk

τgjl +

Ajbl + AlbjFm−1

+ bjbl + (1− τ)AjAl

F 2(m−1)

− ∂

∂xl

τgjk +

Ajbk + AkbjFm−1

+ bjbk + (1− τ)AjAk

F 2(m−1)

)yjyk

],

where

ϕ =b2 + (m− 1)τ − 1

(m− 1)F2 − (m− 2)

(m− 1)F 2.

That is,

Gi=

1

4τ

[gil + yi

(ϕyl − bl

(m− 1)F

)− biyl

(m− 1)F

]×[(

2τ∂gjl∂xk

− τ∂gjk∂xl

+ 2τkgjl − τlgjk

)yjyk + 2

∂

∂xk(Xjl)y

jyk − ∂

∂xl(Xjk)y

jyk], (3.18)

43

where τk = ∂τ∂xk and

Xjl =Ajbl + AlbjFm−1

+ bjbl + (1− τ)AjAl

F 2(m−1). (3.19)

Equation (3.18) can be rewritten as

Gi=

1

4gil(2∂gjl∂xk

− ∂gjk∂xl

)yjyk +

1

4τgil(2τkgjl + 2

∂

∂xk(Xjl)− τlgjk −

∂

∂xl(Xjk)

)yjyk

+1

4τyi[

2τ∂gjl∂xk

+ 2τkgjl + 2∂

∂xk(Xjl)− τ

∂gjk∂xl

− τlgjk −∂

∂xl(Xjk)

yjyk

]×(ϕyl − bl

(m− 1)F

)− biylyjyk

4τ(m− 1)F

2τ∂gjl∂xk

+ 2τkgjl + 2∂

∂xk(Xjl)− τ

∂gjk∂xl

− τlgjk −∂

∂xl(Xjk)

.

Further since

Gi =1

4gil(

2∂gjl∂xk

− ∂gjk∂xl

)yjyk

,

we obtain

Gi= Gi +

1

4τgil(2τkgjl + 2

∂

∂xk(Xjl)− τlgjk −

∂

∂xl(Xjk)

)yjyk

+1

4τyi[

2τ∂gjl∂xk

+ 2τkgjl + 2∂

∂xk(Xjl)− τ

∂gjk∂xl

− τlgjk −∂

∂xl(Xjk)

yjyk

]×(ϕyl − bl

(m− 1)F

)− biylyjyk

4τ(m− 1)F

2τ∂gjl∂xk

+ 2τkgjl + 2∂

∂xk(Xjl)− τ

∂gjk∂xl

− τlgjk −∂

∂xl(Xjk)

.

(3.20)

Now in view of (3.19), we get

Xjkyjyk =

[Ajbk + Akbj

Fm−1+ bjbk + (1− τ)

AjAk

F 2(m−1)

]yjyk.

That is,

Xjkyjyk =

2Aβ

Fm−1+ β2 +

(1− τ)A2

F (2m−2)

= 2Fβ + β2 + (1− τ)F 2

= 2Fβ + β2 + (F − F )F

= 2Fβ + β2 − βF = β(β + F )

= βF

= τβF.

44

Here we have used:

Ajyj = A, bjy

j = β, A = Fm, τ =F

Fand F = F + β.

Substituting the value of gil from equation (3.16) and the value of Xjkyjyk from

above in equation (3.20) we obtain

Gi= Gi +

1

4τ

Fm−2

(m− 1)Ail

[2τ0gjly

j + 2∂

∂xk(Xjl)y

jyk − τlF2 − ∂

∂xl(τβF )

]+yi

4τ

(m− 2)

(m− 1)

yk

F 2

τkF

2 +∂

∂xk(τβF )

+ yi

[ϕyk

4τ

∂

∂xk(F

2)− blyjyk

4τ(m− 1)F

(2∂gjl∂xk

−∂gjk∂xl

)]− biyk

4τ(m− 1)F

∂

∂xk(F

2).

Now from (3.19)

Xjlyj =

Abl + Alβ

Fm−1+ βbl +

(1− τ)AAl

F 2(m−1).

Substituting this value in above equation, we have

Gi= Gi + yi

[ϕyk

4τ

∂

∂xk(F

2)− blyjyk

4τ(m− 1)F

(2∂gjl∂xk

−∂gjk∂xl

)+

1

4τ

(m− 2)

(m− 1)

yk

F 2

τkF

2 +∂

∂xk(τβF )

]+

1

4τ

Fm−2

(m− 1)Ail

[2τ0gjly

j + 2yk∂

∂xk

Abl + Alβ

Fm−1+ βbl +

(1− τ)AAl

F 2(m−1)

− τlF

2 − ∂

∂xl(τβF )

]− biyk

2τ(m− 1)

∂

∂xk(F ).

The above equation can be rewritten as

Gi= Gi + Pyi +Qi, (3.21)

where

P =ϕyk

4τ

∂(F2)

∂xk− blyjyk

4τ(m− 1)F

(2∂gjl∂xk

−∂gjk∂xl

)+

1

4τ

(m− 2)

(m− 1)F 2ykτkF

2 +∂

∂xk(τβF )

and

Qi =1

4τ

Fm−2

(m− 1)Ail

[2τ0yl + 2yk

∂

∂xk

(Abl + Alβ

Fm−1+ blβ +

(1− τ)AAl

F 2(m−1)

)− τlF

2 − ∂

∂xl(τβF )

]− biyk

2τ(m− 1)

∂(F )

∂xk.

45

Further since

Abl + Alβ

Fm−1+ blβ +

(1− τ)AAl

F 2(m−1)=FmblFm−1

+ylF

m−1

F

β

Fm−1+ blβ +

(1− τ)Fm

F 2(m−1)

ylFm−1

F

= Fbl +βylF

+ blβ + (1− τ)yl = (F + β)bl = Fbl,

we have

Qi =1

4τ

Fm−2

(m− 1)Ail

[2τ0yl + 2yk

∂

∂xk(Fbl)− τlF

2 − ∂

∂xl(τβF )

]− biyk

2τ(m− 1)

∂(F )

∂xk.

Now, in view of (3.21), the Finsler metrics F and F are projectively related if

Qi = 0, that is

Fm−2Ail

[2τ0yl + 2yk

∂

∂xk(Fbl)− τlF

2 − ∂

∂xl(τβF )

]= 2biyk

∂(F )

∂xk. (3.22)

Theorem 3.2 The Randers transformed m-th root metric F and m-th root

metric F , on an open subset U ⊂ Rn, are projectively related if equation (3.22) is

satisfied.

3.5 Conformally related Randers transformed m-

th root metric

The conformal transformation between two Finsler metrics F and F are defined

by F (x, y) = eσ(x)F (x, y), where σ is a scalar function on M . We call such two

metrics F and F conformally related. The conformal change is called homothetic

and isometry if σi = ∂σ∂xi = 0 and σ(x) = 0, respectively.

In this section, we prove that if a Randers transformed m-th root metric is con-

formal to a m-th root Finsler metric, then both of them reduce to Riemannian

metrics. More precisely, we prove the following.

Theorem 3.3 Let F = F + β and F = A1m are Randers transformed m-th

root metric and m-th root Finsler metric on an open subset U ⊂ Rn, respectively,

where A := ai1i2...im(x)yi1yi2 ...yim and β = bi(x)y

i. Suppose that F is conformal

to F . Then F and F reduce to Riemannian metrics.

46

In present section, we prove a generalized version of Theorem (3.3). We consider

two Randers transformed m-th root metrics F = F + β and F = F + β, where β

and β are two one forms on M given by β = bi(x)yi and β = bi(x)y

i, which are

conformal. Then we prove the following:

Theorem 3.4 Let F = F + β and F = F + β are two Randers transformed

m-th root Finsler metrics on an open subset U ⊂ Rn, where β = bi(x)yi and

β = bi(x)yi. Suppose that F is non-isometric conformal to F , then F = A

1m is a

Riemannian metric.

Proof: Since F is conformal to F , we have

F = eσF , (3.23)

where F = F + β and F = F + β are Randers transformed m-th root Finsler

metrics on an open subset U ⊂ Rn, with β = bi(x)yi and β = bi(x)y

i.

Then,

F2= e2σF 2.

The metric tensors gij and gij of the two Finsler spaces are related by

gij = e2σgij. (3.24)

Further since F = F + β, in view of equation (3.13) we have

gij = τgij + bibj + (libj + ljbi)−β

Flilj, (3.25)

where τ = FF

and li = Ai

Fm−1 .

Similarly, since F = F + β we have

gij = µgij + bibj +(libj + lj bi

)− β

Flilj, (3.26)

where µ = FF.

Then by (3.24), (3.25) and (3.26), we have

τgij + bibj + (libj + ljbi)−β

Flilj = e2σ

[µgij + bibj +

(libj + lj bi

)− β

Flilj

]

47

gijτ − e2σµ

=(e2σ bibj − bibj

)+(e2σlibj − libj

)(3.27)

+(e2σlj bi − ljbi

)−

(e2σlilj

β

F− lilj

β

F

).

Since σ is not isometry, i.e., σ = 0, then by (3.27), we get

gij =1

τ − e2σµ

(e2σ bibj − bibj

)+(e2σlibj − libj

)(3.28)

+(e2σlj bi − ljbi

)−

(e2σlilj

β

F− lilj

β

F

)which is a function of x alone.

This implies that Cijk = 0 and hence F is Riemannian.

By (3.28), we get the following.

Corollary 3.1 Let F = F + β and F = F + β are two Randers transformed

m-th root Finsler metrics on an open subset U ⊂ Rn, where F = A1m is not

Riemannian, β = bi(x)yi and β = bi(x)y

i. Suppose that F is conformal to F then

F is isometric to F .

Proof of Theorem (3.3) In theorem (3.4), put β = 0 and F = F . Suppose

that the Randers transformed m-th root metric F = F + β is conformal to the

m-th root Finsler metric F = A1m . By Theorem (3.4), F is Riemannian and then

Cijk = 0. Since gij = e2σgij then,

gij + bibj + (libj + ljbi) + (gij − lilj)β

F= e2σgij.

gij

(1− e2σ +

β

F

)= −bibj − (libj + ljbi) + lilj

β

F.

gij =−bibj − (libj + ljbi) + lilj

βF(

1− e2σ + βF

) (3.29)

which yields

gij = e2σ−bibj − (libj + ljbi) + lilj

βF(

1− e2σ + βF

) (3.30)

which is a function of x alone.

Hence

C ijk = Cijk = 0.

Thus C ijk = 0, which implies that F reduces to a Riemannian metric. This

completes the proof.

48

3.6 Locally dually flatness of Randers transformed

m-th root metric

The notion of dually flat Riemannian metrics was introduced by S.-I. Amari and H.

Nagaoka, when they studied the information geometry on Riemannian manifolds.

In Finsler geometry, Z. Shen extended the notion of locally dually flatness for

Finsler metrics. Dually flat Finsler metrics form a special and valuable class of

Finsler metrics in Finsler information geometry, which plays a very important

role in studying flat Finsler information structure. Information geometry has

emerged from investigating the geometrical structure of a family of probability

distributions.

A Randers transformed Finsler metric F = F (x, y) on a manifold M is said to be

locally dually flat, if at any point there is a standard coordinate system (xi, yi) in

TM such that[F

2]xkyl

yk = 2[F

2]xl

. In this case, the coordinate (xi) is called

an adapted local coordinate system. Every locally Minkowskian metric is locally

dually flat.

Consider the Randers transformation F = F +β, where F is an m-th root metric.

We have[F

2]xl=[(F + β)2

]xl = 2 [F + β]

[1

mA

1−mm Axl + βl

]=

2

mA

2−mm Axl + 2A

1mβl +

2

mA

1−mm βAxl + 2ββl. (3.31)

If we put bij = ∂bi∂xj , we have βj = ∂β

∂xj = bijyj.

From (3.31), we get[F

2]xk

=2

mA

2−mm Axk + 2A

1mβk +

2

mA

1−mm βAxk + 2ββk

and [F

2]xkyl

=2

mA

2−mm Axkyl +

2

m

(2−m

m

)A

2−2mm AylAxk (3.32)

+2βk1

mA

1−mm Ayl + 2A

1m blk +

2

mβAxk

(1−m

m

)A

1−2mm Ayl

+2

mβA

1−mm Axkyl +

2

mA

1−mm Axkbl + 2blβk + 2βblk.

49

Let the Finsler metric F is locally dually flat, we have[F

2]xkyl

yk − 2[F

2]xl= 0. (3.33)

Therefore from (3.31), (3.32) and (3.33), we obtained[F

2]xkyl

yk − 2[F

2]xl=

[2

mA

2−mm Axkyl +

2

m

(2−m

m

)A

2−2mm AylAxk

+2βk1

mA

1−mm Ayl + 2A

1m blk +

2

mβAxk

(1−m

m

)A

1−2mm Ayl

+2

mβA

1−mm Axkyl +

2

mA

1−mm Axkbl + 2blβk + 2βblk

]yk

−2

[2

mA

2−mm Axl + 2A

1mβl +

2

mA

1−mm βAxl + 2ββl

]= 0.

The above equation can be written as

Axl

[4

mA

2−mm +

4

mA

1−mm β

]=

2

mA

2−mm A0l +

2

m

(2−m

m

)A

2−2mm A0Ayl

+2βk1

mA

1−mm Ayly

k + 2A1mβl +

2

mβA0

(1−m

m

)A

1−2mm Ayl

+2

mβA

1−mm A0l +

2

mA

1−mm A0bl + 2blβky

k + 2ββl − 4A1mβl − 4ββl,

that is,

Axl

4

mA

1−mm [β + F ] =

2

mA0AylA

1−2mm

(2−m

m

)F + β

(1−m

m

)+

2

mA0lA

1−mm [F + β] + 2βk

1

mA

1−mm Ayly

k

+2

mA

1−mm A0bl + 2blβky

k − 2βl (F + β) .

Therefore F is locally dually flat metric iff

Axl =A0Ayl

2A

[F

mF+

(1−m

m

)]+A0l

2+βkAyly

k

2F+A0bl

2F+mblβky

k

2FA1−mm

− mβl

2A1−mm

.

Thus we have

Theorem 3.5 Let F be a Randers transformed m-th root Finsler metric on

a manifold M . Then, F is locally dually flat metric iff

Axl =A0Ayl

2A

[F

mF+

(1−m

m

)]+A0l

2+βkAyly

k

2F+A0bl

2F+mblβky

k

2FA1−mm

− mβl

2A1−mm

.

50

Chapter 4On Conformal Transformation of m-th root Finsler metric

The purpose of the present chapter is to study the conformal transformation of

m-th root Finsler metric. The spray coefficients, Riemann curvature and Ricci

curvature of conformally transformed m-th root metrics are shown to be certain

rational functions of direction. Further under certain conditions it is shown that

a conformally transformed m-th root metric is locally dually flat if and only if

the transformation is a homothety. Moreover the conditions for the transformed

metrics to be Einstein and Isotropic mean Berwald curvature are also found.

4.1 Introduction

The m-th root Finsler metric has been developed by Shimada (1979), applied to

Biology by Antonelli et al. (1993) and studied by several authors [Srivastava and

Arora (2012), Tayebi et al. (2012), Tayebi and Najafi (2011), Yu and You (2010)

and Prasad and Dwivedi (2002)]. In dimension four, a special fourth root metric

in the form F = 4√y1y2y3y4 is called the Berwald-Moor metric. This metric has

been studied by Balan (2010) and Balan (2006) and considered by physicists as

an important subject for a possible model of space time. Yu and You (2010) have

shown that an m-th root Einstein Finsler metric is Ricci-flat.

The conformal theory of Finsler metric, based on the theory of Finsler spaces

by Matsumoto (1986), has been developed by M. Hashiguchi. Let F and F be

two Finsler metrics on a manifold M such that F = eσ(x)F , where σ is a scalar

function on M , then we call such two metrics F and F are conformally related.

51

More precisely Finsler metric F is called conformally transformed Finsler metric.

A Finsler metric, which is conformally related to a Minkowski metric, is called

conformally flat Finsler metric. The conformal change is said to be a homothety

if σ is a constant.

4.2 Preliminaries

Let F n = (M,F ) be a Finsler space, where M is an n-dimensional C∞-manifold

and F is a Finsler metric.

The normalized supporting element li and angular metric tensor hij of F n are

defined respectively as:

li =∂F

∂yi, hij = F

∂2F

∂yi∂yj. (4.1)

Let F be a Finsler metric defined by F = m√A, where A is given by A :=

ai1i2...im(x)yi1yi2 ...yim , with ai1...im symmetric in all its indices, Shimada (1979).

Then F is called an m-th root Finsler metric. Clearly, A is homogeneous of degree

m in y.

Let

Ai = aii2...im(x)yi2 ...yim =

1

m

∂A

∂yi, (4.2)

Aij = aiji3...im(x)yi3 ...yim =

1

m(m− 1)

∂2A

∂yi∂yj. (4.3)

The normalized supporting element li of F n is given by

li := Fyi =∂F

∂yi=∂ m√A

∂yi=

1

m

∂A∂yi

Am−1m

=Ai

Fm−1. (4.4)

Consider the conformal transformation

F (x, y) = eσ(x)F (x, y)

of m-th root metric F = m√A. Clearly F is also an m-th root Finsler metric on M .

Throughout the chapter we call the Finsler metric F as to conformally transformed

m-th root metric and (M,F ) = Fn as conformally transformed Finser space.

We restrict ourselves for m > 2 throughout the chapter and also the quantities

52

corresponding to the transformed Finsler space F n will be denoted by putting bar

on the top of that quantity, for instance,

A = emσA, Ai = emσAi and Aij = emσAij.

4.3 Fundamental tensor and Spray coefficients of

conformally transformed m-th root metric

The fundamental metric tensor gij of Finsler space F n is given by

gij =1

2

∂2F 2

∂yi∂yj= FFyiyj + FyiFyj .

In view of (4.2), (4.3) and (4.4), we have

gij = (m− 1)Aij

Fm−2− (m− 2)

AiAj

F 2(m−1). (4.5)

The contravariant metric tensor gij of Finsler space F n is given by

gij =Fm−2

(m− 1)Aij +

(m− 2)

(m− 1)

yiyj

F 2, (4.6)

where matrix (Aij) denotes inverse of (Aij), Yu and You (2010). Here we have

used AijAj = Ai = yi.

Since the covariant and contravariant metric tensor of transformed Finsler space

Fn are given by gij = e2σgij and gij = e−2σgij, we have

Theorem 4.1 The covariant metric tensor gij and contravariant metric ten-

sor gij of transformed m-th root Finsler space F n are given as

gij = e2σ((m− 1)

Aij

Fm−2− (m− 2)

AiAj

F 2(m−1)

)(4.7)

and

gij = e−2σ

(Fm−2

(m− 1)Aij +

(m− 2)

(m− 1)

yiyj

F 2

). (4.8)

The geodesics of F n are characterized by a system of equations

d2xi

dt2+Gi

(x,dx

dt

)= 0,

53

where

Gi =1

4gil[F 2]xkyl

yk −[F 2]xl

(4.9)

are called the spray coefficients of F n.

The spray coefficients Gi of F n can be written as

Gi =1

4e−2σgil

∂2(e2σF 2)

∂xk∂ylyk − ∂(e2σF 2)

∂xl

=

1

4e−2σgil

e2σ(F 2

xkylyk − F 2

xl) + 2FFyle2σ2σxkyk − F 2e2σ2σxl

,

i. e.,

Gi = Gi +1

2gil2FFylσxkyk − F 2σxl

, (4.10)

where Gi are given, Yu and You (2010)

Gi =Ail

2(m− 1)

∂Al

∂xkyk − 1

m

∂A

∂xl

. (4.11)

Further in view of equation (4.8) we have

Gi = Gi +1

2

Fm−2

(m− 1)Aij +

(m− 2)

(m− 1)

yiyj

F 2

2Fljσxkyk − F 2σxj

= Gi+

1

2

2F (m−1)yi

(m− 1)Fm−1σxkyk − Fm

(m− 1)Aijσxj +

(m− 2)

(m− 1)

yiyj

F2ljσxkyk − (m− 2)

(m− 1)yiyjσxj

= Gi +

1

2

2yi

(m− 1)σxkyk +

(m− 2)

(m− 1)yiσxjyj − Fm

(m− 1)Aijσxj

.

Here, we have used

Aijlj =yi

Fm−1.

Thus

Gi = Gi +1

2(m− 1)

mσxjyiyj − AAijσxj

. (4.12)

Hence we have

Proposition 4.2 The spray coefficients Gi of the transformed Finsler space

Fn are given by (4.12), where Gi are spray coefficients of Finsler space F n.

In view of equation, (4.11) Gi are rational functions of y, Yu and You (2010).

Hence from equation (4.12), we have

Corollary 4.1 The spray coefficients Gi of the transformed Finsler space F n

are rational functions of y.

54

4.4 Locally dually flat conformally transformed m-

th root metric

Amari and Nagaoka (2000) introduced the notion of dually flat Riemannian met-

rics when they studied the information geometry on Riemannian manifolds. In

Finsler geometry, Shen (2006) extended the notion of locally dually flatness for

Finsler metrics. Dually flat Finsler metrics form a special and valuable class of

Finsler metrics in Finsler information geometry, which play a very important role

in studying flat Finsler information structure. Information geometry has emerged

from investigating the geometrical structure of a family of probability distributions

and has been applied successfully to various areas including statistical inference,

control system theory and information theory [Amari and Nagaoka (2000) and

Amari (1985)].

Definition 4.1 A transformed Finsler metric F = F (x, y) on a manifold M

is said to be locally dually flat, if at any point there is a standard coordinate system

(xi, yi) in TM such that[F

2]xkyl

yk = 2[F

2]xl.

In this case, the coordinate (xi) is called an adapted local coordinate system, Shen

(2006).

For instance, every locally Minkowskian metric is locally dually flat.

Consider the conformal transformation F = eσF , where F is an m-th root metric.

Since

F2

xk = 2e2σσkF2 + e2σF 2

xk = e2σ[F 2xk + 2F 2σk

],

where

σk :=∂σ

∂xk.

We have

F2

xkyl = e2σ[F 2xkyl + 2F 2

ylσk]

F2

xkylyk = e2σ

[F 2xkyly

k + 2Fllσkyk].

55

Therefore

2F2

xl − F2

xkylyk = e2σ

[2F 2

xl + 4F 2σl − F 2xkyly

k − 2ylσ0],

where σ0 := σkyk.

Thus F is locally dually flat metric iff

A(2−2m)

m

m(

2

m− 1)AlA0 + AA0l − 2AAxl

+m

(σ0yl − 2A

2mσl

)= 0, (4.13)

where A0 := Axkyk and A0l := Axkylyk.

The equation (4.13) can be rewritten as

Axl =1

2A

[m(

2

m− 1)AlA0 + AA0l

+m

(σ0yl − 2A

2mσl

)A

(2m−2)m

]. (4.14)

Theorem 4.3 Let F be a conformally transformed m-th root Finsler metric

on a manifold M . Then, F is locally dually flat metric iff (4.14) holds.

Corollary 4.2 If F is locally dually flat metric then the conformally trans-

formed m-th root Finsler metric F is also locally dually flat iff conformal trans-

formation is homothetic.

Proof: In view of Yu and You (2010), F is locally dually flat iff

Axl =1

2A

[(2

m− 1)AlA0 + AA0l

].

Hence F is locally dually flat iff

σ0yl − 2A2mσl = 0. (4.15)

Contracting by yl, we have

σ0F2 − 2F 2σ0 = 0,

i.e. σ0 = 0.

Hence from equation (4.15), σl = 0, i.e. ∂σ∂xl = 0. So σ is constant. Hence the

transformation is homothetic.

The converse is trivial.

56

4.5 Conformally transformed Einstein m-th root

metric

In Finsler geometry, the flag curvature is an analogue of sectional curvature in

Riemannian geometry. A natural problem is to study and characterize Finsler

metrics of constant flag curvature. There are only three local Riemannian metrics

of constant sectional curvature, up to a scaling. However there are lots of non-

Riemannian Finsler metrics of constant flag curvature. For example, the Funk

metric is positively complete and non-reversible with K = −14

and the Hilbert-

Klein metric is complete and reversible with K = −1 [Funk (1929) and Chern and

Shen (2004)].

For a Finsler metric F , the Riemann curvature Ry : TxM → TxM is defined by

Ry(u) = Ri

k(x, y)uk ∂∂xi , u = uk ∂

∂xi , where

Ri

k = 2∂G

i

∂xk− yj

∂2Gi

∂xj∂yk+ 2G

j ∂2Gi

∂yj∂yk− ∂G

i

∂yj∂G

j

∂yk. (4.16)

The Finsler metric F is said to be of scalar flag curvature if there is a scalar

function K = K(x, y) such that

Ri

k = K(x, y)F2δik −

F ykyi

F

. (4.17)

Moreover F is said to be of constant flag curvature if K in equation (4.17) is

constant.

The Ricci curvature of a transformed Finsler metric F on a manifold is a scalar

function Ric : TM → R, defined to be the trace of Ry, i.e.,

Ric(y) := Rk

k(x, y)

satisfying the homogeneity Ric(λy) = λ2Ric(y), for λ > 0. A Finsler metric F

on an n-dimensional manifold M is called an Einstein metric if there is a scalar

function K = K(x) on M such that

Ric = K(n− 1)F2.

57

A Finsler metric is said to be Ricci-flat if Ric = 0. By formula (4.16) and corollary

(4.1), we get the following

Lemma. Ri

k and Ric = Rkk are rational functions in y.

Proposition 4.4 Let F be a non-Riemannian conformally transformed m-th

root Finsler metric with m > 2 on a manifold M of dimension n > 1. If F is an

Einstein metric, then it is Ricci-flat.

Proof: If F is an Einstein metric, i.e. Ric = K(n− 1)F2, and F 2 is an irrational

function, as m > 2 and Ric are rational function of y. Therefore K = 0 and hence

Ric = 0.

Corollary 4.3 Let F = eσ(x)F be a non-Riemannian transformed m-th root

Finsler metric with m > 2 on a manifold M of dimension n > 1. If F is of

constant flag curvature K, then K = 0.

4.6 Conformally transformed m-th root metric with

Isotropic E-curvature

Let if Gi be spray coefficients of a Finsler space F n then the Berwald curvature of

Fn is defined as

Bi

jkl =∂3G

i

∂yj∂yk∂yl.

A transformed Finsler metric F is called a Berwald metric if spray coefficients Gi

are quadratic in y ∈ TxM , for any x ∈ M or equivalently, the Berwald curvature

vanishes. The E-curvature is defined by the trace of the Berwald curvature, i.e.,

Eij =12B

m

mij. A Finsler metric F on an n-dimensional manifold M is said to be

isotropic mean Berwald curvature or of isotropic E-curvature if

Eij =c(n+ 1)

2Fhij, (4.18)

where hij = gij − gipypgjqy

q is the angular metric and c = c(x) is a scalar function

on M . If c = 0, then F is called weakly Berwald metric.

58

From equation (4.7), we have

gij = e2σgij = e2σ((m− 1)

Aij

Fm−2− (m− 2)

AiAj

F 2(m−1)

). (4.19)

The angular metric is given by

hij = gij − lilj = e2σ((m− 1)

Aij

Fm−2− (m− 1)

AiAj

F 2(m−1)

). (4.20)

From equation (4.18) and (4.20), we have

Eij =(n+ 1)c

2Fe2σ(m− 1)

(Aij

Fm−2− AiAj

F 2(m−1)

)

=(n+ 1)c

2eσ(m− 1)

(Aij

Fm−1− AiAj

F (2m−1)

) (F = eσF

)

=(n+ 1)c

2eσ(m− 1)F

(Aij

A− AiAj

A2

) (F = A

1m

)=

(n+ 1)

2A2A

1m eσ(m− 1)c (AijA− AiAj) . (4.21)

In view of equation (4.12), we see that Eij are rational functions with respect to

y. Thus from equation (4.21), we have either c = 0 or

(AijA− AiAj) = 0. (4.22)

Suppose that c = 0. Contracting (4.22) with Ajk yields

Aδki − Aiyk = 0,

which implies that nA = A. This contradicts our assumption n > 1. Therefore

c = 0 and consequently Eij = 0. Thus we have

Proposition 4.5 If F is of isotropic mean Berwald curvature. Then F is

weakly Berwald metric.

59

Chapter 5Transformation of a Finsler Space byNormalised Semi-Parallel Vector Fields

The purpose of the present chapter is to study the properties of a modified Finsler

space obtained by transformation of a Finsler space with the help of two normalised

semi-parallel vector fields.

5.1 Introduction

The semi-parallel vector field in Riemannian Geometry has been introduced by

Fulton (1965), where as in Finsler geometry by Singh and Prasad (1983), for

instance, torse forming vector fields studied by Yano (1944), concurrent vector

fields studied by Yano (1943) and concircular vector fields studied by Adati (1951)

are semi-parallel. The notations and terminology are referred to monograph of

Matsumoto (1986).

Let F n = (M,F ) be a Finsler space, where M is an n-dimensional C∞-

manifold and F is a Finsler metric. The Cartan tensor Cijk of F n is given by

Cijk = 12

∂gij∂yk

. Let CΓ =(F ijk, N

ik, C

ijk

)be Cartan connection on the Finsler Space

F n such that the horizontal and vertical derivative of a vector field X i is written

as

X i|j =

δXi

δxj +XhF ihj and X i|j = ∂Xi

∂yj+XhC i

hj, where δδxj = ∂

∂xj −Nmj

∂∂ym

.

Definition 5.1 [Kitayama (1998)] A normalised vector field X i(x) in a Finsler

space F n is said to be parallel if:

(i) X i is function of coordinate only,

60

(ii) X i|j := δjX

i +XhF ihj = 0, and

(iii) X i|j := ∂jXi +XhCi

hj = XhCihj = 0,

where ∂j stands for ∂∂xj , ∂j stands for ∂

∂yjand δj stands for δ

δxj .

Definition 5.2 [Pandey and Diwedi (1999) and Singh and Prasad (1983)] A

normalised vector field Xi in a Finsler space F n is said to be semi-parallel if:

(i) Xi is function of coordinate only,

(ii) CijkXi = 0, and

(iii) Xi|j = ρ(gij−XiXj), where ρ is a non-zero scalar function of coordinate only.

Consider two normalised semi-parallel vector fields X(α)i , α = 1, 2 in a Finsler

space F n with scalars ρ(α), α = 1, 2 respectively, that is

(i) X(α)i are functions of coordinate only,

(ii) CijkX

(α)i = 0, and

(iii) X(α)i|j = ρ(α)(gij −X

(α)i X

(α)j ), for α = 1, 2.

Kitayama studied a transformed Finsler space F ∗n with metric F ∗(x, y), given by

F ∗2 = F 2 + (β)2, where β = Xiyi and Xi is a parallel vector field; where as Singh

and Prasad (1983) studied a transformed Finsler space F ∗n with metric F ∗(x, y)

given by, F ∗2 = F 2 + (β)2, where β = Xiyi and Xi is a semi-parallel vector field.

The purpose of the present chapter is to study the properties of a transformed

Finsler space F ∗n with the metric F ∗(x, y), given by

F ∗2 = F 2 + β(1)β(2) (5.1)

where β(1) = X(1)i yi and β(2) = X

(2)i yi are two one forms and X

(α)i ; α = 1, 2; are

semi-parallel vector fields satisfying the condition X(1)i X

(2)j = X

(1)j X

(2)i .

Lemma. [Singh and Prasad (1983)] If a Finsler space F n admits normalised

semi-parallel vecror fields X(α)i ; for α = 1, 2, then there exist no functions ξ(α)(x, y)

such that X(α)i = ξ(α)(x, y)yi.

61

5.2 Transformed Finsler space obtained by Nor-

malised semi-parallel vector fields

Consider the transformed Finsler space F ∗n, whose metric F ∗(x, y) is given by

equation (5.1). Differentiating equation (5.1) with respect to yi, the line element

l∗i of F ∗n is given by

2F ∗l∗i = 2Fli + β(1)X(2)i + β(2)X

(1)i . (5.2)

Again differentiating (5.2) with respect to yj, we have the metric tensor g∗ij of F ∗n

as

g∗ij = gij +X(1)i X

(2)j , (5.3)

where gij is the metric tensor of F n.

The reciprocal metric tensor g∗ij of F ∗n is given by

g∗ij = gij − 1

1 + λX(1)iX(2)j, (5.4)

where λ = X(1)i X(2)i.

Again differentiating (5.3) and using (5.4), we have

C∗ijk = Cijk, C

∗ijk = C i

jk. (5.5)

In view of above equation, if F n is Riemannian then F ∗n is also Riemannian. More-

over the (h)hv torsion tensor is invariant under the transformation (5.1). From

definition (5.2) and from Ricci identities for h and v− covariant differentiations

we have

RhijkX(α)h =

ρk + (ρ(α))2X

(α)k

gij−

ρj + (ρ(α))2X

(α)j

gik−X(α)

i

ρkX

(α)j − ρjX

(α)k

,

PhijkX(α)h = ρ(α)Cijk, (5.6)

and

ShijkX(α)h = 0,

where Rhijk, Phijk and Shijk are the components of the h-curvature tensor, hv-

curvature tensor and v-curvature tensor respectively and ρ(α)i = ∂ρ(α)

∂xi . From (5.3),

62

(5.5) and definition (5.2), we have

F ∗ijk = F i

jk +1

2(1 + λ)

[ρ(2)X(1)i

(gjk −X

(2)j X

(2)k

)+ ρ(1)X(2)i

(gjk −X

(1)j X

(1)k

)],

(5.7)

where F ∗ijk and F i

jk are the Cartan’s connection parameters of F ∗n and F n respec-

tively.

From definition (5.2), equation (5.3), (5.4), (5.5) and (5.7), we get the three Car-

tan’s curvature tensors, given by

S∗hijk = Shijk and P ∗

hijk = Phijk and

R∗hijk =Rhijk +

ρ(1)ρ(2)

2(1 + λ)[2(ghjgik − ghkgij) (5.8)

+ ghk

(λ+ 1)X

(1)i X

(1)j +X

(2)i X

(2)j −X

(1)i X

(2)j

− ghj

X

(1)i X

(1)k (λ+ 1) +X

(2)i X

(2)k −X

(1)i X

(2)k

+ gij

(X

(1)h X

(1)k +X

(2)h X

(2)k

)− gik

(X

(1)h X

(1)j +X

(2)h X

(2)j

)]+

1

4(1 + λ)2

[a(λ+ 1)X(1)i + b(X(1)i +X(2)i)

(ghkX

(1)j − ghjX

(1)k )

+c(λ+ 1)X(1)i + d(X(1)i +X(2)i)

(ghkX

(2)j − ghjX

(2)k )]+

λ

λ+ 1

X

(1)i X(2)

m Rmhjk

,

where a = (ρ(2))2 − 2λρ(1)ρ(2), b = 2ρ(1)ρ(2) − λ(ρ(1))2,

c = 2ρ(1)ρ(2) − λ(ρ(2))2 and d = (ρ(1))2 − 2λρ(1)ρ(2).

Theorem 5.1 Let a Finsler space F n admits normalised semi-parallel vecror

fields X(α)i and F ∗n be a modified Finsler space given by (5.1). Then hv and

v-curvature tensors of F n and F ∗n are identical but the h-curvature tensors are

related by (5.8).

Contracting (5.7) by yjyk, we have

G∗i = Gi +1

4(1 + λ)

[ρ(2)X(1)i

F 2 − (β(2))2

+ ρ(1)X(2)i

F 2 − (β(1))2

]. (5.9)

Differentiating (5.9) with respect to yj and yk respectively, we have

G∗ijk = Gi

jk +1

2(1 + λ)

ρ(2)X(1)i

(gjk −X

(2)j X

(2)k

)+ ρ(1)X(2)i

(gjk −X

(1)j X

(1)k

),

(5.10)

63

where

2Gi = F ijky

jyk, X0 = Xjyj and Gi

jk =∂2Gi

∂yj∂yk

are the Berwald’s connection parameters.

Theorem 5.2 If the vector field X(1)i is parallel and X(2)

i is semi-parallel in

Finsler space F n then the vector field X(2)i is semi-parallel also in transformed

Finsler space F ∗n obtained under the transformation (5.1).

Proof. In view of definitions (5.1) and (5.2) X(1)i|j = 0, that is, ρ(1) = 0 and

X(2)i|j = ρ(2)(gij−X(2)

i X(2)j ). Further since Cartan tensor Cijk and Ci

jk are invariant

under transformation (5.1). We need only to show thatX(2)i|∗j = k(gij−X(2)

i X(2)j ) for

some scalar k. Here |∗ denotes h-covariant differentiation with respect to Cartan

connection CΓ∗ of F ∗n. Considering (5.7) we have

X(2)i|∗j = X

(2)i|j − X

(2)h

2(1 + λ)

[ρ(2)X(1)h

(gij −X

(2)i X

(2)j

)+ ρ(1)X(2)h

(gij −X

(1)i X

(1)j

)]= k(gij −X

(2)i X

(2)j ),

(5.11)

where k = (2+λ)ρ(2)

2(1+λ).

Corollary. If the vector fields X(1)i and X

(2)i are parallel in Finsler space F n

then these vector fields are also parallel in transformed Finsler space F ∗n obtained

under the transformation (5.1).

Suppose the spaces F n and F ∗n are in geodesic correspondence [Singh and Prasad

(1983)] then

G∗i = Gi + P (x, y)yi, (5.12)

where P (x, y) is positively homogeneous of degree one in yi.

Comparing (5.9) and (5.12), we get

1

4(1 + λ)

[ρ(2)X(1)i

F 2 − (β(2))2

+ ρ(1)X(2)i

F 2 − (β(1))2

]= Pyi. (5.13)

Contracting above by X(1)i and X(2)i separately, we have

P =ρ(2)

(F 2 − (β(2))2

)(λ− 1)

4 (λβ(2) − β(1))=ρ(1)

(F 2 − (β(1))2

)(λ− 1)

4 (λβ(1) − β(2)), (5.14)

provided λ2 = 1. Thus we have

64

Theorem 5.3 If a Finsler space F n admits normalised semi-parallel vector

fields X(α)i and F ∗n be a modified Finsler space given by (5.1). Then the spaces

F n and F ∗n are in geodesic correspondence if P satisfies (5.14) with λ2 = 1.

Remark. If the two normalised semi-parallel vector fields are same, we have

λ = 1. In view of Theorem (2) of Singh and Prasad (1983), the two spaces are not

in geodesic correspondence.

Next suppose that F n and F ∗n are in conformal correspondence that is g∗ij =

ψ(x)gij. Comparing this with equation (5.3), we have

Theorem 5.4 Let a Finsler space F n (n > 2) admits normalised semi-

parallel vecror fields X(α)i and F ∗n be a modified Finsler space given by (5.1).

Then the spaces F n and F ∗n are not in conformal correspondence.

5.3 Special Finsler spaces with semi-parallel vec-

tor fields

In this section we consider some special Finsler spaces admitting normalized semi-

parallel vector fields. First we consider a Finsler space with T-condition, that

is

Thijk := FChij|k + lhCijk + liChjk + ljChik + lkChij = 0. (5.15)

Contracting (5.15) by Xh, we have XhlhCijk = 0 by Definition (5.2) (ii). Thus we

have Cijk = 0, because Xhlh = β(1)

F= 0. Consequently from (5.5), we have

Theorem 5.5 If a Finsler space F n satisfying T-condition admits a semi-

parallel vector field, then both the Finsler spaces are Riemannian.

The generalised T -condition is defined by

Tij := Tijrsgrs = FCi|j + liCj + ljCi = 0, (5.16)

where the tensor Tij is called the contracted T -tensor and Ci = Cijkgjk is the

torsion vector [Tiwari (2012) and Pandey and Tiwari (1999)].

Contracting (5.16) byX i and using Definition (5.2) (ii), we have Cj = 0. According

to Deiche’s theorem and from (5.5), we have

65

Theorem 5.6 If a Finsler space F n satisfying generalised T -condition ad-

mits a semi-parallel vector field, then both the Finsler spaces F n and F ∗n are

Riemannian.

A Finsler space F n, (n > 2) is called quasi C-reducible if the torsion tensor Cijk

is written as

Cijk = AijCk + AjkCi + AkiCj, (5.17)

where Aij is symmetric tensor satisfying Ai0 = Aijyj = 0.

Contracting (5.17) by X iXj we get ξCk = 0, where ξ = AijXiXj. Thus we have

Theorem 5.7 If a quasi C- reducible Finsler space F n, (n > 2) admits semi-

parallel vector fields then the spaces F n and F ∗n are Riemannian provided ξ = 0.

Now we consider a C-reducible Finsler space which is characterized by

Cijk = AijCk + AjkCi + AkiCj, (5.18)

where Aij =hij

(n+1)from Matsumoto (1972). Since hijX iXj = 0 for a Finsler space

with n > 2.

Corollary. If a C- reducible Finsler space F n, (n > 2) admits semi-parallel vector

fields then the spaces F n and F ∗n are Riemannian.

An n-dimensional Finsler space F n is said to be semi-C-reducible if its (h)hv-

torsion tensor Cijk is written as

Cijk = p[hijCk + hjkCi + hkiCj] + q[CiCjCk], (5.19)

where p, q are scalars satisfying p(n+1)+qC2 = 1 and C2 = gijCiCj. In particular

if p = 0 but C2 = 0, that is, the torsion tensor Cijk is written as Cijk =CiCjCk

C2 ,

F n is said to be C2 − like. Contracting (5.19) by X iXj we have pCk = 0 which

shows p = 0, thus we have

Theorem 5.8 If a semi-C- reducible Finsler space F n, (n > 2) admits semi-

parallel vector fields then the spaces F n and F ∗n are C-2 like.

66

A P-2 like Finsler space F n (n > 2) is characterized by

Phijk = KhCijk −KiChjk, (5.20)

where Kh is a covariant vector field. Contracting above equation by X(α)h and

using (5.6) and Definition (5.2) (ii), we have (X(α)hKh − ρ(α))Cijk = 0. Thus we

have

Theorem 5.9 If a P2-like Finsler space F n, (n > 2) admits semi-parallel

vector fields then the spaces F n and F ∗n are Riemannian provided X(α)hKh = ρ(α)

for any α = 1, 2.

An n-dimensional Finsler space F n is called a Landsberg space if the (v)hv-torsion

tensor Pijk of F n vanishes. Further a Finsler space F n is called P-reducible if

torsion tensor Pijk of F n is written in the form

Pijk =1

n+ 1(hijPk + hjkPi + hkiPj), (5.21)

where Pi = P rir = Ci|0. Contracting above equation by X iXj and using (5.6), we

obtain hijX iXjPk = 0. But since hijX iXj = 0 we have Pk = 0. Thus

Theorem 5.10 If a P-reducible Finsler space F n, (n > 2) admits semi-

parallel vector fields then the spaces F n and F ∗n are Landsberg.

67

Chapter 6Predator-prey model with prey refuges:Jacobi stability vs Linear stability

In this chapter, the dynamics of a predator-prey model is proposed and analyzed.

Three types of refuges: those that protect a constant number of prey population, a

constant proportion of prey population and a function of predator-prey encounters

using refuges are considered. Linear stability analysis based on Lyapunov theory

and Jacobi stability analysis based on KCC theory are carried out. Comparisons

of results obtained in both cases shows that, Jacobi stability analysis of these

models reflects the better ecological interpretation.

6.1 Introduction

Geometry is the link between physical world and its visualization. Nature’s ge-

ometrisation has been the subject of theoretical interest. This contains use of

geometric hypothesis and applications in the natural sciences, such as the general

theory of relativity and Cosmology in physics [Yamasaki and Yajima (2013)]. In

biology, nature’s geometrisation is also of scientific and technological interest. For

example, the KCC theory which are applied to many biological problems, such

as production in the Volterra method (Antonelli et al. (2003) and Antonelli et

al. (1993)), the Volterra-Hamilton system ( Antonelli et al. (2011) and Antonelli

and Bucataru (2001)), Tyson’s model for the cell division cycle (Antonelli et al.

(2002)) and the robustness of biological systems (Sabau (2005)). The notion of the

KCC (Kosambi-Cartan-Chern) theory regard in works of Kosambi (1933), Cartan

(1933) and Chern (1939), and the abbreviation KCC come from the names of

68

these three initiators, who profound the geometric theory of a system of second

order ordinary differential equation (SODE). The first attempt to establish and to

develop systematically the KCC theory is due to Antonelli and Bucataru (2001).

The KCC theory describes the evolution of a dynamical system in geometric terms,

by considering it as a geodesic in a Finsler space. Thus a second order differential

equation can be investigated in geometric terms by KCC-theory inspired by the

geometry of a Finsler space. By associating a non-linear connection and a Berwald

type connection to the differential system, five geometrical invariants are obtained

with the second invariant giving the Jacobi stability of the system.

There are so many Mathematicians and Ecologists among them the prey-

predator model is a subject of great interest. The linear and non-linear asymp-

totic stability conditions of model in a homogeneous habitat have been obtained.

It is observed that many species have already become extinct and many others

are at the stage of extinction due to several reasons such as over exploitation,

mismanagement of resources, indiscriminate harvesting, over predation, loss of

habitat and environmental pollution, etc. To save the species from getting ex-

tinction we are taking measures like improving conditions of their habitat, reduce

the interaction of the species with external agents which tend to decrease their

numbers, impose restriction on harvesting, creating refuges, establish protected

areas etc. so that the species grow in these protected areas without any exter-

nal disturbances and hence the protected population can improve their numbers.

Chattopadhyay et al. (2000) studied a prey-predator model with some cover on

prey species. Kar (2006) proposed a predator-prey model incorporating a prey

refuge and independent harvesting on either species. Recently, (Devi (2013) and

Devi (2012)) studied the effect of prey refuge on a ratio-dependent predator-prey

model with stage-structure and non constant harvesting of prey population, re-

spectively. In these models she observed that prey refuge parameter plays a very

crucial role in the analysis.

In the general theory of relativity and Cosmology we can geometrise the inter-

action between masses but in this chapter we geometrise the interaction between

living things. The purpose of present chapter is to consider the differential geo-

69

metric structure for some dynamical systems of the predator-prey model based on

KCC theory.

Organization of this chapter is as follows:

KCC theory and Jacobi Stability theory are described in section 6.2. Section 6.3

presents the applications of geometric theory to second order systems. Mathe-

matical models and their analyses are given in subsection 6.4.1, 6.4.2 and 6.4.3

of section 6.4. Numerical simulations and discussions are presented in section 6.5

followed by conclusion in section 6.6.

6.2 Preliminary

6.2.1 About Kosambi-Cartan-Chern (KCC) theory and Ja-

cobi stability

In present section, we recall the basics of KCC theory and Jacobi stability based

on the papers of Bohmer et al. (2010), Yajima and Nagahama (2007) and Antonelli

and Bucataru (2001).

Let M be a real smooth n-dimensional manifold, and (TM, π,M) be its tangent

bundle, where π : TM −→ M is a projection from the total space TM to the

base manifold M . A point x ∈ M has local coordinates (xi), where i = 1, ..., n.

The local chart of a point in TM is denoted by (xi, xi), where xi = dxdt

and t is an

absolute invariant.

The equations of motion of a dynamical system can be derived from a Lagrangian

L via the Euler-Lagrange equations. For a regular Lagrangian the Euler-Lagrange

equations are equivalent to a system of second-order differential equations

xi +Gi(x, x) = 0, (6.1)

where Gi(x, x) is a smooth function. Vary the trajectories xi(t) of the system (6.1)

into nearby ones according to

xi(t) = xi(t) + ϵui(t), (6.2)

70

where ui(t) are components of a contravariant vector field along the path xi(t)

and ϵ denotes a scalar parameter with small value |ϵ|. Substituting from (6.2) into

equations (6.1) and taking the limit ϵ→ 0 we obtain the variational equations

ui +∂Gi

∂xjuj +

∂Gi

∂xjuj = 0. (6.3)

By using the KCC-covariant differential we can write equation (6.3) in the covari-

ant formD2ui

Dt2= P i

juj, (6.4)

where D(...)/Dt is a covariant differential defined by

Dui

Dt=dui

dt+N i

juj, (6.5)

N ij are coefficient of the non-linear connection given by

N ij =

1

2

∂gi

∂xj, (6.6)

P ij is the deviation curvature tensor

P ij = − ∂gi

∂xj+∂N i

j

∂xkxk −Gi

jkgk +N i

kNkj (6.7)

and Gijk is a Finsler (Berwald) connection given by Antonelli (2003),

Gijk =

∂N ij

∂xk. (6.8)

Equation (6.4) is called the Jacobi equations, or the variation equations attached

to the system of SODE, and P ij is called the second KCC-invariant or the deviation

curvature tensor. When the system (6.1) describes the geodesic equations in either

Riemann or Finsler geometry, equation (6.4) is the usual Jacobi equation.

The first term of (6.7): ∂gi

∂xj is the curvature when Finsler (Berwald) connection

and coefficient of the non-linear connection become zero, we get

Zij =

∂gi

∂xj. (6.9)

The deviation curvature tensor P ij gives the stability of whole trajectories via the

following theorem, Antonelli et al. (1993) :

71

Theorem 6.1 The trajectories of (6.1) are Jacobi stable if and only if the real

parts of the eigenvalues of the deviation curvature tensor P ij are strictly negative

everywhere, and Jacobi unstable, otherwise.

In particular, the trajectories of the one-dimensional system are Jacobi stable

when P 11 ≤ 0, and Jacobi unstable when P 1

1 > 0.

The third, fourth and fifth invariants of the system (6.1) are given by Antonelli

(2003)

P ijk ≡

1

3

(∂P i

j

∂xk− ∂P i

k

∂xj

), P i

jkl ≡∂P i

jk

∂xl, Di

jkl ≡∂Gi

jk

∂xl. (6.10)

The third invariant is a torsion tensor, while the fourth and fifth invariants are the

Riemann-Christoffel curvature tensor, and the Douglas tensor, respectively given

by Antonelli (2003). The second invariant is expressed using the third invariant

and the h−covariant derivative of the first invariant where as the vertical compo-

nent of a semispray with respect to non-linear connection, gives the first invariant

of semispray, called the deviation tensor. The third, fourth and fifth tensors al-

ways exist and they describe the geometrical properties of a system of SODE (6.1)

(Sabau (2005) and Antonelli (2003)).

Throughout this chapter, Einstein’s summation convention is used and we call Zij

as the zero-connection curvature.

6.3 Applications of geometric theory to second or-

der system

In present section, we derive deviation curvature for some dynamical systems of

the predator-prey model. Linear stability analysis are applied for the predator-

prey model, which is the theory of local stability around a point on the tangent

space. In this case, the equation (6.1) is a first-order differential equation with

respect to x and equation (6.3) is the Jacobi stability which reduces to an equation

in a linear stability theory given by Yajima and Nagahama (2008). Therefore, the

Jacobi stability gives a global stability than the linear stability. The Jacobian

matrix of the linearised system plays an important role in linear stability analysis.

72

It would be interesting to correlate linear stability with Jacobi stability. In other

words, to compare the signs of the eigenvalues of the Jacobian matrix J at a

fixed point with the signs of the eigenvalues of the deviation curvature tensor P ij

evaluated at the same point. We express the geometric quantities of KCC theory

in terms of the Jacobian matrix of the linearised system.

6.3.1 Geometric quantities and Jacobian matrix

Let us consider a two-dimensional vector field described by

xi = f i(x), (6.11)

where i = 1, 2, ..., n and f i denote a given function. Equation (6.11) is approx-

imated by a linear system around an equilibrium point xi0 using the relation

xi = xi0 + ηi, where ηi is a small quantity. That is,

ηi = J ij(x0)η

j, (6.12)

where J ij(x0) is the Jacobian matrix of f i evaluated at point x0.

Even though this should be possible in the general case but here we consider the

two-dimensional case (i = 1, 2)

η = J(x0)η, (6.13)

where

η =(η1, η2

)T, (6.14)

J(x0) = J =

∂1f1(x0) ∂2f

1(x0)

∂1f2(x0) ∂2f

2(x0)

(6.15)

and ∂i = ∂(...)/∂xi. The simultaneous differential equation (6.13) can be rewritten

as a SODE. When we consider the coordinate system (ηi, ηi), we have the following

equation for i = 1 :

η1 − tr[J ]η1 + det[J ]η1 = 0. (6.16)

This is a particular case of (6.1) for g1 = −tr[J ]η1+det[J ]η1. Therefore, equations

(6.6), (6.9) and (6.7) give the non-linear connection, the zero-connection curvature

73

and the deviation curvature of the linearised system, respectively as:

N = −1

2tr[J ], (6.17)

Z = det[J ], (6.18)

P =1

4

(tr[J ])2 − 4det[J ]

= N2 − Z, (6.19)

where N = N11 , Z1

1 = Z and P = P 11 (we use same notations throughout the

chapter). From equation (6.8), the Finsler connection vanishes in this linearised

system. Now, equations (6.17), (6.18) and (6.19) show that the geometric quanti-

ties of the linearised system can be easily calculated when the Jacobian matrix of

the system is obtained. The system is

linear stable for N > 0,

linear unstable for N < 0.

The left term of equation (6.19) is related to the Jacobi stability, i.e.

Jacobi stable for P ≤ 0,

Jacobi unstable for P > 0.

6.4 Mathematical models

6.4.1 Predator-prey model for a constant number of prey

using refuges with exponential growth rate

Consider a habitat where prey and predator species are living together. The

predator-prey interaction between any two species occurs when one species (the

predator) feeds on the other species ( the prey ).

In 1925 the American biophysicist Alfred James Lotka and Italian Mathematician

Vito Volterra describe a simple Mathematical model for the interaction between

predators and their prey by means of the following non-linear differential equa-

tions,

x(t) = rx− a(x−m)y, x(0) = x0 > m > 0, (6.20)

74

y(t) = b(x−m)y − cy, y(0) = y0 > 0. (6.21)

Here, x(t) and y(t) are densities of prey and predator population respectively

at time t. r is the natural growth rate of the prey population, m is constant

prey population, a is the rate at which predators capture prey, b is the rate at

which predators increase by consuming prey and c is the natural death rate of the

predator population. All parameters r,m, a, b and c are assumed to be positive.

This system can be approximated by a linear system around an equilibrium point

(x01, y01). From (6.15), the Jacobian matrix of the system is

J =

r − ay01 −a(x01 −m)

by01 b(x01 −m)− c

. (6.22)

Next, we consider the deviation curvature in two cases: (i) when both species

coexist, (ii) when both species extinct.

In the case when the two species coexist, i.e. x01 = 0 = y01, we have equilibrium

point (x01, y01), where x01 = m+ cband y01 = r

ac(bm+c). Therefore, (6.22) becomes

J =

− rbmc

−acb

rbac(bm+ c) 0

. (6.23)

From (6.17) and (6.18), the non-linear connection and the zero-connection curva-

ture are given by N = 12bmrc> 0 and Z = r(bm + c) > 0. Then, from (6.19), the

deviation curvature of coexistence is

P =1

4

b2m2r2

c2− r(bm+ c). (6.24)

Here system is always linear stable because N > 0.

In the case of extinction, i.e. x01 = 0 = y01, (6.22) becomes

J =

r am

0 −bm− c

. (6.25)

In this case, since N = (bm+c−r)2

and Z = −(bm+ c)r < 0, the deviation curvature

of extinction is

P = N2 − Z =

[(bm+ c+ r)

2

]2> 0. (6.26)

This is always positive, i.e. the extinction state is always Jacobi unstable where

as system is linear stable if N > 0, i. e., if bm + c > r, and linear unstable if

bm+ c < r.

75

6.4.2 Predator-prey model with constant proportion of prey

using refuges

Consider the predator-prey model model with constant proportion of prey using

refuges. The standard form of the system is given by

x(t) = rx− a(1− η)xy, x(0) = x0 > 0, (6.27)

y(t) = b(1− η)xy − cy, y(0) = y0 > 0. (6.28)

0 < η < 1,

where η is constant proportion of prey which is assumed to be positive and other

parameters r, a, b and c have same meaning as given in section (6.4.1).

This system can be approximated by a linear system around an equilibrium point

(x02, y02). From (6.15), the Jacobian matrix of the system is

J =

r − ay02 + aηy02 −ax02 + aηx02

by02 − bηy02 bx02 − bηx02 − c

. (6.29)

Next, we consider the deviation curvature in two cases: (i) when both species

coexist, (ii) when both species extinct.

In the case when the two species coexist, i.e. x02 = 0 = y02, we have equilibrium

point (x02, y02), where x02 = cb(1−η)

and y02 = ra(1−η)

. Therefore, (6.29) becomes

J =

0 −acb

bra

0

. (6.30)

From (6.17) and (6.18), the non-linear connection and the zero-connection curva-

ture are given by

N = 0 and Z = cr, respectively. Then, from (6.19), the deviation curvature for

the two species coexisting is

P = −cr. (6.31)

In the case of extinction, i.e. x02 = 0 = y02, (6.29) becomes

J =

r 0

0 −c

. (6.32)

76

In this case, since N = −12(r − c) and Z = −rc, the deviation curvature for the

case of extinction is

P =1

4(r + c)2. (6.33)

For the Jacobi stable, P should be negative or zero, otherwise Jacobi unstable but

here P is always positive. Therefore the extinction state is the Jacobi unstable

where as system is linear stable if N > 0, i. e., if r < c, and linear unstable if

r > c.

6.4.3 Predator-prey model with the function of predator-

prey encounters for prey refuge

Consider the predator-prey model with a function of predator-prey encounters for

prey refuge. The standard form of the system is given by

x(t) = rx− a(1− ϵy)xy, x(0) = x0 > 0, (6.34)

y(t) = b(1− ϵy)xy − cy, y(0) = y0 > 0. (6.35)

where ϵ > 0 is a proportionality constant or predator-prey encounters function and

all other parameters r, a, b and c have same meaning as given in section (6.4.1).

This system can be approximated by a linear system around an equilibrium point

(x03, y03). From (6.15), the Jacobian matrix of the system is

J =

r − a(1− ϵy03)y03 −ax03 + 2ϵax03y03

b(1− ϵy03)y03 bx03 − 2bϵx03y03 − c

. (6.36)

Next, we consider the deviation curvature in two cases: (i) when both species

coexist, (ii) when both species extinct.

In the case when the two species coexist, i.e. x03 = 0 = y03, we have equilibrium

point (x03, y03), where x03 = 2ac

b[a∓√a2−4aϵr]

and y03 = a∓√a2−4aϵr2aϵ

.

Taking positive sign, we get an equilibrium point

(x03, y03) =

(2ac

b[a+

√a2 − 4aϵr

] , a+√a2 − 4aϵr

2aϵ

). (6.37)

77

Therefore, Jacobian matrix (6.36) for equilibrium point (x03, y03) becomes

J =

0 2ac√a2−4aϵr

b(a+√

a2−4aϵr)

bra

−ac−3c√a2−4aϵr

a+√a2−4aϵr

. (6.38)

From (6.17) and (6.18), the non-linear connection and the zero-connection curva-

ture for equilibrium point (x03, y03) are given by

N = 12ac+3c

√a2−4aϵr

a+√a2−4aϵr

and Z = −2cr√a2−4aϵr

a+√a2−4aϵr

, respectively.

Then, from (6.19), the deviation curvature for the two species coexisting is

P =c

4(a+√a2 − 4aϵr)2

[c(a+ 3

√a2 − 4aϵr)2 + 8r

√a2 − 4aϵr(a+

√a2 − 4aϵr)

](6.39)

In this case system is linear stable if a ≥ 4ϵr.

In the case of extinction, i.e. x03 = 0 = y03, (6.36) becomes

J =

r 0

0 −c

. (6.40)

In this case, since N = −12(r − c) and Z = −rc, the deviation curvature for the

case of extinction is

P =1

4(r + c)2. (6.41)

For the Jacobi stable, P should be negative or zero, otherwise Jacobi unstable but

here P is always positive. Therefore the extinction state is the Jacobi unstable

where as system is linear stable if N > 0, i. e., if r < c, and linear unstable if

r > c.

6.5 Numerical simulations and discussion

6.5.1 Table

78

Cases Linear Stability Jacobi Stability

Fixed no. of prey Co-existence Always linear Jacobi stable if

using refuges of species stable b2m2r ≤ 4c2(bm+ c)

Extinction Linear stable if Always Jacobi

of species bm+ c > r unstable

Constant proportion of Co-existence bifurcation Always Jacobi

prey using refuges of species stable

Extinction Linear stable if Always Jacobi

of species r < c unstable

Function of predator-prey Co-existence Linear stable if Jacobi unstable

encounter using refuges of species a ≥ 4ϵr if a ≥ 4ϵr

Extinction Linear stable if Always Jacobi

of species r < c unstable

From table (6.5.1), we notice the following observations:

(i) For co-existence of species, when fixed number of prey using refuges, it is

observed that system is always linearly stable whereas Jacobi stability if system

depends on the prey refuge parameter m. Inequality b2m2r ≤ 4c2(bm + c), may

not be satisfied for large values of m. This reflects the ecological meaning of

predator-prey system that if more prey population will use refuge, then predator

population will go to extinct.

(ii) Again, for co-existence of species, when constant proportion of prey using

refuges, it is observed that in case of linear stability bifurcation occurs, whereas

system is always Jacobi stable.

(iii) Lastly, when a function of predator-prey encounter using refuges, system

is linearly stable if a > 4ϵr, whereas under the same condition system is Jacobi

unstable. From a > 4ϵr, we note that this inequality may be satisfied for larger

values of ϵ, i. e. if predator-prey encounters are occur more frequently then prey

population will go towards extinction.

79

From all these observations, we can say that, study of Jacobi stability behavior

of systems reflects the better ecological interpretation.

To substantiate our all analytical findings numerically, we consider the fol-

lowing set of parameter values r = 10, c = 3, a = 1.5 and b = 0.8.

Values of parameters are hypothetical and do not necessarily have a biological

meaning.

Figures 1-8, are the plots of prey and predator population versus time for

different values of prey refuge parameter m(= 0.2, 0.5, 0.8, and 3.0) when constant

numbers of prey are using refuges. From these figures, we observe that oscil-

latory behavior of these populations decreases with increase in value of m and

consequently populations attain their equilibrium levels in less time.

Figures 09-14, are the plots of prey and predator population versus time

for different values of η(= 0.2, 0.5, and 0.8) when constant proportion of prey

using refuges. From these figures, it can be depicted that both population never

attain their equilibrium level, because oscillatory behavior and also peak value of

oscillations increases with time. This is the case of bifurcation.

Figures 15-18, are the plots of prey and predator populations against time

for different values of ϵ(= 0.02 and 0.03) prey refuge parameter when function of

prey and predator encounters using refuges. These figures show that, equilibrium

values of both populations increase with ϵ. It is also observed that oscillatory

behavior decreases with time and ultimately populations attain their equilibrium

level.

Figures 19-22, are the phase planes of prey versus predator population for

different values of m(= 0.2, 0.5, 0.8, and 3.0). For these values of m, we obtain

stable spirals showing that populations eventually attain their equilibrium level.

This is the case of global stability for any system.

Figures 23-25, are the phase planes of prey versus predator population for

different values of η(= 0.2, 0.5, and 0.8). Here, we note that bifurcation (neither

stable nor unstable spirals) occurs for each value of η.

Figures 26-27, are the phase planes of prey versus predator population for

different values of ϵ(= 0.02 and 0.03). Here, we obtain stable spirals for these

80

0 5 10 15 201

2

3

4

5

6

7

8

Time (t)

prey

(x)

m=0.2

Fig. 1: Time vs prey for m=0.2

0 5 10 15 202

3

4

5

6

7

8

9

Time (t)

prey

(x)

m=0.5

Fig. 2: Time vs prey for m=0.5

0 5 10 15 202

3

4

5

6

7

8

9

10

Time (t)

prey

(x)

m=0.8

Fig. 3: Time vs prey for m=0.8

0 5 10 15 202

4

6

8

10

12

14

16

Time (t)

prey

(x)

m=3

Fig. 4: Time vs prey for m=3

0 5 10 15 204

5

6

7

8

9

10

11

Time (t)

pred

ator

(y)

m=0.2

Fig. 5: Time vs predator for m=0.2

0 5 10 15 204

5

6

7

8

9

10

11

12

Time (t)

pred

ator

(y)

m=0.5

Fig. 6: Time vs predator for m=0.5

81

0 5 10 15 204

5

6

7

8

9

10

11

12

Time (t)

pred

ator

(y)

m=0.8

Fig. 7: Time vs predator for m=0.8

0 5 10 15 202

4

6

8

10

12

14

16

18

Time (t)

pred

ator

(y)

m=3

Fig. 8: Time vs predator for m=3

0 5 10 15 200

2

4

6

8

10

12

14

Time (t)

prey

(x)

η=0.2

Fig. 9: Time vs prey for η = 0.2

0 5 10 15 200

5

10

15

20

25

30

35

Time (t)

prey

(x)

η=0.5

Fig. 10: Time vs prey for η = 0.5

0 5 10 15 200

50

100

150

Time (t)

prey

(x)

η=0.8

Fig. 11: Time vs prey for η = 0.8

0 5 10 15 202

4

6

8

10

12

14

16

Time (t)

pred

ator

(y)

η=0.2

Fig. 12: Time vs predator for η = 0.2

82

0 5 10 15 200

5

10

15

20

25

30

35

Time (t)

pred

ator

(y)

η=0.5

Fig. 13: Time vs predator for η = 0.5

0 5 10 15 200

20

40

60

80

100

120

140

Time (t)

pred

ator

(y)

η=0.8

Fig. 14: Time vs predator for η = 0.8

0 5 10 15 202

3

4

5

6

7

8

9

10

Time (t)

prey

(x)

ε=0.02

Fig. 15: Time vs prey for ϵ = 0.02

0 5 10 15 202

4

6

8

10

12

Time (t)

prey

(x)

ε=0.03

Fig. 16: Time vs prey for ϵ = 0.03

0 5 10 15 204

5

6

7

8

9

10

11

12

13

Time (t)

pred

ator

(y)

ε=0.02

Fig. 17: Time vs predator for ϵ = 0.02

0 5 10 15 204

6

8

10

12

14

16

Time (t)

pred

ator

(y)

ε=0.03

Fig. 18: Time vs predator for ϵ = 0.03

83

0 2 4 6 84

5

6

7

8

9

10

11

prey (x)

pred

ator

(y)

m=0.2

Fig. 19: Prey vs predator for m=0.2

2 4 6 8 104

5

6

7

8

9

10

11

12

prey (x)

pred

ator

(y)

m=0.5

Fig. 20: Prey vs predator for m=0.5

2 4 6 8 104

5

6

7

8

9

10

11

12

prey (x)

pred

ator

(y)

m=0.8

Fig. 21: Prey vs predator for m=0.8

0 5 10 15 202

4

6

8

10

12

14

16

18

prey (x)

pred

ator

(y)

m=3

Fig. 22: Prey vs predator for m=3

0 5 10 152

4

6

8

10

12

14

16

prey (x)

pred

ator

(y)

η=0.2

Fig. 23: Prey vs predator for η = 0.2

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

prey (x)

pred

ator

(y)

η=0.5

Fig. 24: Prey vs predator for η = 0.5

84

0 50 100 1500

20

40

60

80

100

120

140

prey (x)

pred

ator

(y)

η=0.8

Fig. 25: Prey vs predator for η = 0.8

2 4 6 8 104

6

8

10

12

14

prey (x)

pred

ator

(y)

ε=0.02

Fig. 26: Prey vs predator for ϵ = 0.02

2 4 6 8 10 124

6

8

10

12

14

16

prey (x)

pred

ator

(y)

ε=0.03

Fig. 27: Prey vs predator for ϵ = 0.03

85

values of ϵ.

6.6 Conclusions

In this chapter, we have proposed and analyzed predator-prey models in three

cases: 1) when constant number of preys are using refuges, 2) when constant

proportion of preys are using refuges and 3) when a function of predator-prey

encounter using refuges. Analyses of these models have been done using the Lya-

punov and KCC theory of stability. Jacobi stability behaviors have been studied

by obtaining the deviation curvature in each case. Stability results have been com-

pared for both cases. In case when constant proportion of prey is using refuges, we

have observed that Jacobi stability and linear stability behaviors of populations

completely differ from each other. In the case when a function of prey-predator

encounters using refuges, it has been observed that, inequality under which the

system is linearly stable also holds for Jacobi instability of the system. From all

these findings, we have also pointed out a very important observation that Jacobi

stability analysis reveals the better ecological interpretation of any system. This

is so because Jacobi stability analysis also includes the non-linear terms of differ-

ential equations but in linear stability analysis we omit non-linear terms.

The models presented in this chapter are of great use to maintain the stability of

parks where prey and predator population live together.

86

Chapter 7Finsler-Randers Cosmological modelsin Modified Gravity Theories

In this chapter, Finsler-Randers cosmological models in modified gravity theories

have been investigated. The de Sitter, power law and general exponential solu-

tions are assumed for the scale factor in the corresponding cosmological models.

For each scenario, we have discussed all energy conditions in detail. We have

also investigated the behaviour of FR cosmological models in modified theories of

gravity like Einstein theory, Hoyle-Narlikar Creation field theory, Lyra geometry

and General class of scalar-tensor theories.

7.1 Introduction

Particular attention over the last decade has been paid on the so-called Finsler-

Randers (hereafter FR) cosmological model [Randers (1941)]. In general metrical

extensions of Riemann geometry can provide a Finslerian geometrical structure

in a manifold which leads to generalized gravitational field theories. During the

last decade there is a rapid development of applications of Finsler geometry in its

FR context, mainly in the topics of general relativity, astrophysics and cosmology.

The spatially homogeneous cosmological models allow extension of cosmological

investigations to distorting and rotating universes, giving estimates of effects of

anisotropy on primordial element production and on the measured CMBR spec-

trum anisotropy [Ellis and Elst (1999)]. Apart from observational reasons, there

are various theoretical considerations that have motivated the study of anisotropic

cosmologies. Among these are (i) some kind of singularity in our “past" is strongly

87

indicated if certain reasonable conditions hold [Hawking and Ellis (1973)]. How-

ever, it could differ greatly from the type found in FRW models [Belinski et al.

(1970)]. (ii) The “Chaotic Cosmology” programme of Misner (1968) sought mech-

anism to explain why the observed isotropy and homogeneity should exist regard-

less of the initial conditions [MacCallum (1979), Ellis (1993) and Kolb and Turner

(1990)].

There exist wide class of anisotropic cosmological models, which are often

studied in cosmology [Misner et al. (1973)]. There are theoretical arguments that

sustain the existence of an anisotropic phase that approaches an isotropic case

[Misner (1968)] (Chaotic Cosmology). Also, anisotropic cosmological models are

found a suitable candidate to avoid the assumption of specific initial conditions

in FRW models. The early Universe could also be characterized by irregular ex-

pansion mechanism. Therefore, it would be useful to explore cosmological models

in which anisotropies, existing at early stage of expansion, are damped out in

the course of evolution. Interest in such models have received attention [Hu and

Parker (1978)].

Stavrinos (2008) have studied the Friedman-like Robertson-Walker model in

generalized metric space time with weak anisotropy. Recently, Basilakos and

Stavrinos (2013) have studied cosmological equivalence between the Finsler-Randers

space time and the DGP gravity model. This motivates the researchers to con-

sider the model of Universe with Finsler-Randers space time cosmology. Given

the growing interest of cosmologists, here, we propose to study the evolution of

the Universe within the framework of Finsler-Randers cosmology. In this chapter

we have studied the Friedman Robertson Walker model in the Finsler-Randers

cosmology. The out line of the chapter is as follows: In section 1, a brief introduc-

tion is given. In section 2, the metric and the basic field equations are described

in Finsler-Randers cosmology. Section 3 deals with the FR cosmological model in

Lyra geometry. FR cosmological model in general class of scalar tensor theories

and C-field theory are given in section 4 and 5, respectively. Finally, conclusions

are summarized in the last section 6.

88

7.2 Finsler-Randers Cosmological Model in Ein-

stein Theory

The FR cosmic scenario is based on the Finslerian geometry which extends the

Riemannian geometry. Notice that a Riemannian geometry is also a particular

case of Finslerian. Bellow we discuss only the main features of the theory [for

more details see Rund (1959), Miron and Anastasiei (1994), Bao et al. (2000)

and Vacaru et al. (2005)]. Generally, a Finsler space is derived from a generating

function F (x, y) on the tangent bundle TM of a manifold M . The generating

function F is differentiable on TM0 = TM \ 0 and continuous on the zero

cross section. The function F is also positively homogeneous of degree-one in

y = x = dxdt

. In other words, F introduces a structure on the space-time manifold

M that is called Finsler space-time. In the case of a FR space-time we have

F (x, y) =√aij(x)yiyj + bi(x)y

i, (7.1)

where aij are component of a Riemannian metric and bi = (b0, 0, 0, 0) is a weak

primordial vector field with |bi| << 1. Now the Finslerian metric tensor gij is

constructed by the Hessian of F 2

2

gij =1

2

∂2F 2

∂yi∂yj. (7.2)

It is interesting to mention that the Cartan tensor Cijk = 12

∂gij∂yk

= 14

∂3F 2

∂yi∂yj∂yk

is a significant ingredient of the Finsler geometry. Indeed it has been found by

Stavrinos (2008) that b0 = 2C000.

The Finslerian-Randers field equations are given by

Rij −1

2gijR = −8πG

c4Tij, (7.3)

where Rij is the Finslerian Ricci tensor, Tij is the energy-momentum tensor

and T is the trace of the energy-momentum tensor. Modelling the expanding

Universe as a Finslerian perfect fluid with four-velocity Ui for co-moving observers,

we have

89

Tij = −pgij + (ρ+ p)UiUj, (7.4)

where ρ and p are the total energy density and pressure of the cosmic fluid

respectively.

Thus the energy momentum tensor becomes

Tij = diag(ρ,−pg11,−pg22,−pg33). (7.5)

Following the work of Kolassis et al. (1988), Chatterjee and Banerjee (2004) and

Bali and Saraf (2013), we discuss briefly week, dominant and strong energy con-

ditions in the context of Finslerian cosmology for our model.

We have T 00 = ρ, T 1

1 = T 22 = T 3

3 = −p in the locally Minkowskian frame.

Obviously the roots of matrix equation

|Tij − rgij| = dia[(ρ− r), (r + p), (r + p), (r + p)] = 0, (7.6)

give the eigenvalues r for our energy momentum tensor as r0 = ρ and r1 = −p =

r2 = r3. The energy conditions for our model are given by :

• Null Energy Condition (NEC):

ρ+ p ≥ 0. (7.7)

• Weak Energy Condition (WEC):

r0 ≥ 0 ⇒ ρ ≥ 0; r0 − ri ≥ 0 ⇒ ρ+ p ≥ 0. (7.8)

• Strong Energy Condition (SEC):

r0 −∑

ri ≥ 0 ⇒ ρ+ 3p ≥ 0 and ρ+ p ≥ 0. (7.9)

• Dominant Energy Condition (DEC):

90

r0 ≥ 0 ⇒ ρ ≥ 0; −r0 ≤ −ri ≤ r0 ⇒ ρ± p ≥ 0. (7.10)

In the context of a FRW metric

aij = diag

(1,− a2

1− kr2,−a2r2,−a2r2sin2θ

), (7.11)

where a is the function of time t only and k is the curvature parameter

having values +1, 0,−1 for closed, flat and open models respectively. The non-

zero components of the Finslerian Ricci tensors are

R00 = 3

(a

a− 3

4

a

au0

)(7.12)

and

Rii = −

(aa+ 2a2 + 2k + 11

4aau0

ii

), (7.13)

where 11 = 1− kr2, 22 = r2 and 33 = r2sin2θ.

The gravitational FR field equations (7.1), for co-moving observers, FRW Einstein

Field equations are

a

a+

3

4

a

aZt = −4πG

3(ρ+ 3p) (7.14)

a

a+ 2

(a

a

)2

+ 2k

a2+

11

4

a

aZt = 4πG (ρ− p) , (7.15)

where the over-dot denotes derivative with respect to the cosmic time ‘t’ and

Zt = b0 < 0 [Stavrinos (2008)].

From equations (7.2) and (7.3), we get

H2 +k

a2+HZt =

8πG

3ρ. (7.16)

Obviously, the extra term H(t)Zt in the modified Friedmann equation (7.16)

affects the dynamics of the Universe. If we consider b0 ≡ 0 or (C000 ≡ 0, Fα= 1),

which implies Zt = 0, then the field equations (7.14) and (7.15) reduce to the

nominal Einstein’s equations, a solution of which is the usual Friedman equation.

91

Here we discuss two different physically viable cosmologies, which have physical

interests to describe the decelerating and accelerating phases of universe.

Case 1: de Sitter solution

de Sitter solutions are well known in Cosmology because the current epoch,

wherein the Universe expansion is being accelerated, can be described approxi-

mately with a de Sitter solutions. This kind of solution consists of an exponential

expansion of the scale factor, which yields a constant Hubble parameter.

We consider the following form for a scale factor by Arbab. (2007): a = ceγt, where

c and γ are constants. For γ2 > 0 it gives an accelerating Universe.

Now, Hubble parameter

H =a

a=γceγt

ceγt= γ. (7.17)

Using equations (7.4) and (7.5), the energy density evolving as

ρ =3

8πG

[γ2 + γZt +

k

c2e2γt

]. (7.18)

From equations (7.2) and (7.6), the pressure is given by

p = − 3γ2

8πG− 5

16πGγZt −

k

8πGc2e2γt. (7.19)

Using equations (7.6) and (7.7), we obtain

ρ+ p =γZt

16πG+

1

4πG

k

c2e2γt. (7.20)

and

ρ− p =3γ2

4πG+

11γZt

16πG+

1

2πG

k

c2e2γt. (7.21)

Again, from equations (7.6) and (7.7), one can get

ρ+ 3p = − 3γ2

4πG− 9

16πGγZt. (7.22)

It is noticed that, the equation (7.16) reduces to standard Friedmann equa-

tion by considering Zt = 0. The current observations says that the Universe

92

is anisotropic at early stage and becomes isotropic at late time (i.e. at present

epoch). So keeping these things in mind, we consider the physically variable

choices of Zt < 0 in two different scenario. i.e. (i) Zt = −e−t and (ii) Zt = −t−n.

Subcase 1a: When Zt = −e−t

Substituting the value of Zt in equations (7.18) and (7.19), we can determine

ρ and p respectively as

ρ =3

8πG

[γ2 − γe−t +

k

c2e2γt

]. (7.23)

p =5γe−t

16πG− 3γ2

8πG− k

8πGc2e2γt. (7.24)

From equations (7.23) and (7.24), we obtain

ρ+ p = − γe−t

16πG+

1

4πG

k

c2e2γt. (7.25)

and

ρ− p =3γ2

4πG− 11γe−t

16πG+

1

2πG

k

c2e2γt. (7.26)

Again, from equations (7.23) and (7.24), one can obtain easily

ρ+ 3p = − 3γ2

4πG+

9γe−t

16πG. (7.27)

From equations (7.23)-(7.27), it is observed that the NEC is satisfied if

c2 ≤ 4kγe(2γ+1)t = A1, WEC is satisfied for c2 ≤ min

k

e2γt[γe−t−γ2], 4kγe(2γ+1)t

= A2,

DEC is satisfied if c2 ≤ min

ke2γt[γe−t−γ2]

, 4kγe(2γ+1)t ,

8ke2γt[11γe−t−12γ2]

= A3 and SEC

is satisfied if 0 < γ ≤ 34et

.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in this

case if c2 ≤ min A1, A2, A3 whereas SEC is satisfied in this model if 0 < γ ≤ 34et

.

However, we also observed that for large cosmic time ‘t’, NEC, WEC and DEC

are satisfied whereas SEC is violated, which is responsible for current accelerated

93

expansion of Universe.

Subcase 1b: When Zt = −t−n

Substituting the value of Zt in equations (7.18) and (7.19), we can determine

ρ and p respectively as

ρ =3

8πG

[γ2 − γ

tn+

k

c2e2γt

](7.28)

and

p =5γ

16πGtn− 3γ2

8πG− k

8πGc2e2γt. (7.29)

From equations (7.28) and (7.29), we obtain

ρ+ p = − γt−n

16πG+

1

4πG

k

c2e2γt(7.30)

and

ρ− p =3γ2

4πG− 11γt−n

16πG+

1

2πG

k

c2e2γt. (7.31)

Again, from equations (7.28) and (7.29), we can obtain easily

ρ+ 3p =9γt−n

16πG− 3γ2

4πG. (7.32)

From equations (7.28)-(7.32), it is observed that the NEC is satisfied if

c2 ≤ 4kγe2γttn

= A4, WEC is satisfied for c2 ≤ min

ke2γt[γt−n−γ2]

, 4kγtne(2γ)t

= A5,

DEC is satisfied if c2 ≤ min

ke2γt[γt−n−γ2]

, 4kγtne(2γ)t

, 8ke2γt[11γt−n−12γ2]

= A6 and SEC

is satisfied if 0 < γ ≤ 34tn

.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in this

case if c2 ≤ min A4, A5, A6 whereas SEC is satisfied in this model if 0 < γ ≤ 34tn

.

However, we also observed that for large cosmic time ‘t’, NEC, WEC and DEC

are satisfied but SEC is violated.

94

Case 2: Power-law solution

We know that the power law solutions are very important in the standard

Cosmology, because this type of solution provides a framework for establishing

the behaviour of more general cosmological solutions in different histories of our

Universe, such as radiation-dominant, matter-dominant, or dark energy-dominant

eras. Let us consider a Universe with a power law by Padmanabhan (2002):

a = ctδ, where c and δ are constants. For δ > 1 it gives an accelerating Universe.

Now, Hubble parameter

H =a

a=cδtδ−1

ctδ=δ

t. (7.33)

Using equations (7.16) and (7.33), the energy density evolving as

ρ =3

8πG

[δ2

t2+δ

tZt +

k

c2t2δ

]. (7.34)

From equations (7.14) and (7.34), the pressure is given by

p =−3δ2 + 2δ

8πGt2− 5δZt

16πGt− 1

8πG

k

c2t2δ. (7.35)

From equations (7.34) and (7.35), we obtain

ρ+ p =1

4πG

δ

t2+

1

16πG

δ

tZt +

1

4πG

k

c2t2δ(7.36)

and

ρ− p =1

4πG

δ(3δ − 1)

t2+

11

16πG

δ

tZt +

1

2πG

k

c2t2δ. (7.37)

Again, from equations (7.34) and (7.35), we can obtain easily

ρ+ 3p =3

4πG

δ(1− δ)

t2− 9

16πG

δ

tZt. (7.38)

Subcase 2a: When Zt = −e−t

Substituting the value of Zt in equations (7.34) and (7.35), we can determine

ρ and p respectively as

95

ρ =3

8πG

[δ2

t2− δ

tet+

k

c2t2δ

], (7.39)

p =2δ − 3δ2

8πGt2+

5δ

16πGtet− 1

8πG

k

c2t2δ. (7.40)

From equations (7.39) and (7.40), we obtain

ρ+ p =1

4πG

δ

t2− 1

16πG

δ

tet+

1

4πG

k

c2t2δ(7.41)

and

ρ− p =δ(3δ − 1)

4πGt2− 11

16πG

δ

tet+

1

2πG

k

c2t2δ. (7.42)

Again, from equations (7.39) and (7.40), we obtain ρ+ 3p, where

ρ+ 3p =3δ(1− δ)

4πGt2+

9

16πG

δ

tet. (7.43)

From equations (7.39)-(7.43), it is observed that the NEC is satisfied if

c2 ≤ 4kδt(2δ−1)e−t−4δ2t(2δ−2) = B1, WEC is satisfied for

c2 ≤ min

kδt2δ−1e−t−δ2t2δ−2 ,

4kδt(2δ−1)e−t−4δ2t(2δ−2)

= B2, DEC satisfied if

c2 ≤ min

kδt2δ−1e−t−δ2t2δ−2 ,

4kδt(2δ−1)e−t−4δ2t(2δ−2) ,

8k11δt2δ−1e−t−4δ(3δ−1)t2δ−2

= B3 and

SEC satisfied if 1 < δ ≤ 1 + 3t4et

. It is observed that, for any value of ‘t’, NEC,

WEC and DEC are satisfied in this case if c2 ≤ min B1, B2, B3 whereas SEC is

satified in this model if 1 < δ ≤ 1 + 3t4et

. It is also observed that for large cosmic

time ‘t’, NEC, WEC, DEC are stasfied but SEC is violated.

Subcase 2b: When Zt = −t−n

Substituting the value of Zt in equations (7.34) and (7.35), we can determine

ρ and p respectively as

ρ =3

8πG

[δ2

t2− δ

tn+1+

k

c2t2δ

], (7.44)

96

p =2δ − 3δ2

8πGt2+

5δ

16πGtn+1− 1

8πG

k

c2t2δ. (7.45)

From equations (7.44) and (7.45), we obtain

ρ+ p =1

4πG

δ

t2− 1

16πG

δ

tn+1+

1

4πG

k

c2t2δ(7.46)

and

ρ− p =δ(3δ − 1)

4πGt2− 11

16πG

δ

tn+1+

1

2πG

k

c2t2δ. (7.47)

Again, from equations (7.44) and (7.45), we can obtain easily

ρ+ 3p =3δ(1− δ)

4πGt2+

9

16πG

δ

tn+1. (7.48)

From equations (7.44)-(7.48), it is observed that the NEC is satisfied if

c2 ≤ 4kδt2δ−n−1−δt2δ−2 = B4, WEC is satisfied for

c2 ≤ min

kδt2δ−n−1−δ2t2δ−2 ,

4kδt2δ−n−1−δt2δ−2

= B5, DEC is satisfied if

c2 ≤ min

kδt2δ−n−1−δ2t2δ−2 ,

4kδt2δ−n−1−δt2δ−2 ,

8k11δt2δ−n−1−4δ(3δ−1)t2δ−2

= B6 and SEC is

satisfied if 1 < δ ≤ 1 + 34tn−1 .

It is observed that, for any value of ‘t’, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min B4, B5, B6 whereas SEC is satisfied in this model for

1 < δ ≤ 1 + 34tn−1 . It is also observed that for large cosmic time ‘t’, NEC, WEC,

DEC are stasfied but SEC is violated.

7.3 Finsler-Randers cosmological model in Lyra ge-

ometry

The field equation in this theory are given by

Rij −1

2Rgij +

3

2ϕiϕj −

3

4gijϕkϕ

k = −Tij, (7.49)

where ϕij are vector displacement field. Here we take

97

ϕi = (0, 0, 0, β(t)) .

In this theory Field equations are

(a

a

)2

+k

a2+HZt =

8πG

3ρ+

1

2β2, (7.50)

2a

a+

(a

a

)2

+k

a2+

5

2HZt = 8πGp− 1

2β2. (7.51)

From equations (7.50) and (7.51), we obtain

a

a+

3

4

a

aZt = −4πG

3(ρ+ 3p)− 1

2β2. (7.52)

Case 1: de Sitter solution

Let a = ceγt, where γ2 > 0.

Now, Hubble parameter

H =a

a=γceγt

ceγt= γ. (7.53)

From equations (7.52) and (7.53), the energy density evolving as

ρ =3

8πG

[γ2 + γZt +

k

c2e2γt− 1

2β2

]. (7.54)

From equations (7.52) and (7.54), the pressure is given as

p = − 3γ2

8πG− 5

16πGγZt −

k

8πGc2e2γt− 1

16πGβ2. (7.55)

From equations (7.54) and (7.55), we obtain

ρ+ p =γZt

16πG+

1

4πG

k

c2e2γt− 1

4πGβ2 (7.56)

and

ρ− p =3γ2

4πG+

11γZt

16πG+

1

2πG

k

c2e2γt− 1

8πGβ2. (7.57)

Again, from equations (7.54) and (7.55), we can obtain easily

98

ρ+ 3p = − 3γ2

4πG− 9

16πGγZt −

3

8πGβ2. (7.58)

In this section, we discuss two different cases of physically viable cosmologies.

Subcase 1a: When Zt = −e−t :

Substituting the value of Zt in equations (7.54) and (7.55), we can determine

ρ and p respectively as

ρ =3

8πG

[γ2 − γe−t +

k

c2e2γt− 1

2β2

], (7.59)

p =5γe−t

16πG− 3γ2

8πG− k

8πGc2e2γt− 1

16πGβ2. (7.60)

From equations (7.59) and (7.60), we obtain

ρ+ p = − γe−t

16πG+

1

4πG

k

c2e2γt− 1

4πGβ2 (7.61)

and

ρ− p =3γ2

4πG− 11γe−t

16πG+

1

2πG

k

c2e2γt− 1

8πGβ2. (7.62)

Again, from equations (7.59) and (7.60), we obtain easily

ρ+ 3p = − 3γ2

4πG+

9γe−t

16πG− 3

8πGβ2. (7.63)

From equations (7.59)-(7.63), it is noticed that the NEC is satisfied if

c2 ≤ 4ke2γt[γe−t−4β2]

= A1, WEC is satisfied for c2 ≤ min

ke2γt[γe−t−γ2+β2/2]

, 4ke2γt[γe−t−4β2]

=

A2 DEC is satisfied if c2 ≤ min

ke2γt[γe−t−γ2+β2/2]

, 4ke2γt[γe−t−4β2]

, 8ke2γt[11γe−t−12γ2−2β2]

=

A3 and SEC is satisfied if 0 < β2 ≤ 3γ2et

− 2γ2.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min A1, A2, A3 whereas SEC is satisfied in this model for

0 < β2 ≤ 3γ2et

− 2γ2. However, we also observed that for large cosmic time ‘t’,

NEC, WEC and DEC are satisfied but SEC is violated.

99

Subcase 1b: When Zt = −t−n :

Substituting the value of Zt in equations (7.54) and (7.55), we can determine

ρ and p respectively as

ρ =3

8πG

[γ2 − γ

tn+

k

c2e2γt− 1

2β2

], (7.64)

p =5γ

16πGtn− 3γ2

8πG− k

8πGc2e2γt− 1

16πGβ2. (7.65)

From equations (7.64) and (7.65), we obtain

ρ+ p = − γt−n

16πG+

1

4πG

k

c2e2γt− 1

4πGβ2 (7.66)

and

ρ− p =3γ2

4πG− 11γt−n

16πG+

1

2πG

k

c2e2γt− 1

8πGβ2. (7.67)

Again, from equations (7.64) and (7.65), we obtain

ρ+ 3p =9γt−n

16πG− 3γ2

4πG− 3

8πGβ2. (7.68)

From equations (7.64)-(7.68), it is noticed that the NEC is satisfied if

c2 ≤ 4ke2γt[γt−n+4β2]

= A4, WEC is satisfied for c2 ≤ min

ke2γt[γt−n−γ2+β2/2]

, 4ke2γt[γt−n+4β2]

=

A5, DEC is satisfied if c2 ≤ min

ke2γt[γt−n−γ2+β2/2]

, 4ke2γt[γt−n+4β2]

, 8ke2γt[11γt−n−12γ2+2β2]

=

A6 and SEC is satisfied if 0 < β2 ≤ 3γ2tn

− 2γ2

It is observed that, for any valu of t, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min A4, A5, A6 whereas SEC is satisfied in this model if

0 < β2 ≤ 3γ2tn

− 2γ2. However, we also observed that for large cosmic time ‘t’,

NEC, WEC and DEC are satisfied but SEC is violated.

Case 2: Power-law

Let a = ctδ, where δ > 1.

Now, Hubble parameter

100

H =a

a=cδtδ−1

ctδ=δ

t. (7.69)

From equations (7.50) and (7.69), we have

ρ =3

8πG

[δ2

t2+δ

tZt +

k

c2t2δ− 1

2β2

]. (7.70)

From equations (7.50) and (7.70), we get

p =−3δ2 + 2δ

8πGt2− 5δZt

16πGt− 1

8πG

k

c2t2δ− 1

16πGβ2. (7.71)

From equations (7.70) and (7.71), we obtain

ρ+ p =1

4πG

δ

t2+

1

16πG

δ

tZt +

1

4πG

k

c2t2δ− 1

4πGβ2 (7.72)

and

ρ− p =1

4πG

δ(3δ − 1)

t2+

11

16πG

δ

tZt +

1

2πG

k

c2t2δ− 1

8πGβ2. (7.73)

Again, from equations (7.70) and (7.71), we obtain

ρ+ 3p =3

4πG

δ(1− δ)

t2− 9

16πG

δ

tZt −

3

8πGβ2. (7.74)

Subcase 2a: When Zt = −e−t :

Substituting the value of Zt in equations (7.70) and (7.71), we can determine

ρ and p respectively as

ρ =3

8πG

[δ2

t2− δ

tet+

k

c2t2δ− 1

2β2

], (7.75)

p =2δ − 3δ2

8πGt2+

5δ

16πGtet− 1

8πG

k

c2t2δ− 1

16πGβ2. (7.76)

From equations (7.75) and (7.76), we obtain

ρ+ p =1

4πG

δ

t2− 1

16πG

δ

tet+

1

4πG

k

c2t2δ− 1

4πGβ2 (7.77)

101

and

ρ− p =δ(3δ − 1)

4πGt2− 11

16πG

δ

tet+

1

2πG

k

c2t2δ− 1

8πGβ2. (7.78)

Again, from equations (7.75) and (7.76), we obtain

ρ+ 3p =3δ(1− δ)

4πGt2+

9

16πG

δ

tet− 3

8πGβ2. (7.79)

From equations (7.75)-(7.79), it is observed that the NEC is satisfied if

c2 ≤ 4kδt(2δ−1)e−t−4δ2t(2δ−2)+4β2t2δ

= B1, WEC is satisfied for

c2 ≤ min

kδt2δ−1e−t−δ2t2δ−2+t2δβ2/2

, 4kδt(2δ−1)e−t−4δ2t(2δ−2)+4β2t2δ

= B2,

DEC is satisfied if

c2 ≤ min

kδt2δ−1e−t−δ2t2δ−2+t2δβ2/2

, 4kδt(2δ−1)e−t−4δ2t(2δ−2)+4β2t2δ

, 8k11δt2δ−1e−t−4δ(3δ−1)t2δ−2+2β2t2δ

=

B3 and SEC is satisfied if 0 < β2 ≤ 2δ(1−δ)t2

+ 3δ2tet

.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min B1, B2, B3 whereas SEC is satisfied in this model if

0 < β2 ≤ 2δ(1−δ)t2

+ 3δ2tet

. However, we also observed that for large cosmic time

‘t’, NEC, WEC and DEC are satisfied but SEC is violated.

Subcase 2b: WhenZt = −t−n

Substituting the value of Zt in equations (7.70) and (7.71), we can determine

ρ and p respectively as

ρ =3

8πG

[δ2

t2− δ

tn+1+

k

c2t2δ− 1

2β2

], (7.80)

p =2δ − 3δ2

8πGt2+

5δ

16πGtn+1− 1

8πG

k

c2t2δ− 1

16πGβ2. (7.81)

From equations (7.80) and (7.81), we obtain

ρ+ p =1

4πG

δ

t2− 1

16πG

δ

tn+1+

1

4πG

k

c2t2δ− 1

4πGβ2 (7.82)

and

102

ρ− p =δ(3δ − 1)

4πGt2− 11

16πG

δ

tn+1+

1

2πG

k

c2t2δ− 1

8πGβ2. (7.83)

Again, from equations (7.80) and (7.81), we obtain

ρ+ 3p =3δ(1− δ)

4πGt2+

9

16πG

δ

tn+1− 3

8πGβ2. (7.84)

From equations (7.80)-(7.84), it is observed that the NEC is satisfied if

c2 ≤ 4kδt2δ−n−1−δt2δ−2+t2δβ2/2

= B4, WEC is satisfied for

c2 ≤ min

kδt2δ−n−1−δ2t2δ−2+4t2δβ2 ,

4kδt2δ−n−1−δt2δ−2+t2δβ2/2

= B5,

DEC satisfied if

c2 ≤ min

kδt2δ−n−1−δ2t2δ−2+4t2δβ2 ,

4kδt2δ−n−1−δt2δ−2+t2δβ2/2

, 8k11δt2δ−n−1−4δ(3δ−1)t2δ−2+2t2δβ2

=

B6 and SEC is satisfied if 0 < β2 ≤ 2δ(1−δ)t2

+ 3δ2tn+1 .

It is observed that, for any value of t, NEC, WEC, and DEC are satisfied in

this case if c2 ≤ min B4, B5, B6 whereas SEC is satisfied in this model if

0 < β2 ≤ 2δ(1−δ)t2

+ 3δ2tn+1 . However, we also observed that for large cosmic time ‘t’,

NEC, WEC and DEC are satisfied but SEC is violated.

7.4 Finsler-Randers Cosmological Model in Gen-

eral Class of Scalar-tensor theory

The field equation of this theory are given by

Gij = Rij −1

2gijR = −8πGTij + 2

(ϕ,iϕ,j −

1

2gijϕ,kϕ

k

), (7.85)

where G is the gravitational constant and ϕ is the scalar field.

In this theory Field equation’s are

3

[H2 +

k

a2+HZt

]= 8πGρ+ ϕ2, (7.86)

2a

a+

(a

a

)2

+k

a2+

5

2HZt = −8πGp− ϕ2. (7.87)

From equations (7.86) and (7.87), we obtain

103

3

[a

a+

3

4

a

aZt

]= −4πG (ρ+ 3p)− 2ϕ2. (7.88)

Case 1: de Sitter solution

Let a = ceγt, where γ2 > 0.

Now, Hubble parameter

H =a

a=γceγt

ceγt= γ. (7.89)

From equations (7.88) and (7.89), we have

ρ =3

8πG

[γ2 + γZt +

k

c2e2γt− ϕ2

3

]. (7.90)

From equations (7.88) and (7.90), we get

p = − 3γ2

8πG− 5

16πGγZt −

k

8πGc2e2γt− 1

8πGϕ2. (7.91)

From equations (7.90) and (7.91), we obtain

ρ+ p =γZt

16πG+

1

4πG

k

c2e2γt− 1

4πGϕ2 (7.92)

and

ρ− p =3γ2

4πG+

11γZt

16πG+

1

2πG

k

c2e2γt. (7.93)

Again, from equations (7.90) and (7.91), we obtain

ρ+ 3p = − 3γ2

4πG− 9

16πGγZt −

1

2πGϕ2. (7.94)

In this section, we discuss two different cases of physically viable cosmologies.

Subcase 1a: When Zt = −e−t

Substituting the value of Zt in equations (7.90) and (7.91), we can determine

ρ and p respectively as

104

ρ =3

8πG

[γ2 − γe−t +

k

c2e2γt− ϕ2

3

], (7.95)

p =5γe−t

16πG− 3γ2

8πG− k

8πGc2e2γt− 1

8πGϕ2. (7.96)

From equations (7.95) and (7.96), we obtain

ρ+ p = − γe−t

16πG+

1

4πG

k

c2e2γt− 1

4πGϕ2 (7.97)

and

ρ− p =3γ2

4πG− 11γe−t

16πG+

1

2πG

k

c2e2γt. (7.98)

Again, from equations (7.95) and (7.96), we obtain

ρ+ 3p = − 3γ2

4πG+

9γe−t

16πG− 1

2πGϕ2. (7.99)

From equations (7.95)-(7.99), it is noticed that the NEC is satisfied if

c2 ≤ 4ke2γt[γe−t+4ϕ2]

= A1, WEC is satisfied for c2 ≤ min

ke2γt[γe−t−γ2+ϕ2/3]

, 4ke2γt[γe−t+4ϕ2]

=

A2, DEC is satisfied if c2 ≤ min

ke2γt[γe−t−γ2+ϕ2/3]

, 4ke2γt[γe−t+4ϕ2]

, 8ke2γt[11γe−t−12γ2]

=

A3 and SEC is satisfied if 0 < ϕ2 ≤ 9γ8et

− 32γ2.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in this

case if c2 ≤ min A1, A2, A3 whereas SEC is satisfied if 0 < ϕ2 ≤ 9γ8et

− 32γ2.

However, we also observed that for large cosmic time ‘t’, NEC, WEC and DEC

are satisfied but SEC is violated.

Subcase 1b: When Zt = −t−n

Substituting the value of Zt in equations (7.54) and (7.55), we can determine

ρ and p respectively as

ρ =3

8πG

[γ2 − γ

tn+

k

c2e2γt− ϕ2

3

], (7.100)

105

p =5γ

16πGtn− 3γ2

8πG− k

8πGc2e2γt− 1

8πGϕ2. (7.101)

From equations (7.100) and (7.101), we obtain

ρ+ p = − γt−n

16πG+

1

4πG

k

c2e2γt− 1

4πGϕ2 (7.102)

and

ρ− p =3γ2

4πG− 11γt−n

16πG+

1

2πG

k

c2e2γt. (7.103)

Again, from equations (7.100) and (7.101), one can obtain easily

ρ+ 3p =9γt−n

16πG− 3γ2

4πG− 1

2πGϕ2. (7.104)

From equations (7.100)-(7.104), it is observed that the NEC is satisfied if

c2 ≤ 4ke2γt[γt−n+4ϕ2]

= A4, WEC is satisfied for c2 ≤ min

ke2γt[γt−n−γ2+ϕ2/3]

, 4ke2γt[γt−n+4ϕ2]

=

A5, DEC is satisfied if c2 ≤ min

ke2γt[γt−n−γ2+ϕ2/3]

, 4ke2γt[γt−n+4ϕ2]

, 8ke2γt[11γt−n−12γ2]

=

A6 and SEC is satisfied if 0 < ϕ2 ≤ 9γ8tn

− 32γ2.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min A4, A5, A6 whereas SEC is satisfied in this model if

0 < ϕ2 ≤ 9γ8tn

− 32γ2. However, we also observed that for large cosmic time ‘t’,

NEC, WEC and DEC are satisfied but SEC is violated.

Case 2: Power-law

Let a = ctδ, where δ > 1.

Now, Hubble parameter

H =a

a=cδtδ−1

ctδ=δ

t. (7.105)

From equations (7.88) and (7.105), we have

ρ =3

8πG

[δ2

t2+δ

tZt +

k

c2t2δ− ϕ2

3

]. (7.106)

106

From equations (7.88) and (7.106), we get

p =−3δ2 + 2δ

8πGt2− 5δZt

16πGt− 1

8πG

k

c2t2δ− 1

8πGϕ2. (7.107)

From equations (7.106) and (7.107), we obtain

ρ+ p =1

4πG

δ

t2+

1

16πG

δ

tZt +

1

4πG

k

c2t2δ− 1

4πGϕ2 (7.108)

and

ρ− p =1

4πG

δ(3δ − 1)

t2+

11

16πG

δ

tZt +

1

2πG

k

c2t2δ. (7.109)

Again, from equations (7.106) and (7.107), we can obtain easily

ρ+ 3p =3

4πG

δ(1− δ)

t2− 9

16πG

δ

tZt −

1

2πGϕ2. (7.110)

Subcase 2a: When Zt = −e−t

Substituting the value of Zt in equations (7.105) and (7.106), we can determine

ρ and p respectively as

ρ =3

8πG

[δ2

t2− δ

tet+

k

c2t2δ− ϕ2

3

], (7.111)

p =2δ − 3δ2

8πGt2+

5δ

16πGtet− 1

8πG

k

c2t2δ− 1

8πGϕ2. (7.112)

From equations (7.111) and (7.112), we obtain

ρ+ p =1

4πG

δ

t2− 1

16πG

δ

tet+

1

4πG

k

c2t2δ− 1

4πGϕ2 (7.113)

and

ρ− p =δ(3δ − 1)

4πGt2− 11

16πG

δ

tet+

1

2πG

k

c2t2δ. (7.114)

Again, from equations (7.111) and (7.112), we can obtain easily

ρ+ 3p =3δ(1− δ)

4πGt2+

9

16πG

δ

tet− 1

2πGϕ2. (7.115)

107

From equations (7.111)-(7.115), it is observed that the NEC is satisfied if

c2 ≤ 4kδt(2δ−1)e−t−4δt(2δ−2)+4ϕ2t2δ

= B1, WEC is satisfied for

c2 ≤ min

kδt2δ−1e−t−δ2t2δ−2+t2δϕ2/3

, 4kδt(2δ−1)e−t−4δt(2δ−2)+4ϕ2t2δ

= B2,

DEC satisfied if

c2 ≤ min

kδt2δ−1e−t−δ2t2δ−2+t2δϕ2/3

, 4kδt(2δ−1)e−t−4δt(2δ−2)+4ϕ2t2δ

, 8k11δt2δ−1e−t−4δ(3δ−1)t2δ−2

=

B3 and SEC is satisfied if 0 < ϕ2 ≤ 3δ(1−δ)2t2

+ 9δ8tet

.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min B1, B2, B3 whereas SEC is satisfied in this model for

0 < ϕ2 ≤ 3δ(1−δ)2t2

+ 9δ8tet

. However, we also observed that for large cosmic time ‘t’,

NEC,WEC and DEC are satisfied but SEC is violated.

Subcase 2b: When Zt = −t−n

Substituting the value of Zt in equations (7.105) and (7.106), we can determine

ρ and p respectively as

ρ =3

8πG

[δ2

t2− δ

tn+1+

k

c2t2δ− ϕ2

3

], (7.116)

p =2δ − 3δ2

8πGt2+

5δ

16πGtn+1− 1

8πG

k

c2t2δ− 1

8πGϕ2. (7.117)

From equations (7.116) and (7.117), we obtain

ρ+ p =1

4πG

δ

t2− 1

16πG

δ

tn+1+

1

4πG

k

c2t2δ− 1

4πGϕ2 (7.118)

and

ρ− p =δ(3δ − 1)

4πGt2− 11

16πG

δ

tn+1+

1

2πG

k

c2t2δ. (7.119)

Again, from equations (7.116) and (7.117), one can obtain easily

ρ+ 3p =3δ(1− δ)

4πGt2+

9

16πG

δ

tn+1− 1

2πGϕ2. (7.120)

From equations (7.116)-(7.120), it is observed that the NEC is satisfied if

c2 ≤ 4kδt2δ−n−1−4δt2δ−2+4t2δϕ2 = B4, WEC is satisfied for

108

c2 ≤ min

kδt2δ−n−1−δ2t2δ−2+t2δϕ2/3

, 4kδt2δ−n−1−4δt2δ−2+4t2δϕ2

= B5, DEC is satisfied

if c2 ≤ min

kδt2δ−n−1−δ2t2δ−2+t2δϕ2/3

, 4kδt2δ−n−1−4δt2δ−2+4t2δϕ2 ,

8k11δt2δ−n−1−4δ(3δ−1)t2δ−2

=

B6 and SEC is satisfied if 0 < ϕ2 ≤ 3δ(1−δ)2t2

+ 9δ8tn+1 .

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min B4, B5, B6 whereas SEC is satisfied in this model for

0 < ϕ2 ≤ 3δ(1−δ)2t2

+ 9δ8tn+1 . However, we also observed that for large cosmic time ‘t’,

NEC, WEC and DEC are satisfied whereas SEC is violated.

7.5 Finsler-Randers Cosmological Model in C-field

theory

The field equations in this theory

Rij −1

2gijR = −8πG

[Tij − f

(cicj −

1

2gijckc

k

)], (7.121)

where c(x, t) = c(t), c = dcdt

and f ≥ 0.

In this theory field equation’s are

(a

a

)2

+k

a2+HZt =

8πG

3

(ρ− 1

2f c2), (7.122)

2a

a+

(a

a

)2

+k

a2+

5

2HZt = −8πG

(p− 1

2f c2). (7.123)

From equations (7.29) and (7.30), we obtain

a

a+

3

4

a

aZt = −4πG

3(ρ+ 3p) +

8πG

3f c2. (7.124)

Case 1: de Sitter solution

Let a = αeγt, where γ2 > 0. Here α and γ are constants.

Now, Hubble parameter

H =a

a=γαeγt

αeγt= γ. (7.125)

109

From equations (7.124) and (7.125), we have

ρ =3

8πG

[γ2 + γZt +

k

α2e2γt

]+

1

2f c2. (7.126)

From equations (7.124) and (7.126), we get

p = − 3γ2

8πG− 5

16πGγZt −

k

8πGα2e2γt− 1

2f c2. (7.127)

From equations (7.126) and (7.127), we obtain

ρ+ p =γZt

16πG+

1

4πG

k

α2e2γt(7.128)

and

ρ− p =3γ2

4πG+

11γZt

16πG+

1

2πG

k

α2e2γt+ f c2. (7.129)

Again, from equations (7.126) and (7.127), we obtain ρ+ 3p, where

ρ+ 3p = − 3γ2

4πG− 9

16πGγZt − f c2. (7.130)

In this section, we discuss two different cases of physically viable cosmologies.

Subcase 1a: When Zt = −e−t

Substituting the value of Zt in equations (7.126) and (7.127), we can determine

ρ and p respectively as

ρ =3

8πG

[γ2 − γe−t +

k

c2e2γt

]+

1

2f c2, (7.131)

p =5γe−t

16πG− 3γ2

8πG− k

8πGc2e2γt− 1

2f c2. (7.132)

From equations (7.131) and (7.132), we obtain

ρ+ p = − γe−t

16πG+

1

4πG

k

c2e2γt(7.133)

and

110

ρ− p =3γ2

4πG− 11γe−t

16πG+

1

2πG

k

c2e2γt+ f c2. (7.134)

Again, from equations (7.131) and (7.132), we obtain

ρ+ 3p = − 3γ2

4πG+

9γe−t

16πG− f c2. (7.135)

From equations (7.131)-(7.135), it is noticed that the NEC is satisfied if

c2 ≤ 4ke2γt[γe−t]

= A1, WEC is satisfied for c2 ≤ min

ke2γt[γe−t−γ2]

, 4ke2γt[γe−t]

= A2,

DEC is satisfied if c2 ≤ min

ke2γt[γe−t−γ2]

, 4ke2γt[γe−t−4β2]

, 8ke2γt[11γe−t−12γ2]

= A3 and

SEC is satisfied if 0 < c2 ≤ 9γ16πGfet

− 3γ2

4πGf.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min A1, A2, A3 whereas SEC is satisfied in this model for

0 < c2 ≤ 9γ16πGfet

− 3γ2

4πGf. However, we also observed that for large cosmic time ‘t’,

NEC, WEC and DEC are satisfied whereas SEC is violated.

Subcase 1b: When Zt = −t−n

Substituting the value of Zt in equations (7.126) and (7.127), we can determine

ρ and p respectively as

ρ =3

8πG

[γ2 − γ

tn+

k

c2e2γt

]+

1

2f c2, (7.136)

p =5γ

16πGtn− 3γ2

8πG− k

8πGc2e2γt− 1

2f c2. (7.137)

From equations (7.136) and (7.137), we obtain

ρ+ p = − γt−n

16πG+

1

4πG

k

c2e2γt(7.138)

and

ρ− p =3γ2

4πG− 11γt−n

16πG+

1

2πG

k

c2e2γt+ f c2. (7.139)

Again, from equations (7.136) and (7.137), we obtain

111

ρ+ 3p =9γt−n

16πG− 3γ2

4πG− f c2. (7.140)

From equations (7.136)-(7.140), it is observed that the NEC is satisfied if

c2 ≤ 4ke2γt[γt−n]

= A4, WEC is satisfied for c2 ≤ min

ke2γt[γt−n−γ2]

, 4ke2γt[γt−n]

= A5,

DEC is satisfied if c2 ≤ min

ke2γt[γt−n−γ2]

, 4ke2γt[γt−n]

, 8ke2γt[11γt−n−12γ2]

= A6 and

SEC is satisfied if 0 < c2 ≤ 9γ16πGftn

− 3γ2

4πGf.

It is observed that, for any value of t, NEC, WEC and DEC are satisfied in

this case if c2 ≤ min A4, A5, A6 whereas SEC is satisfied in this model for

0 < c2 ≤ 9γ16πGftn

− 3γ2

4πGf. However, we also observed that for large cosmic time ‘t’,

NEC, WEC and DEC are satisfied whereas SEC is violated.

Case 2: Power-law

Let a = αtδ, where δ > 1. Here α and δ are constants.

Now, Hubble parameter

H =a

a=αδtδ−1

αtδ=δ

t. (7.141)

From equations (7.124) and (7.141), we have

ρ =3

8πG

[δ2

t2+δ

tZt +

k

α2t2δ

]+

1

2f c2. (7.142)

From equations (7.124) and (7.141), we get

p =−3δ2 + 2δ

8πGt2− 5δZt

16πGt− 1

8πG

k

α2t2δ− 1

2f c2. (7.143)

From equations (7.142) and (7.143), we obtain

ρ+ p =1

4πG

δ

t2+

1

16πG

δ

tZt +

1

4πG

k

α2t2δ(7.144)

and

ρ− p =1

4πG

δ(3δ − 1)

t2+

11

16πG

δ

tZt +

1

2πG

k

α2t2δ+ f c2. (7.145)

Again, from equations (7.142) and (7.143), we obtain ρ+ 3p, where

112

ρ+ 3p =3

4πG

δ(1− δ)

t2− 9

16πG

δ

tZt − f c2. (7.146)

Subcase 2a: When Zt = −e−t

Substituting the value of Zt in equations (7.142) and (7.143), we can determine

ρ and p respectively as

ρ =3

8πG

[δ2

t2− δ

tet+

k

c2t2δ

]+

1

2f c2, (7.147)

p =2δ − 3δ2

8πGt2+

5δ

16πGtet− 1

8πG

k

c2t2δ− 1

2f c2. (7.148)

From equations (7.147) and (7.148), we obtain

ρ+ p =1

4πG

δ

t2− 1

16πG

δ

tet+

1

4πG

k

c2t2δ(7.149)

and

ρ− p =δ(3δ − 1)

4πGt2− 11

16πG

δ

tet+

1

2πG

k

c2t2δ+ f c2. (7.150)

Again, from equations (7.147) and (7.148), we obtain

ρ+ 3p =3δ(1− δ)

4πGt2+

9

16πG

δ

tet− f c2. (7.151)

From equations (7.147)-(7.151), it is noticed that the NEC is satisfied if

c2 ≤ 4kδt(2δ−1)e−t−4δ2t(2δ−2) = B1, WEC is satisfied for

c2 ≤ min

kδt2δ−1e−t−δ2t2δ−2 ,

4kδt(2δ−1)e−t−4δ2t(2δ−2)

= B2, DEC is satisfied if c2 ≤

min

kδt2δ−1e−t−δ2t2δ−2 ,

4kδt(2δ−1)e−t−4δ2t(2δ−2) ,

8k11δt2δ−1e−t−4δ(3δ−1)t2δ−2

= B3 and SEC is

satisfied if 0 < c2 ≤ 3δ(1−δ)4πGft2

+ 9δ16πGftet

.

It is observed that, for any value of t, NEC, DEC and SEC are satisfied in

this case if c2 ≤ min B1, B2, B3 whereas SEC is satisfied in this model for

0 < c2 ≤ 3δ(1−δ)4πGft2

+ 9δ16πGftet

. However, we also observed that for large cosmic time

‘t’, NEC, WEC and DEC are satisfied but SEC is violated.

113

Subase 2b: When Zt = −t−n

Substituting the value of Zt in equations (7.142) and (7.143), we can determine

ρ and p respectively as

ρ =3

8πG

[δ2

t2− δ

tn+1+

k

c2t2δ

]+

1

2f c2, (7.152)

p =2δ − 3δ2

8πGt2+

5δ

16πGtn+1− 1

8πG

k

c2t2δ− 1

2f c2. (7.153)

From equations (7.152) and (7.153), we obtain

ρ+ p =1

4πG

δ

t2− 1

16πG

δ

tn+1+

1

4πG

k

c2t2δ(7.154)

and

ρ− p =δ(3δ − 1)

4πGt2− 11

16πG

δ

tn+1+

1

2πG

k

c2t2δ+ f c2. (7.155)

Again, from equations (7.152) and (7.153), we obtain

ρ+ 3p =3δ(1− δ)

4πGt2+

9

16πG

δ

tn+1− f c2. (7.156)

From equations (7.116)-(7.120), it is observed that the NEC is satisfied if c2 ≤4k

δt2δ−n−1−δt2δ−2 = B4, WEC is satisfied for c2 ≤ min

kδt2δ−n−1−δ2t2δ−2 ,

4kδt2δ−n−1−δt2δ−2

=

B5, DEC is satisfied if c2 ≤ min

kδt2δ−n−1−δ2t2δ−2 ,

4kδt2δ−n−1−δt2δ−2 ,

8k11δt2δ−n−1−4δ(3δ−1)t2δ−2

=

B6 and SEC is satisfied if 0 < c2 ≤ 3δ(1−δ)4πGft2

+ 9δ16πGftn+1 .

It is observed that, for any value t, NEC, WEC and DEC are satisfied in this

case if c2 ≤ min B4, B5, B6 whereas SEC is satisfied in this model for 0 < c2 ≤3δ(1−δ)4πGft2

+ 9δ16πGftn+1 . However, we also observed that for large cosmic time ‘t’, NEC,

WEC and DEC are satisfied but SEC is violated.

7.6 Conclusions

In this chapter we have discussed the Finsler-Randers cosmological models in

modified theories of gravity. The present chapter dealt with the de Sitter solu-

114

tion, power law solution and general expansion solution. We have discussed the

behaviour of FR cosmological model in modified theory of gravity (Hoyle-Narlikar

creation field theory, Lyra geometry, General class of scalar-tensor theories and

Einstein theory) by considering Zt = −e−t and −t−n. We have also discussed

null energy condition (NEC), Weak energy condition (WEC), dominant energy

condition (DEC) and Strong energy condition (SEC) and find under what condi-

tions our FR cosmological model is physically stable in different modify theories

of gravitation. It is seen that all energy conditions are satisfied for some suitable

value of constant but for large cosmic time t (i.e. at present epoch), NEC, WEC

and DEC are satisfied but SEC is violated in all modify gravity theories, which is

responsible for current accelerated expansion of Universe. At t→ ∞ we obtained

Zt → 0, Finsler-Randers cosmological model tend to Friedman Robertson-Walker

model. The model represents an expanding Universe, which approaches isotropy

for large values of t. Results of this chapter are in favour of the observational

features of the Universe.

115

Bibliography

Adati, T, On sub projective spaces, I, Tohoku Mathematical Journal, 3 (2), 159-

173, 1951.

Amari, S, Differential-Geometrical Methods in Statistics, Lecture Notes in Statis-

tics, Springer-Verlag, 28, 1985.

Amari, S and Nagaoka, H, Methods of Information Geometry, AMS Translation

of Mathematical Monographs, Oxford University Press, 191, 2000.

Antonelli, P L, Handbook of Finsler geometry, Kluwer Academic, Dordrecht, 1,

2003.

Antonelli, P L and Bucataru, I, Volterra-Hamilton production models with dis-

counting: general theory and worked examples, Nonlinear Analysis: Real World

Applications, 2, 337-356, 2001.

Antonelli, P L, Rutz, S F and Hirakawa, C E, The mathematical theory of en-

dosymbiosis I, Nonlinear Analysis: Real World Applications, 12, 3238-3251,

2011.

Antonelli, P L, Bevilacqua, L and Rutz, S F, Theories and models in symbiogen-

esis, Nonlinear Analysis: Real World Applications, 4, 743-753, 2003.

Antonelli, P L, Rutz, S F and Sabau, V S, A transient-state analysis of Tyson’s

model for the cell division cycle by means of KCC-theory, Open Syst. Inf. Dyn.,

9, 223-238, 2002.

116

Antonelli, P L, Ingarden R S and Matsumoto, M, The Theory of Sprays and Finsler

Spaces with Applications in Physics and Biology, Kluwer Academic Publishers

Netherlands, 58, 1993.

Arbab, A I, Viscous dark energy models with variable G and lambda, Chin. Phys.

Letter, 25 (10), 3834-3836, 2008.

Asanov, G S, Finslerian Extension of General Relativity, Reidel, Dordrecht, 1984.

Bacso, S and Matsumoto, M, On Finsler spaces of Douglas type. A generalization

of the notion of Berwald space, Publ. Math. Debrecen, 51 (3), 385-406, 1997.

Balan, V, Notable sub manifolds in Berwald-Moor spaces, BSG Proc., Geometry

Balkan Press, 17, 21-30, 2010.

Balan, V, CMC and minimal surfaces in Berwald-Moor spaces, Hyper complex

Numbers in Geometry and Physics, 2 (6), 113-122, 2006.

Balan, V and Brinzei, N, Einstein equations for (h, v) -Berwald -Moor relativistic

models, Balkan. J. Geom. Appl., 11 (2), 20-26, 2006.

Balan, V and Brinzei, N, Berwald-Moor-type (h, v)-metric physical models, Hyper

complex Numbers in Geometry and physics, 2 (4), 114-122, 2005.

Bali, R and Saraf, S, C-field cosmological model for barotropic fluid distribution

with bulk viscosity and decaying vacuum energy (Λ) in FRW space time, Cana-

dian Journal of Physics, 91 (9), 728-732, 2013.

Bao, D, Chern, S S and Shen, Z, An Introduction to Riemann-Finsler Geometry,

Springer, New York, 2000.

Basilakos, S and Stavrinos, P, Cosmological equivalence between the Finsler-

Randers space-time and the DGP gravity model, arXiv:1301.4327v2 [gr-qc] 2013.

Belinski, V A, Khalatnikov, I M and Lifshits, E M, Oscillatory Approach to a

Singular Point in the Relativistic Cosmology, Advances in Physics, 19, 525,

1970.

117

Berwald, L, Ueber Finslersche und Cartansche Geometrie IV. Projektivkrummung

allgemeiner affiner Raume und Finslersche Raume skalarere Krumming, Ann.

of Math., 48 (2), 755-781, 1947.

Blaschke, W, Vorlesungen Uber Differential geometrie, Springer, 2, 1923.

Bliss, G A, A generalization of the notion of angles, Trans. Amer. Math. Soc., 7

(2), 184-196, 1906.

Bohmer, C G, Harko, T and Sabau, S V, Jacobi stability analysis of dynamical

systems-applications in Gravitation and Cosmology, 1-37, 2010.

Brinzei, N, Projective relations for m-th root metric spaces, Journal of the Calcutta

Mathematical Society, 5, 21-35, 2009.

Cartan, E, Les Espaces de Finsler, Exposees de Geometrie, Paris, 2, 41, 1934.

Cartan, E, Observations sur le memoire precedent, Math. Zeitschrift, 37, 619-622,

1933.

Chatterjee, S and Banerjee, A, C-Field Cosmology in Higher Dimensions, General

Relativity and Gravitation, 36 (2), 303-313, 2004.

Chattopadhyay, J, Bairagi, N and Sarkar, R R, A predator-prey model with some

cover on prey species, Nonlin. Phenom. Complex Systems, 3 (4), 407-420, 2000.

Chern, S S, On the Euclidean connections in a Finsler space, Proc. National Acad.

soc., 29, 33-37, 1943.

Chern, S S, Sur la geometrie d’un systeme d’equations differentielles du second

ordre, Bull. Sci. Math., 63, 206-212, 1939.

Chern, S S and Shen, Z, Riemann-Finsler Geometry, Nankai Tracts in Mathemat-

ics, World Scientific, 6, 2004.

Douglas, J, The general geometry of paths, Annals of Math., 29, 143-168, 1927-28.

118

Devi, Sapna, Effects of prey refuge on a ratio-dependent predator-prey model with

stage-structure of prey population, Applied Mathematical Modelling, 37, 4337-

4349, 2013.

Devi, Sapna, Non constant prey harvesting in ratio-dependent predator prey sys-

tem incorporating a constant prey refuge, International Journal of Biomathe-

matics, 5 (2), 1250021 [20 pages] DOI: 10.1142/S1793524511001635, 2012.

Dubey, B, A prey-predator model with a reserved area, Nonlinear Analysis: Mod-

elling and Control, 12, 479-494, 2007.

Ellis, G F R and Elst, H V, Cosmological models, Proceedings of the NATO

Advanced Study Institute on Theoretical and Observational Cosmology, Kluwer

Academic, 541, 1-116, 1999.

Ellis, G F R, Exact and inexact solutions of the Einstein field equations, The

Renaissance of General Relativity and Cosmology, 20, 1993.

Finsler, P, Uber Kurven und Flachen in allgemeinen Raumen, Dissertation, Uni-

versity of Gottingen, 1918: Verlag Birkhauser, Basel, 1951.

Fulton, G M, Parallel vector fields, Proc. Amer. Math. Soc., 16, 136-137, 1965.

Funk, P, Uber Geometrien, bei denen die Geraden die Kurzesten sind, Math. Ann.,

101, 226-237, 1929.

Hashiguchi, M, On conformal transformation of Finsler metrics, J. Math. Kyoto

University, 16, 25-50, 1976.

Hawking, S W and Ellis, G F R, The large scale structure of space-time, Cambridge

University Press, U.K, 1973.

Hu, B L and Parker, L, Anisotropy damping through quantum effects in the early

universe, Phys. Rev. D, 17, 933-945, 1978.

Ingarden, R S, Differential geometry and Physics, Tensor N. S, 30, 201-209, 1976.

119

Ingarden, R S, On the Geometrically Absolute Optimal Representation in the Elec-

tron Microscope, Trav. Soc. Sci. Letter., Wroclaw, B45, 1957.

Kar, T K, Modelling the analysis of a harvested prey-predator system incorporating

a prey refuge, J. Comp. Appl. Math., 185, 19-33, 2006.

Kitayama, Masashi, Finsler Spaces Admitting a Parallel Vector Field, Balkan

Journal of Geometry and Its Applications, 3, 29-36, 1998.

Kolassis, C A, Santos, N O and Tsoubelis, D, Energy conditions for an imperfect

fluid, Classical and Quantum Gravity, 5, 1329, 1988.

Kolb, E W and Turner, M S, The Early Universe, Addison-Wesley, 1990.

Kosambi, D D, Parallelism and path space, Math. Zeitschrift, 37, 608-618, 1933.

Lammerzahl, Claus , Perlick, Volker and Hasse, Wolfgang, Observable effects in a

class of spherically symmetric static Finsler spacetimes, Physical Review D, 86,

104042, 2012.

Landsberg, G, Ueber dia Krilmmung in der Variation srechnung, Math. Ann., 65,

313-349, 1908.

Lebedev, S V, The generalized Finslerian metric tensors, Space-Time Structure,

Algebra and Geometry (Lilia-Print, Moscow), 174-181, 2007.

Lee, Il-Yong and Lee, Myung-Han, On Weakly-Berwald spaces of special (α, β)-

Metric, Bull. Korean Math. Soc., 43 (2), 425-441, 2006.

Lee, Il-Yong and Park, Hong-Suh, Finsler spaces with infinite series (α, β)-metric,

J. Korean Math. Soc., 41 (3), 567-589, 2004.

Li, B and Shen, Z, Projectively flat fourth root Finsler metrics, Canadian Mathe-

matical Bulletin, 55, 138-145, 2012.

Li, B, Shen, Y and Shen, Z, On a class of Douglas metrics in Finsler geometry,

Studia Sci. Math. Hungarica, 46 (3), 355-365, 2009.

120

MacCallum, M A H, Anisotropic and inhomogeneous relativistic cosmologies, Gen-

eral Relativity: An Einstein Centenary Survey, Cambridge University Press,

U.K., 533-580, 1979.

Matsumoto, M, Theory of Finsler spaces with (α, β)-metric, Rep. Math. Phys.,

31, 43-83, 1992.

Matsumoto, M, The Berwald connection of a Finsler space with an (α, β)-metric,

Tensor N. S., 50, 18-21, 1991.

Matsumoto, M, Foundations of Finsler Geometry and Special Finsler Spaces, Kai-

seisha Press, Saikawa, Otsu, 520 Japan, 1986.

Matsumoto, M and Numata, S, On Finsler spaces with a cubic metric, Tensor N.

S., 38, 153-162, 1982.

Matsumoto, M, On C-reducible Finsler spaces, Tensor N. S., 24, 29-37, 1972.

Miron, R and Anastasiei, M, The geometry of Lagrange spaces: Theory and Ap-

plications, Kluwer Academic Dordrecht, 2, 97-115, 1994.

Misner, C W, Thorne, K S and Wheeler, J A, Gravitation, W.H. Freeman and

Co., San Francisco: New York, 1279, 1973.

Misner, C W, The Isotropy of the Universe, Astrophysical Journal, 151, 431-457,

1968.

Padmanabhan, T, Accelerated expansion of the universe driven by tachyonic mat-

ter, Phys. Rev. D, 66, 1301, 2002.

Pandey, T N and Diwedi, D K, Normalised semi-parallel Ch-vector field in special

Finsler Spaces, Indian. J. Pure. Appl. Math., 30 (3), 307-315, 1999.

Pandey, P N and Tiwari, B, Cv-reducible Finsler spaces, Tensor, 61, 212-215, 1999.

Park, H and Lee, Y, Projective changes between a Finsler space with (α, β)-metric

and the associated Riemannian metric, Tensor, 24, 163-188, 1984.

121

Pavlov, D G, Space-Time Structure, Algebra and Geometry, Collected papers,

TETRU, 2006.

Pavlov, D G, Four-dimensional time, Hypercomplex Numbers Geom. Physics, 1,

31-39, 2004.

Prasad, B N and Dwivedi, Ashwini Kumar, On conformal transformation of

Finsler spaces with m-th root metric, Indian J. pure appl. math., 33 (6), 789-796,

2002.

Randers, G, On an asymmetric metric in the four-space of general relativity, Phys.

Rev., 59, 195-199, 1941.

Riemann, G F B, ‘Uber die Hyphothesen welche der Geometrie zu Grunde liegen’,

Habilitation thesis, University of Gottingen, 1854.

Rund, H, The Differential Geometry of Finlser Spaces, Springer Berlin, 1959.

Sabau, S V, Systems biology and deviation curvature tensor, Nonlinear Analysis:

Real World Applications, 6, 563-587, 2005.

Singh, U P, and Prasad, B N, Modification of a Finsler Space by a Normalised

Semi-Parallel Vector Field, Periodica Mathematica Hungarica, 14 (1), 31-41,

1983.

Shen, Z, Riemann-Finsler geometry with applications to information geometry,

Chinese Annals of mathematics, 27, 73-94, 2006.

Shen, Z and Yildirim, G Civi, On a Class of Projectively Flat metrics with Con-

stant Flag Curvature, Canadian Journal of Mathematics, 60, 443-456, 2008.

Shimada, H, On Finsler spaces with the metric L = m√ai1......imy

i1 ....yim , Tensor

N. S., 33, 365-372, 1979.

Stavrinos, P C, Kouretsis, A P and Stathakopoulos, M, Friedman-like Robertson

Walker model in generalized metric space-time with weak anisotropy, General

Relativity and Gravitation, 40 (7), 1403-1425, 2008.

122

Srivastava, Asha and Arora, Priya, Randers change of m-th root metric, Indian J.

pure appl. math., 43 (1), 37-48, 2012.

Tayebi, A and Najafi, B, On m-th root Finsler metrics, Journal of Geometry and

Physics, 61, 1479-1484, 2011.

Tayebi, A, Peyghan, E and Nia, M Shahbazi, On generalized m-th root Finsler

metrics, Linear Algebra and its Applications, 437, 675-683, 2012.

Tiwari, B, On special reducible Finsler spaces, Differential Geometry-Dynamical

Systems, 14, 132-137, 2012.

Tiwari, B, Zermelo Navigation Problem and the Finsler-Poincare Disc, Mathe-

matics Newsletter, Ramanujan Mathematical Society, 22 (2), 161-167, 2012.

Vacaru, S, Stavrinos, P C, Gaburov, E and Gonta, D, Clifford and Riemann-

Finsler structures in geometric mechanics and gravity, Geometry Balkan Press,

2005.

Yajima, T and Nagahama, H, Nonlinear dynamical systems and KCC theory,

Acta Math. Acad. Paedagog. Nyhazi., 24, 179-189, 2008.

Yajima, T and Nagahama, H, KCC-theory and geometry of the Rikitake system,

J. Phys. A, 40, 2755-2772, 2007.

Yamasaki, Kazuhito and Yajima, Takahiro, Lotka-Volterra system and KCC the-

ory: Differential geometric structure of competitions and predations, Nonlinear

Analysis: Real World Applications, 14, 1845-1853, 2013.

Yano, K, On the torse forming directions in Riemannian spaces, Proc. Imp. Acad.

Tokyo, 20, 340-345, 1944.

Yano, K, Sur le parallelisme et la concourance dans l’espaces de Riemann, Proc.

Imp. Acad. Tokyo, 19, 149-197, 1943.

Yu, Y and You, Y, On Einstein m-th root metrics, Differential Geometry and its

Applications, 28, 290-294, 2010.

123

Zhou, L, A Local classification of a class of (α, β)-metric with constant flag cur-

vature, Differential Geometry and its Applications, 28, 170-193, 2010.

124

List of Published/ Accepted/ Communicated Papers

List of Published Papers

1. Bankteshwar Tiwari and Manoj Kumar, On Randers change of a Finsler

space with m-th root metric, International Journal of Geometric Methods in

Modern Physics (World Scientific), Vol. 11 (2014) DOI: 10.1142/S021988781450087X.

2. Bankteshwar Tiwari and Manoj Kumar, Transformation of a Finsler Space

by Normalised Semi-Parallel Vector Fields, Journal of Tensor Society of

Japan, 74, 156-163, 2013.

List of Accepted Paper

1. Bankteshwar Tiwari and Manoj Kumar, On Finsler space with a special

(α, β)-metric, Journal of Indian Mathematical Society, (Accepted in 2014).

List of Communicated Papers

1. Bankteshwar Tiwari and Manoj Kumar, On Conformal Transformation

of m-th root Finsler metric, (Communicated).

2. Sapna Devi, Bankteshwar Tiwari and Manoj Kumar, Predator-prey model

with prey refuges: Jacobi stability versus Linear stability, (Communicated).

3. Bankteshwar Tiwari, Manoj Kumar and Ghanashyam Kumar Prajapati,

On the Projective change between two special Finsler spaces of (α, β)-metrics,

(Communicated).

4. R. Chaubey, Bankteshwar Tiwari, Anjani Kr. Shukla and Manoj Kumar,

Finsler-Randers Cosmological models in Modified Gravity Theories, (Com-

municated).

Papers presented in Conferences

1. International Conference on Differential Geometry and Relativity, (in collab-

oration with the Tensor Society) participated and presented a paper entitled

“On Finsler space with a special (α, β)-metric”, held at the Department of

Mathematics, Aligarh Muslim University, Aligarh, during November 20-22,

2012.

2. The 12th International Conference of Tensor Society (Japan) on Differential

Geometry and Its Applications, participated and presented a paper enti-

tled “Transformation of a Finsler Space by Normalised Semi-Parallel Vector

Fields”, held at the Department of Pure Mathematics, University of Cal-

cutta, during December 17-21, 2012.

3. International Conference on Differential Geometry and Relativity (ICDGR-

2013), presented a paper entitled “On Randers change of a Finsler space with

m-th root metric”, held at the Department of Mathematics and Statistics,

DDU Gorakhpur University, Gorakhpur, during November 09-11, 2013.

PERSONAL PROFILE

Manoj Kumar,

S/O Sri Harish Chandra Verma

Basdevpur, Bikapur

Distt.-Faizabad-224204

Uttar Pradesh, India.

Mobile: 91-9451788931

E-mail: mvermamath@gmail.com

Objective: To further continue my research and to get a teaching position in

some recognized institution offering opportunity for career advancement and pro-

fessional growth, which will help me to upgrade my knowledge and research work.

Educational Profile:

• M.Sc. (Mathematics) from Dr.R. M. L. Awadh University, Faizabad in 2008.

(Percentage: 62.86)• B.Sc. (Physics, Chemistry, Mathematics) from K. N. I. P. S. S., Sultanpur

in 2006. (Percentage: 57.83%)• Intermediate from G. V. I. C., Faizabad in 2003. (Percentage: 56.00%)• High school from G. V. I. C., Faizabad in 2001. (Percentage: 56.16%)

Software Skills: Ms-office, Maple, Matlab, Mathematica and LaTeX.

Area of Interest: Finsler Geometry, Modelling and Cosmology.

Declaration:

I hereby declare that the information furnished above is true to the best of my

knowledge.

Manoj Kumar