Shortest and Straightest geodesics in Sub-RiemannianGeometry

Dmitri Alekseevsky ∗

A.A.Kharkevich Institute for Information Transmission ProblemsB.Karetnuj per.,19, 127051, Moscow, Russia

and Faculty of Science University of Hradec KraloveRokitanskeho 62, Hradec Kralove, 50003, Czech Republic

Dedicated to Jubilee of Joseph Krasil’shchik

Abstract

There are several different, but equivalent definitions of geodesics in a Rieman-nian manifold, based on two characteristic properties: geodesics as shortest curvesand geodesics as straightest curves. They are generalized to sub-Riemannian man-ifolds, but become non-equivalent. We give an overview of different approach todefinition, study and generalisation of sub-Riemannian geodesics and interrelationsbetween them. For Chaplygin transversally homogeneous sub-Riemannian manifoldQ, we prove that straightest geodesics (defines as geodesics of the Schouten partialconnection) coincides with shortest geodesics. defined as the projection to Q ofintegral curves (with trivial initial covector) of the sub-Riemannian Hamiltoniansystem.This gives a hamiltonization of Chaplygin systems in non-holonomic mechanics.We describe some classes of homogeneous sub-Riemannian manifolds, where straight-est geodesics coincides with shortest geodesics. In particular, we give a descriptionof sub-Riemannian symmetric spaces in terms of affine symmetric spaces.

Contents

1 Introduction 3

2 Sub-Riemannian geodesics as shortest curves 42.1 Euler-Lagrange variational definition of sub-Riemannian geodesics

(EL-geodesics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Pontryagin optimal control definition of sub-Riemannian geodesics

(P-geodesics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Hamiltonian definition of sub-Riemannian geodesics (H-geodesics) of

a sub-Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . . 5

∗The work was partially supported by the grant no. 18-00496S of the Czech Science Foundation

1

2.4 Pontryagin Maximum Principle for bracket generated sub-Riemannianmanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1 Normal P-geodesics as H-geodesics . . . . . . . . . . . . . . . 62.4.2 Abnormal P-geodesics . . . . . . . . . . . . . . . . . . . . . . 7

3 Sub-Riemannian geodesics as straightest curves 93.1 d’Alemebert’s definition of sub-Riemannian geodesics (dA-geodesics) 93.2 Levi-Civita-Schouten-Synge-Vranceanu definition of sub-Riemannian

geodesics (S-geodesics) . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.1 Levi-Civita definition of Riemannian geodesics . . . . . . . . 103.2.2 Schouten partial connection of a sub-Riemannian manifold . 103.2.3 Schouten-Synge-Vranceasnu definition of sub-Riemannian geodesics

(S-geodesics) and non-holonomic mechanics . . . . . . . . . . 11

4 Cartan-Tanaka-Morimoto frame bundle definition of sub-Riemanniangeodesics (M-geodesics) 124.1 Cartan definition of a Riemannian geodesics . . . . . . . . . . . . . 12

4.1.1 Cartan connection and generalized Cartan geodesics (C-geodesics)12

4.1.2 Generalized geodesics for a G-structures of finite type . . . . 134.2 Cartan-Tanaka -Morimoto definition of sub-Riemannian geodesics

on a regular sub-Riemannian manifold . . . . . . . . . . . . . . . . . 144.2.1 The symbol algebra . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Regular distributions and graded tangent bundle . . . . . . . . . . . 144.4 Regular sub-Riemannian manifolds . . . . . . . . . . . . . . . . . . 15

4.4.1 Regular sub-Riemannian structure as Tanaka structure . . . 154.4.2 Tanaka and Morimoto theorems . . . . . . . . . . . . . . . . 164.4.3 Sub-Riemannian M-geodesics as projection of constant vector

fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Sub-Riemannian Chaplygin metric on principal bundle 185.1 Principal connection and its curvature . . . . . . . . . . . . . . . . . 185.2 Chaplygin metric and its standard extension . . . . . . . . . . . . . 20

5.2.1 S-geodesics of a Chaplygin metric . . . . . . . . . . . . . . . 205.2.2 H-geodesics of a Chaplygin metric . . . . . . . . . . . . . . . 215.2.3 Bi-invariant extension of Chaplygin metric and Yang-Mills

dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Homogeneous sub-Riemannian manifolds 256.1 Chaplygin system on Lie group associated to a homogeneous Rie-

mannian manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Sub-Riemaniian homogeneous manifold associated with a homoge-

neous Riemannian manifold with non-simple stabilizer . . . . . . . . 256.2.1 Homogeneous contact sub-Riemannian manifolds . . . . . . . 26

6.3 Symmetric sub-Riemannian manifolds . . . . . . . . . . . . . . . . . 266.3.1 Sub-Riemannian symmetric spaces associated with an affine

symmetric space M = G/H . . . . . . . . . . . . . . . . . . . 266.3.2 Compact sub-Riemannian symmetric space associated to a

graded complex semisimple Lie algebra . . . . . . . . . . . . 27

2

1 Introduction

The important role of Riemannian geometry in applications is based on the factthat many important equations , arising in mechanics, mathematical physics , bi-ology, economy, information theory etc., can be reduced to the geodesic equation.Moreover, Riemannian geometry gives an effective tool for investigation of geodesicequation and other equations associated with the metric ( Laplace, wave, heat andSchrodinger equation, Einstein equation, Yang-Mills equation etc).There are many equivalent definitions of geodesics in a Riemannian manifold. Theyare naturally generalised to sub-Riemannian manifold, but become non-equivalent.H.R. Herz remarked that there are two main approaches for definition of geodesics:geodesics as shortest curves based on Mopertrui’s principle of least action (variational approach ) and geodesics as straightest curves based on d’Alembert’sprinciple of virtual work ( which leads to geometric descriptions based on the notionof parallel transport).

We consider three variational definitions of sub-Riemannian geodesics as (lo-cally) shortest curves (Euler-Lagrange (EL-geodesics), Pontyagin (P-geodesics) andHamilton (H-geodesics)) andthree definitions of sub-Riemannian geodesics as straightest or autoparallel curves,used in nonholonomic mechanics. (d’Alembert (dA-geodesics), Levi-Civita-Schouten-Synge-Vrancheanu (S-geodesics) and Cartan-Tanaka-Morimoto (M-geodesics) ) anddiscuss interrelations between them.

A. Vershik and L. Faddeev [V-F] had formulated the problem of characteri-sation of sub-Riemannian manifolds such that straightest S-geodesics ”coincides”(more precisely, consistent) with shortest H-geodesics in the following sense.It is known that an S-geodesic of a sub-Riemannian manifold (Q,D, gD) is deter-mined by the initial velocity v ∈ Dq ⊂ TqQ, and the initial data for an H-geodesicis a pair (v, λ) ∈ Dq × D0

q where D0 := AnnD ⊂ T ∗Q is the codistribution ((theannihilator of the distribution D).Due to this , following [V-F], we say that straightest geodesics coincides with short-est geodesics if the class of straightest geodesics coincides with the class of shortestgeodesics with zero codistribution covector λ.

Vershik and Faddeev showed that for a generic sub-Riemaniian manifold allshortest geodesics are different from straightest geodesics. They gave first example( left invariant distribution on a compact Lie group with a bi-invariant metric) whenshortest geodesics coincides with straightest geodesics (with zero initial covector λ ).

We generalised this example and show that this is true for any G-invariant sub-Riemannian Chaplygin metric (D = ker(ϖ), gD) on a principal bundle π : Q →M = Q/G with a principal connection ϖ : TQ → g, associated to a Riemannianmetric gM on M . Moreover, sub-Riemannian H-geodesics of gD are the horizontallifts of the projection to M of geodesics of the standard extension of gD to a metricgQ of Q , defined by any left invariant metric of G. The (straightest ) S-geodesicare horizontal lifts of geodesics of the Riemannian manifold (M, g).

This is a generalization of results by R. Montgomery [Mont], [Mont1] who con-sidered the bi-invariant extension gQ of gD, associated to a bi-invariant metric onG.We give a simple proof of Wong results on description of the evolution of charge

3

particle in a classical Yang-Mills field in terms of geodesics of the bi-invariant ex-tension gQ of the Chaplugin sub-Riemannian metric and clarify relations betweensub-Riemannian geodesics and geodesics of gQ.

We describe some classes of invariant sub-Riemannian structures on homoge-neous manifolds, where straightest geodesics coincides be shortest ones.In particular, we give a description of all bracket generated symmetric sub-Riemannianmanifolds , introduced by R.S.Strichartz [Str], and indicate a construction of com-pact sub-Riemannian symmetric space associated to a graded complex semisimpleLie algebra.

Acknowledgment. I thank A.M. Vershik for his comments on non-holonomicgeometry and explanation of his joint with L.D. Faddeev results and A. V. Bolsi-nov for information about his results on integrability of geodesic flows and usefuldiscussions.

2 Sub-Riemannian geodesics as shortest curves

2.1 Euler-Lagrange variational definition of sub-Riemanniangeodesics (EL-geodesics)

Recall that a distribution D ⊂ TM on a connected n dimensional manifold M iscalled bracket generated ( or totally nonholonomic ) if the space ΓD of sectionsgenerates the Lie algebra X(M) of vector fields.According to Rashevsky-Chow theorem , any two points on such manifold can bejoint by a horizontal (i.e. tangent to D) curve. Then any Lagrangian L ∈ C∞(TQ)defines a nonholonomic variational problem:Find a curve q(t), t ∈ [0, T ] in the space CD(q0, q1) of horizontal curves,joints pointsq0, q1 ∈ Q, which delivers a minimum or, more generally, a critical point, for theaction functional

A(γ) =

∫ T

0L(q(t), ˙q(t))dt.

The Lagrangian L(q, q) determines a horizontal 1-form

FL = (δL)idqi := (

d

dtLqi − Lqi)dq

i

on TQ, called the Lagrangian force [Sy] [Sy1], [V-F]. Denote by (ω1, · · · , ωm) acoframe of D0 such that the 1-form

ωλ =∑

λa(q, q)ωai (q)dq

i

vanishes on D for any functions λ(q, q) = (λ1, · · · , λm) on TQ.Then critical points q(t) of A(q(t)) are solution of the Euler-Lagrange equations

(δL)iqi(t) = (

d

dtωλ)iq

i (1)

ωλ(q) = 0

for unknown curve q(t) ∈ CL(q0, q1) and vector function λ(q(t), q(t)). The r.h.s.of (1) can be written as

d

dtωλ = Lq(t)ωλ = λaω

a + λaqydωa.

4

An sub-Riemannian manifold (Q,D, gD) is a manifold with a distribution D anda Riemannian metric gD on D . We assume that D is bracket generated.An Euler-Lagrange (shortly, EL) non-parametrized geodesic of an sub-Riemannianmanifold is a critical point of the length functional (L =

√g(q, q)) and a naturally

parametrized EL geodesic is a critical point of the energy functional ( L = 12g(q, q))

in CD(q0, q1). It is known that any curve in CD(q0, q1) of minimal length is an ELgeodesic.

2.2 Pontryagin optimal control definition of sub-Riemanniangeodesics (P-geodesics)

Let (Q,D, gD) be a bracket generated sub-Riemannian manifold as above. Denoteby (X1, · · · , Xm) a field of orthonormal frames in D. Then any horizontal curveq(t) ∈ CD(q0, q1) is a solution of the first order ODE

q(t) =m∑a=1

ua(t)Xa(q(t)), q(0) = q0. (2)

where qi are coordinates in Q and vector-functionu(t) = (u1(t), · · · , um(t)) ( called the control ) consists of the coordinates of q withrespect to the frame (Xa) .

The length ℓ(u) =∫ T0

√∑ua(t)2dt of the curve q(t) depends only on the control

u(t) and is called the cost function. It is the functional of the space of admissi-ble control u(t) such that the associated equation has solution qu(t) ∈ CD(q0, q1) .

Pontryagin (or P) geodesic ( resp. minimal P-geodesic ) is the integral curvequ(t) ∈ CD(q0, q1) of ODE with a control u(t) which is a critical point ( resp., min-imum) of the cost functional in the space of admissible control.

P-geodesics coincide with EL-geodesics.

For D = TQ, P-geodesics are geodesics of the Riemannian manifold (Q, gQ).

2.3 Hamiltonian definition of sub-Riemannian geodesics(H-geodesics) of a sub-Riemannian manifold

We will denote by ξD the restriction of a covector ξ ∈ T ∗q Q to D and define the

cometric as the contravariant symmetric tensor g∗D ∈ ΓS2(TQ), given by

g∗(ξ, ξ) = (gD)−1(ξD, ξD) ∈ C∞(T ∗Q).

The sub-Riemannian Hamiltonian is defined as

h = hgD :=1

2g∗(ξ, ξ).

H-geodesics are projection to Q of orbits of Hamiltonian vector field h =ω−1dh ∈ X(T ∗Q) with quadratic ( degenerate ) sub-Riemannian Hamiltonian h =12g

∗(ξ, ξ). Here ω = dpi ∧ dqi is the standard symplectic form of T ∗Q.

5

2.4 Pontryagin Maximum Principle for bracket gener-ated sub-Riemannian manifold

Recall that vector fields X = Xi∂qi bijectively corresponds to fiberwise linear func-tions

pX : T ∗Q → R, ξ = pi∂qi 7→ p(X) = Xipi

on the cotangent bundle T ∗Q. The function pX is the Hamiltonian of the Hamilto-nian vector field

pX = Xi∂qi − pi∂qjXi∂pj ,

which is the complete lift of X onto T ∗Q. The map

X(Q) ∋ X → pX ∈ X(T ∗Q)

is an isomorphism of the Lie algebra X(Q) onto the Lie algebra X(T ∗Q)1 of fiberwiselinear vector fields on T ∗Q.

Theorem 1 (Pontryagin Maximim Principle) Let q(t) ∈ CD(q0, q1) be a minimalP-geodesic on a sub-Riemannian manifold (Q,D, gD) with natural parametrization(s.t. |q(t)| = const), which corresponds to a control u(t) = (ua(t)) :

q(t) = ua(t)Xa(q(t)). (3)

Denote by φt the flow , generated by the non-autonomous vector field Xu = ua(t)Xa.Then for some covector ξ0 ∈ T ∗

q0Q the curve

ξ(t) := φ∗−tξ0 := ξ0 ◦ φ−t∗ ∈ T ∗

q(t)

satisfies the equationξ(t) = ua(t)pa(q(t)) (4)

where pa := pXa and one of the following conditions holds

(N) ua(t) ≡< ξ(t), Xa(q(t)) >(A) 0 ≡< ξ(t), Xi(q(t)) > .

Here the bracket < ξ,X > denotes the pairing between covectors and vectors.

An extremal curve ξ(t) ⊂ T ∗Q, which satisfy (N) ( resp., (A)) is called normal( resp., abnormal) extremal, and its projection q(t) ⊂ Q is called a normal (resp., abnormal) P-geodesic.

2.4.1 Normal P-geodesics as H-geodesics

Corollary 2 Normal extremal ξ(t) ⊂ T ∗Q for (Q,D, gD) is an integral curve ofthe Hamiltionian equation on T ∗Q with the sub-Riemannian Hamiltonian hgD(ξ) =12g

∗D(ξ, ξ) =

12

∑(gD)ijpDi p

Dj where p = pidx

i ∈ T ∗Q, pDi = pi|D.

Proof: In the case of normal geodesic, the equation (4) take the form

ξ(t) =∑

pa(ξ(t))pa(ξ(t)) = ω−1d1

2

∑p2a(ξ(t)) = hgD .

6

�

Since the Hamiltonian vector field H preserves the Hamiltonian, the normal ex-tremal ξ(t) belongs to a level set Lc = {H = c} ⊂ T ∗Q of the Hamiltonian.

A curve ξ(t) ⊂ L on a submanifold L ⊂ T ∗Q is called characteristic if itsvelocity ξ(t) belongs to the kernel ker ω|L of the restriction of the symplectic formto L.

Corollary 3 Assume that an extremal ξ(t) ⊂ Lc belongs to a regular level set ofthe Hamiltonian h = hgQ, i.e. Lc ⊂ T ∗Q is a smooth hypersurface. Then ker ω|Lc

is the 1-dimensional distribution generated by H. In particular, the extremals ξ(t)are the characteristic curves of Lc.

Proof: The tangent space of the level set Lc is described as follows

TξLc = {w ∈ Tξ(T∗Q), 0 =< dhξ, w >=< ω(h)ξ, w >}.

This shows that TξLc consists of all ω-orthogonal to hξ vectors. Since the ω|TξLc

has 1-dimensional kernel, it is generated by hξ. �

2.4.2 Abnormal P-geodesics

Denote by CD(q0) = {γ : [0, T ] → Q, γ(0) = q0, γ ∈ D} the space of horizontalcurves, starting from q0 , where D ⊂ TQ is a bracket generated distribution. Acurve γ(t) ∈ CD(q0) is called singular (regular) if the end-point map

ε : CD(q0) → Q, γ(t) 7→ g(T )

is singular ( resp., regular).The following theorem by R. Montgomery shows that abnormal geodesics coincideswith skingular curves and they are projection on Q of characteristic curves of thecodistribution D0, considered as a submanifold of the symplectic manifold (T ∗Q,ω).

Theorem 4 ( [Mont], [Mont1] ) i) Abnormal geodesics of any sub-Riemannianmetric gD on D are exactly singular horizontal curves in Q.ii) A horizontal curve γ ⊂ Q is singular if and only if it is a projection to Q of a

characteristic curve of the submanifold D0 := D0 \ {zero section} ⊂ T ∗Q.

Now we give a description of characteristic curves in D0, following [Mont]. De-note by τ : T ∗Q → Q, ρ : T ∗(T ∗Q) → T ∗Q the natural projections.

We fix a complementary toD distribution V such that TQ = D+V . LetXi be anorthonormal frame inD , Ya a frame in V and θi ∈ ΓD∗ = ΓV 0 and ηa ∈ ΓV ∗ = ΓD0

the dual coframes. The Liouville tautological 1-form θξ = ξ ◦ τ∗ in T ∗Q ( whereτ∗ : T (T

∗Q) → TQ is the induces projection) at a point ξ = hiθi + kaη

a ∈ T ∗Q canbe written as

θξ = hiθi + kaη

a.

Then the restrictions θ0 := θ|D0 , ω0 = dθ|D0 of the forms θ, ω = dθ to D0 are givenby

θ0ξ = ξ|D = kaηa, ω0 = dθ0 = dka ∧ ηa + kadη

a.

7

By definition, characteristic curves are non-vanishing curves η(t) ⊂ D0 tangent tothe distribution (with singularities )

ch(D0) := ker ω0 = T (D0)⊥ ∩ T (D0),

where the vector bundle T (D0)⊥ ⊂ T (T ∗Q) is the ω-orthogonal complement to thetangent bundle T (D0) ⊂ T (T ∗Q)|D0 .

We consider hi, ka as fiberwise coordinates in τ : T ∗Q → Q.

Lemma 5 The vector bundle T (D0)⊥ = span{ηi, i = 1, · · · ,m} and the projectionτ∗ : T (T

∗Q) → TQ) induces for any η ∈ D0 the isomorphism

τ∗ : Tη(D0)⊥ → Dτ(η),

uihi 7→ uiXi|q, q = τ(η).

Proof: The submanifold D0 = {η = kaηa} is defined by the equations hi = 0, i =

1, · · · ,m. Hence, TηD0 = {v ∈ Tη(T

∗Q), 0 =< dhi, v >= ω(hi, v)}. �Since D0 = {η = kaη

a}, ka are fiberwise coordinate in D0 → Q. Hence∂ka , Xi, Ya is a frame in the tangent bundle T (D0) and the tangent vector to acurve γ(t) = ka(t)η

a(t) ⊂ D0 can be written as

γ(t) = ka∂ka + γiXi(γ(t)) + γaYa(γ(t)).

The restriction to D0 ⊂ T ∗Q of the canonical form θ = hiθi + kaη

a is given byθ0 = kaη

a and the form ω0 = dθ0 = dka∧ηa+kadηa. The 2-form dηa can be written

asdηa = cabdη

b ∧ ηd + caibθi ∧ ηb + caijθ

i ∧ θj

where caij = −ηa([Xi, Xj ]), cabi = −ηa([Yb, Xi]), c

abd = −ηa(Yb, Yd).

Calculating the contraction iγω0 = Aadka +Baη

a + Ciθi of the tangent vector

γ = kα∂kα + γiXi + γaYa ∈ T (D0)

with 2-form ω0 , we obtain the following conditions that γ is in ch(D0):

i) Aa = γa = 0,

ii) Ba = ka + kbcbiaγ

i = 0,iii) Ci = kac

aij γ

j = 0.

Denote by d : Dx → Λ2Dx the linear map, defined by

dη = dη|Λ2Dx,

where η is an extensions of η . Then

dη(X,X ′) = −η([X, X ′]).

This map does not depends on extensions ξ, X, X ′ of the 1-form η ∈ D0x ⊂ T ∗

xQ andthe vectors X,X ′ ∈ Dx. The above formulas implies the following

8

Theorem 6 [Mont] Let η(t) = kaηa ⊂ D0 be a characteristic curve. Then

i) The projection γ(t) := τ(η(t)) is a horizontal curve of the form γ(t) = γiXi.ii) The velocity vector

η(t) = ka∂ka + γiXi = kbcbaiγ

i + γiXi

is completely determines by the vector η(t) and the velocity γ(t) of the projection .iii) The velocity vector γ(t) belong to the kernel ker dη(t) ⊂ Dγ(t) of the 2-formdη(t) on Dγ(t).iv) The characteristic distribution ( with singularites) at a point η ∈ D0 has rankr = dimker dη.

P-geodesics are exhausted by normal geodesics , which are exactly H-geodesics,and abnormal geodesics, which depend only on distribution D.

3 Sub-Riemannian geodesics as straightest

curves

3.1 d’Alemebert’s definition of sub-Riemannian geo-desics (dA-geodesics)

To define dA-geodesics, we extend the sub-Riemannian metric gD to a Riemannianmetric gQ.We will see that in the case, when the sub-Riemannian structure is regular the deepresults by N. Tanaka and T. Morimoto provide a canonical extension gQ.

The d’Alembert’s principle of virtual dispalacements for a mechanical systemmay be formulated as follows, see [V-F].1) The evolution of a mechanical system with a (smooth) configuration space Q isdescribed by projection to Q of integral curves of a special vector field X ∈ X(TQ)on the tangent bundle TQ. The field is special if it corresponds to a second orderequation , that is π∗X(q,q) = q where π : TQ → Q is the projection.2) The vector fieldX is determined by the Lagrangian force defined as the horizontal1-form FL := (δL(q, q))idq

i on TQ, associated with the Lagrangian L(q, q), and theexternal force.3) d’Alembert’s principle states that the real trajectories of the system ( defined bythe special vector field X) is determined by the condition that the Lagrangian forceequal to the external force.Assume thati)the Lagrangian of the system is quadratic in velocities and positively defined (that is can be written as L = 1

2g(q, q) where g is a Riemannian metric in Q ) andthatii) the only external force is the reaction of a non-holonomic constrain, defined by adistribution D = ker η1 ∩ · · · ∩ ker ηk, where ηa = ηai dq

i, a = 1, cdots, k is a coframeof the codistribution D0 . The reaction of the constrain is the horizontal 1-formϕλ = λa(q, q)η

a defined by the condition that the equation FL−ϕλ = 0 correspondsto a vector field X ∈ X(TQ) tangent to the distribution D ⊂ TQ.In coordinates, this equation take form [V-G1]

d

dtLqi − Lqi = λa(q(t), q(t))η

ai

9

ord

dtLqi − Lqi ≡ 0(modD0).

The projection to Q of integral curves of this equation is called dA-geodesic of thesub-Riemannian metric (D, gD) , associated with an extension of gD to a Rieman-nian metric g on Q. In general, the equation of dA-geodesics is neither Lagrangianno Hamiltonian.

3.2 Levi-Civita-Schouten-Synge-Vranceanu definitionof sub-Riemannian geodesics (S-geodesics)

3.2.1 Levi-Civita definition of Riemannian geodesics

Recall that Levi-Civita associated to a Riemannian manifold (Q, g) the canonicaltorsion free connection ∇g, which preserves the metric (called the Levi-Civita con-nection) and defined a geodesic as an autoparallel curve , that is a curve q(t) withsatisfies the geodesic equation

∇gγ γ ≡ qi(t) + Γi

jk(qj(t))qj(t)qk(t) = 0

where qi(t) are coordinates of the curve q(t) (with respect to local coordinate qi)and Γi

jk the Christoffel symbols.The extension of this notion to the sub-Riemannian manifold (Q,D, gD) had beenproposed independently by J.A. Schouten, J.L.Synge and G. Vranceanu.

3.2.2 Schouten partial connection of a sub-Riemannian manifold

Let D ⊂ TQ be a distribution. A partial D-connection in D is a bilinear map

∇D : ΓD × ΓD → ΓD, (X,Y ) 7→ ∇DXY

which is C∞(Q) linear in X and satisfies the Leibnitz rule∇D

X(fY ) = f∇DXY +X · fY in Y ∈ ΓD.

Let ea, a = 1, · · · , k be a frame of D defined in a neighborhood of a horizontal curveq(t) .The Christoffel symbols are the local functions Γa

bc(q) on Q defined by

(∇ebec)a = Γa

bc(q)ea.

The value of the functions Γabc(t) := Γa

bc(q(t)) on the curve q(t) depends only of theframe ea(t) := ea(q(t)) along the curve q(t). Due to this , the partial connectiondefines a parallel transport of a vector Y0 ∈ Dq0 along the curve q(t) as the solutionY (t) = Y c(t)ec(t) ∈ Dq(t) of the equation

0 = ∇q(t)Y (t) = ∇q(t)(Ya(t)ea(t)) = [Y a(t) + Γa

bc(t)qb(t)Y c(t)]ea(t)

where q = qaea.

I.A. Schouten showed that a complementary to D distribution V on a sub-Riemannian manifold ()Q,D, gD) ( called a rigging) defines a partial connection

10

∇S in D with preserves the metric gD and has zero torsion T . The torsion tensoris defined by

T (X,Y ) = ∇SXY −∇S

Y X − [X,Y ]D, X, Y ∈ ΓD, X, Y ∈ ΓD

where XD is the horizontal part of the vector

X = XD +XV ∈ TqQ = Dq + Vq.

In coordinate free way, the Schouten partial connection of (Q,D, gD) , associatedto a rigging V is defined by the Koszul formula

2g(∇SXY, Z) = X · g(Y,Z) + Y · g(X,Z)− Z · g(X,Y )+

g([X,Y ]D, Z)− g(Y, [X,Z]D)− g(X, [Y, Z]D),X, Y, Z ∈ X(Q).

3.2.3 Schouten-Synge-Vranceasnu definition of sub-Riemannian geodesics(S-geodesics) and non-holonomic mechanics

Schouten-Synge-Vranceasnu geodesic (S-geodesic) of a sub-Riemannian manifold(Q,D, gD) asocciated to a rigging V a is a horizontal curve γ(t) with parallel (w.r.t.Schouten connection) tangent vector field g(t), i.e. a solution of the equation

∇Sγ γ = 0.

Schouten defined also the curvature tensor R ∈ so(D)⊗Λ2T ∗M of the Schoutenconnection by

R(X,Y )Z = [∇X ,∇Y ]Z − [[X,Y ]V , Z]D, X, Y, Z ∈ ΓD.

V.V. Wagner generalized this notion and defined Wagner curvature tensor, suchthat the vanishing of the Wagner tensor is equivalent to the flatness of the Schoutenconnection (such the the associated parallel transport does not depend on the path,jointed two points ), see [B], [D-G].

Assume that the sub-Riemannian metric gD is extended to a Riemannian metricg on Q such that g(D,V ) = 0. Then the Levi-Civita connection ∇g induces aconnection ∇D in D . Using the projection prD : TQ → D , it is defined by

∇DXY = πD∇g

XY = (∇gXY )D, X ∈ TQ, Y ∈ ΓD.

The connection ∇D in the distribution D is an extension of the partial Schoutenconnection ∇S .

Theorem 7 ( Vershik-Faddeev)[V-F], [V-F1] Let (Q,D, gD) be a sub-Riemannianmanifold, g is an extension of gD to a metric in Q and V = D⊥ the orthogonalcomplement to D. Then S-geodesics coincides with dA-geodesics and they describeevolution of the free mechanical system with kinetic energy g in configuration spaceQ with nonholonomic linear constrains D.

11

4 Cartan-Tanaka-Morimoto frame bundle def-

inition of sub-Riemannian geodesics (M-geodesics)

4.1 Cartan definition of a Riemannian geodesics

An important definition of Riemannian geodesics as straightest curves had beenproposed by E. Cartan. It is easily generalized to the case of G-structures of finitetype and to sub-Riemannian manifolds ( and other Tanaka structures).A Riemannian metric g on a manifold Q can be consider as a G = SOn-structure,i.e. a principal G-subbundle π : P → Q = P/G of orthonormal coframes (i.e.isometries f : TxQ → Rn = V ) with the tautological soldering form

θ : TP → son, θf (X) := f(π∗X).

Remark 8 If f : TqQ → V is a coframe, then f−1 : V → TqQ is a frame. Due tothis a G-structure can be consider also as the principal G-bundle of frame with theright action of the linear group G.

The total space P of a SOn-structure admits a canonical SOn-equivariant abso-lute parallelism (Cartan connection)

κ = θ + ω : TP → V + so(V ),

which is an extension of the vertical parallelism ip : T vp P ≃ son, ∀p ∈ P (defined

by the free action of SOn on P ). Here ω : TP → son is the connection form of theLevi-Civita connection.Geodesics are projection to Q of constant horizontal vector fields X ∈ κ−1(V ) ∈X(P ), see [K-N].

4.1.1 Cartan connection and generalized Cartan geodesics (C-geodesics)

We recall the definition of a Cartan connection and discuss definition of generalisedgeodesics associated with a Cartan connection.Let M0 = L/G be a homogeneous n-dimensional manifold.A Cartan connection of type M0 = L/G on n-dimensional manifold Q is aprincipal G-bundle π : P → Q = P/G together with a l-valued G-equivariant (s.t.r∗gκ = Ad−1

g ◦ κ, g ∈ G ) kernel free 1-form

κ : TP → l

which extends the vertical parallelism T vp P ≃ g.

The form κ defines an absolute parallelism TpP ≃ l . Hence, tensor fields on P maybe identified with tensor-valued functions.In particular, the curvature 2-form dκ + 1

2 [κ, κ] on P is horizontal and can beidentified with a function K : P → C2(V, l) := l⊗Λ2V ∗ where V is complementarysubspace to g in l.Assume that the homogeneous manifold M0 = L/G is reductive, i.e. there is areductive decomposition l = g+ V , where V is an AdG-invariant complement to g.Then the Cartan connection κ is decomposed as

κ = θ + ω := prV ◦ κ+ prg ◦ κ

12

into the horizontal form θ with ker θ = T vP and the vertical part ω with H :=ker ω = span(κ−1V ).Moreover, θ is a soldering form, which allows identify π : P → Q with a G-structure, i.e. a principal bundle of coframes, see definition below. The form ω is a connec-tion form, which defines a principal connection with horizontal distribution H. Itindices a linear connection ∇ in Q with the holonomy group Hol ⊂ G, see [K-N].We say that the G-structure (π : P → Q) with connection , defined by (θ, ω) is theG-structure, associated to a Cartan connection of reductive type L/G. We defineC-geodesics for such Cartan connection as projection to Q of the orbits of con-stant (horizontal) vector fields from κ−1(V ).These geodesics are geodesics of the linear connections ∇.This construction will be used for definition of Morimoto sub-Riemannian geodesics.

There is another important case, when the complementary distribution V canbe canonically chosen and one can define C-geodesics as above.LetM0 = L/G be a n-dimensional flag manifold , i.e. the coset space of a semisimpleLie group (real or complex) modulo parabolic subgroup G . The parabolic subal-gebra g ⊂ l defines the generalized Gauss decomposition l = l− + l0 + l+ such thatg := l0 + l+ and g− := l− + l0 are the opposite parabolic subalgebras.A parabolic geometry is the geometry of the Cartan connection (π : P → P/G =Q, κ : TP → l) of type L/G. We chose the complementary to g subspace V := l−

and defines C-geodesics as the projection to Q of constant vector fields from κ−1V .(This definition is too general and has to be specified).

The principal bundle π : P → Q admits many Cartan connections. One ofthe problems is to define some normalization conditions of the curvature functionK : P → C2(V, ℓ), which guaranties the existence of unique Cartan connection.The standard condition is that at any point p ∈ P the 2-form Kp ∈ C2(V, l) iscoclosed δ∗Kp = 0 with respect to an appropriate codifferential , dual to the Spencerdifferential δ : Ci(V, l) → Ci+1(V, l) for cohomology of a Lie algebra V with valuein V -module, see [C-S], [A-D]. Existence and uniqueness of such normal Cartanconnection for parabolic geometry is known [C-S]. A very general normalisationconditions for existence and uniqueness of normal Cartan connections is given in[M1] and [C].

4.1.2 Generalized geodesics for a G-structures of finite type

Given a linear group G ⊂ GL(V ), V = Rn.A G -structure can be defines as a G-principal bundle π : P → Q = P/G with asoldering 1-form

θ : TP → V

i.e. a horizontal G-equivariant form with ker θ = T vertP .

Such bundle is naturally identified with a principal G-bundle of coframes on TQ.Assume that the group G is of finite type k, that is its Lie algebra g has non trivialk-th prolongation g(k) and g(k+1) = 0 . Then ([Stern]) one can prolong π to a bundleP (k) → Q with absolute parallelism

κ : TP (k) → g∞ = V + g+ g(1) + · · ·+ g(k).

Generalised C-geodesics for the G-structure are defined as the projection oforbits of constant vector fields κ−1V to Q.

13

In the case k = 0 when the first prolongation

g(1) = g⊗ V ∗ ∩ V ⊗ S2V ∗

of the linear Lie algebra g is trivial , the absolute parallelism

κ : TP → V + g

is a Cartan connection. It defines a linear connection in Q with holonomy groupH ⊂ G and generalised C-geodesics are geodesics of this connection.Let g ⊂ l(V ) be an irreducible linear Lie algebra of type 2. Then the full prolonga-tion g∞ = V +g+g(1) = g−1+g(0)+g(1) is a simple 3-graded Lie algebra. All such3-graded Lie algebra are known. The corresponding G-structure P → M admitsunique normal Cartan connection κ : TP → V + g + g(1) , see [C-S], and corre-sponding generalized C-geodesics form an interesting class of distinguished curvesin Q.For example, for the conformal structure , which can be considered as R+ · SOn-structure, generalized geodesics are conformal circles.

The theory of such generalised geodesics for different parabolic geometries hadbeen developed by A. Cap, J. Slovak, V. Zadnik [C-S-Z], B. Doubrov and I. Zelenko[D-Z], M Herzlich [H] and others.

4.2 Cartan-Tanaka -Morimoto definition of sub-Riemanniangeodesics on a regular sub-Riemannian manifold

4.2.1 The symbol algebra

A distribution D ⊂ TQ on a manifold defines an decreasing filtration

0 ⊂ D−1 = ΓD ⊂ D−2 := D−1 + [D−1,D−1] ⊂ · · ·

in the Lie algebra X (Q) of vector fields and for q ∈ Q a flag

0 ⊂ D−1q ⊂ D−2

q ⊂ · · ·

in TqQ. The Lie bracket induces in the associated graded space

mq = m−1q +m−2

q + · · ·+m−kq , m−i

q := D−iq /D−i−1

q

a structure of a negatively graded Lie algebra , called the symbol algebra at apoint q.By construction, the graded Lie algebra is fundamental, i.e. it is generated bym−1.

4.3 Regular distributions and graded tangent bundle

The distribution D ⊂ TQ is called bracket generated or totally nonholonomicof depth k if k is the minimum integer, s.t. D−k

q = TqQ for any q ∈ Q.Then the negatively grades Lie algebra

T grq Q = mq = m−1

q + · · ·+m−kq = D−1

q +D−2q /D−1

q + · · ·+ TqQ/D−(k−1)q

14

is called the graded tangent space at q ∈ Q.If, moreover the symbol algebra mq at any point is isomorphic to a fixed negativelygraded Lie algebra m = m−1 + · · · + m−k, the distribution D is called a regulardistribution of type m. For a regular distribution D, the associated filtrationD−1 ⊂ D−2 ⊂ D−3 consists of the space of sections of distributions D = D−1 ⊂D−2 ⊂ · · · that is D−i = ΓD−i.

4.4 Regular sub-Riemannian manifolds

A sub-Riemannian manifold (Q,D, gD) with bracket generated distribution D iscalled bracket generated. Then the graded tangent space T gr

q Q = mq hasthe structure of a metric negative graded Lie algebra, i.e. a graded Lie algebramq =

∑−ki=−1m

iq with an Euclidean metric gmq such that the graded spaces mi

q aremutually orthogonal.

The metric gmq is a natural extension of the sub-Riemannian metric gDq in Dq,which is described in the following Lemma

Lemma 9 Let m = m−1+· · ·+m−k be a negatively graded fundamental Lie algebra.Then an Euclidean metric g on m−1 has a natural extension to an Euclidean metricgm in m.

Proof: The construction of the extension is inductive. Denote by β : Λ2m−1 → m−2

the linear map defined by the Lie bracket. Then m−2 = m−1 + β(Λ2(m−1)) . Themetric in m−1 induces a metric

in Λ2(m−1) and on β(Λ2(m−1)). Now we can canonically construct a metric atm−2, which is represented as a (not direct) sum of two Euclidean subspaces. Thenwe iterate this construction. �A sub-Riemannian manifold (Q,D, gD) with a regular distribution D of type m iscalled a regular sub-Riemannian manifold of type (m, gm) if all metric Lie alge-bras (mq, g

mq ) are isomorphic to the metric graded Lie algebra (m, gm) ( called the

metric symbol of the sub-Riemannian manifold).

Denote by g0 = der(m, gm) the Lie algebra of skew-symmetric (graded preserv-ing) derivation of the metric graded Lie algebra. Then

m = g0 +m−1 + · · ·+m−k

is a non-positively graded Lie algebra. If (mq, gmq ) is a metric symbol, then mq is

called the extended symbol of a sub-Riemannian manifold.Note that a derivation A ∈ der(m) is skew-symmetric if its restriction to m−1 isskew-symmetric.

4.4.1 Regular sub-Riemannian structure as Tanaka structure

Let D ⊂ TQ be a regular rank m distribution of type m, and Aut(m) the group ofgraded preserving automorphisms of m.

An admissible frame of D is an isomorphism

f : T grq Q = mq → m

15

of graded Lie algebras. The group Aut(m) acts freely and properly on the manifoldFr(D) of admissible frames on D with the orbit space Fr(D)/Aut(m) = Q. Hence,Fr(D) → Q is a principal bundle ( called the bundle of admissible frames on D).A Tanaka structure ( or relative G-structure) is a G -principal subbundle π : P →Q = P/G of the bundle of admissible frames on D.

The classical identification of a Riemannian manifold with a G-structure withthe orthogonal group G = SO(n) is extended to the sub-Riemannian case:

Proposition 10 A regular sub-Riemannian manifold of a type (m, gm) is identifiedwith a Tanaka G-structure with the structure group G = Aut(m, gm) .

Proof: Let D be a regular distribution of type m =∑−k

i=−1mi and

gm is the Euclidean metric in m generated by an Euclidean metric g in m−1.Let π : P → Q be a Tanaka G-structure with G = Aut(m, gm). Then the associatedsub-Riemannian metric is defined by the condition that for any admissible framef ∈ P , its restriction

fm−1 : m−1 → Dq = m−1q

is an isometry.The converse statement is obvious. �

4.4.2 Tanaka and Morimoto theorems

N. Tanaka generalised the theory ofG-structures to Tanaks structures. In particular,he defined the full prolongation of a non positively graded Lie algebra g = g0+g−1+· · · g−k as a maximal graded algebra of the form

g∞ = g+ g(1) + g(2) + · · ·

such that for any X ∈ gi, i > 0 the condition [X, g−1] = 0 implies X = 0.

Theorem 11 (Tanaka) [T], ,see also [C-S], [Z],[A-D] . Let π : P → Q be a TanakaG-structure on (Q,D) where D is a regular distribution of type m = m−1+· · ·+m−k.Assume that the full prolongation m∞ of the extended symbol algebra m = g + mis finite dimensional. Then there is a canonical bundle P∞ → Q , constructed bysuccessive prolongations, with an absolute parallelism κ : TP∞ → m∞ .If the first prolongation g(1) = 0, then the absolute parallelism κ : TP → m+ g is aCartan connection.

Theorem 12 (T. Morimoto) see [M], [C], [A-M-S]. The first prolongation of thegraded Lie algebra g = g0 + m, which is the extended symbol algebra of a regularsub-Riemannian manifold, is trivial.Moreover, let (Q,D, gD) be a regular sub-Riemannian manifold of type (m, gm) andπ : P → Q the associated G0 = Aut(m, gm) Tanaka structure. Then π admits aunique normal Cartan connection

κ : T (P ) → m = g0 +m.

16

This remarkable theorem has important applications. According tosubsection 4.1.1, the Tanaka G0-structure π : P → Q has the canonical G0-structurewith the soldering form θ := prmκ,and connection, defined by the connectionform ω := prg0κ. Due to this, the group A = Aut(Q,D, gD) of sub-Riemannianautomorphisms naturally acts on P as the group of automorphisms of the Cartanconnection. i.e. the group of automorphisms of the principal bundle π, which pre-serves the absolute parallelism κ. By Kobayashi theorem [Stern], A is a Lie group,which acts freely on P with closed orbits. Also it preserves the Riemannian metricon Q, which is the projection to Q of the G0-invariant metric on the horizontaldistribution H = span{κ−1(m)} , defined by gm.We get

Theorem 13 Let (Q,D, gD) be a regular sub-Riemannian manifold of types (m, gm).Theni) The sub-Riemannian metric gD of a regular sub-Riemannian manifold (Q,D, gD)admits a canonical extension to a Riemannian metric on Q.ii) The group A = Aut(Q,D, gD) of automorphisms of type m is a Lie group ofdimension dimA ≤ dim g0(m) + n and the stability subgroup Aq of a point q ∈ Q iscompact.

A sub-Riemannian manifold (Q,D, gQ) is called homogeneous, if a Lie group Aof automorphisms acts on Q transitively. Since any bracket generated homogeneoussub-Riemannian manifold is regular, we get

Corollary 14 i) Let (Q,D, gQ) be a bracket generated homogeneous sub-Riemannianmanifold. Then Q is identified with a coset space Q = A/H of a Lie group of auto-morphisms by a compact subgroup H.Let a = h+p be a reductive decomposition such that p is identified with the tangentspace ToQ. Then Do ⊂ p is AdH-invariant subspace with AdH-invariant metricgDo .iii) Conversely, let Q = A/H be a homogeneous manifold with compact stabilizer Hand a reductive decomposition a = h+p. Then invariant sub-Riemannian structuresbijectively correspond to AdH-invariant subspaces D0 ⊂ p with AdH-invariant Eu-clidean metric. Invariant sub-Riemannian metrics on D bijectively correspond toAdH-invariant metrics on D0. The distribution D is bracket generated if and onlyif the subalgebra, generated by D0, contains m.

4.4.3 Sub-Riemannian M-geodesics as projection of constant vec-tor fields

Let (Q,D, gD) be a regular sub-Riemannian manifold of type (m, gm) and π : P →Q the asslociated Tanaka G0-structure with the normal Cartan connection

κ : T (P ) → m = g0 +m.

M-geodesics of a regular sub-Riemannian manifold (Q,D, gD) are the projec-tion to Q of the orbit of constant vector fields from κ−1(m−1) ⊂ X(P ).

Proposition 15 The M-geodesics of a regular sub-Riemannian manifold (Q,D, gD)are horizontal curves.

17

The result follows from the explicit description of the first prolongation of a Tanakastructures, given in [A-D]. �

It is not easy to construct the normal Cartan connection for the Tanaka G0-structure π : P → Q associated with a regular sub-Riemannian manifold. Butthere is a simple construction of a Cartan connection (see [A-M-S]), associated toa rigging V , which is consistent with the filtration D−i = ΓD−i defined by D .Thismeans that V has the form

V = V2 + V3 + · · ·+ · · ·Vk

where D−i = D−(i−1) + Vi for i = 2, 3, · · · .Such rigging V defines an isomorphism of the graded tangent bundle T grQ with thetangent bundle TQ, which defines a Riemannian metric in Q and induces an iso-morphism of the Tanaka G0-structure π : P → Q with a G0-structure π0 : P0 → Qon Q.The Cartan connection κ on the Tanaka structure is defined in terms of solderingform θ0 and the connection form ω0 of the G0-structure π0, which has trivial firstprolongation since G0 ⊂ SO(m). We still can define the sub-Riemannian geodesicsas the projection to Q of the constant vector fields from κ−1(m). It would be inter-esting to study these geodesics from point of vierw of non-holonomic mechanics.

5 Sub-Riemannian Chaplygin metric on prin-

cipal bundle

5.1 Principal connection and its curvature

Let π : Q → M = Q/G be a G-principal bundle with a right action Rgq = qg of aLie group G.For a ∈ g = Lie(G) ,we denote by a∗ : q 7→ qa := d

dtq exp(ta)|t=0 ∈ TqQ the funda-mental vector field ( the velocity vector field of Rexp ta).

Recall that the principal connection is an G-equivariant g-valued 1-form ϖ :TQ → g , which is an extension of the vertical parallelism, defined by g ∋ a 7→ a∗q .The equivariancy means that

R∗gϖ = Ad−1

g ◦ϖ, g ∈ G.

The connection ϖ is completely determined by the horizontal G-invariant dis-tribution D = ker ϖ.

The horizontal lift XD ∈ ΓD of a vector field X ∈ X(M) is defined by theisomorphisms π∗ : Dq → Tπ(q)TM .

Recall formulas, [K-N],

[a∗, b∗] = [a, b]∗, [a∗, XD] = 0, [XD, Y D] = [X,Y ]D + 2A(XD, Y D). (5)

Here

A : ΓD × ΓD → ΓT vQ, X → AXY = A(X,Y ) =1

2[X,Y ]v, X, Y ∈ ΓD

18

is the O’Neil tensor, which describes the deviation of the distribution from non-holonomicity. We denote by X = Xv +Xh ∈ TQ = T vQ+D the decomposition ofX ∈ TQ into vertical and horizontal parts.

The tensor A is related to the g-valued horizontal curvature 2-form F = dϖ +12 [ϖ,ϖ] of the connection by

−2A(XD, Y D) = F (XD, Y D)∗ := F (XD, Y D)ae∗a.

where ea is a basis of g.We denote by A∗

X : Γ(T vQ) → ΓD the C∞(Q) linear map, dual to the O’Neil mapAX : ΓD → ΓT vQ. Then

gQ(AXDa∗, Y D) = gQ(a∗, A(XD, Y D)) = −1

2gg(a, F (X,Y )).

Denote by λ = gQ ◦ a∗ ∈ ΓD0 the 1-form dual to fundamental vector field a∗. Thenthe horizontal vector field

A∗XDa

∗ = −1

2F ∗Xλ

where F ∗Xq

: g → Dq is the linear map, dual to the linear map FXq : Dq →g, Yq 7→ FXqYq = F (Xq, Yq), Xq, YQ ∈ Dq.

To write formulas in coordinates , we fix a (local) section s : M → Q . It definesa trivialization

M ×G = Q (x, g) = s(x)g

of the principal bundle , where the group G acts on M ×G as

Rg(x, g1) = (x, g1g).

A fundamental vector field a∗ is identified with the left invariant vector fields

aL : (x, g) 7→ (x, ga).

Similarly, aR : (x, g) 7→ (x, ag) stands for the right invariant vector field.

The connection ϖ is an extension of the left invariant Maurer-Cartan form µL

and can be written as

ϖ = µL + η = (eaL + ηai (x)dxi)⊗ ea,

where the potential η is a g-valued 1-form on M and ea is a basis of g.The horizontal lift of a vector field X ∈ X(M) is given by

XD = Xi(∂Di ) = Xi(∂i − ηai (x)e

La ).

The commutator of XD, Y D is given by

[XD, Y D] = [Xi(∂i − ηai (x)eLa , Y

j(∂j − ηbj(x)eLb )]

= [X,Y ]D − (X · η)(Y )a + (Y · η)(Y )a + [η(X), η(Y )]aeLa= [X,Y ]D + 2A(XD, Y D).

The curvature F = F a ⊗ ea = dϖ + 12 [ϖ,ϖ] of ϖ is a g-valued horizontal 2-form.

Its value on X,Y ∈ X(M) is

F (X,Y ) = F (X,Y )aea ={X · (ηai )Y i − Y · (ηai )Xi +

1

2ηai [X,Y ]i

}ea.

19

5.2 Chaplygin metric and its standard extension

The metric gM on the base manifoldM defines a canonical invariant sub-Riemannianmetric gD on D such that the projection π∗ : Dq → Tπ(q)M is an isometry.

The sub-Riemannian metric (D, gD) is called a Chaplygin metric and the sub-Riemannian manifold (Q,D, gD) is called aChaplygin system or a transversallyhomogeneous sub-Riemannian manifold.

Let (Q,D, gD) be a Chaplygin system associated to a principal bundle (π : Q →M,ϖ) as above and gg an Euclidean metric on g . It defines a degenerate metric

gF (X,Y ) = gg(ϖ(X), ϖ(Y ))

with kernel D, whose restriction to a fiber Fx = π−1(x) is a Riemannian metric.We will consider also gD as a degenerate metric on Q with ker gD = T vQ.

ThengQ = gF + gD (6)

is a Riemannian metric in Q which is called the standard extension of the Chap-lygin metric gD.Note that the metric gQ is G-invariant if and only if the degenerate metric gF isinvariant which means that the metric gg is AdG-invariant. In this case the metricgQ is called the bi-invariant extension of the sub-Riemannian matric gD.Denote by ∇M (resp., ∇Q) the Levi-Civita connection of gM (resp., gQ), and by ∇F

the Levi-Civita connection of the induced metric gF on a fiber, which is a totallygeodesic submanifold.The Koszul formula implies the following O’Neil formulas for the covariant derivativeof fundamental vector field b∗ and the horizontal lift XD of a vector field X ∈ X(M),see [Besse].

i) ∇Qa∗b

∗ = ∇Fa∗b

∗,

ii) ∇Qa∗X

D = ∇QXDa

∗ = (∇Qa∗X

D)h = A∗XDa

∗,

iii) ∇QXDY

D = (∇MX Y )D +AXDY D.

The connection ∇F describes in terms of Lie brackets as follows

2g(∇Fa∗b

∗, c∗) = gg([a, b], c)− gg(b, [a, c])− gg(a, [b, c]), a, b, c ∈ g.

5.2.1 S-geodesics of a Chaplygin metric

The O’Neil formulas implies the following relations between S-geodesics of the Chap-lygin sub-Riemannian metric and geodesics of the Riemannian metrics gQ, gM , seealso [Besse].

Theorem 16 i) The principal bundle π : Q → M with a standard metric gQ asso-ciated to a Riemannian metric gM is a Riemannian submersion with totally geodesicfibers.ii)A Riemannian geodesic γ(s) of (Q, gQ) which is horizontal at one point is hori-zontal and it projects onto a geodesic πγ(t) of the base manifold (M, gM ).iii) S-geodesics of the Chaplygin metric are precisely horizontal geodesics of (Q, gQ)and they are horizontal lifts of geodesics of the base manifold.

20

Proof: Recall that S-geodesics are geodesics of the Schouten connection ∇SXY =

prD∇QXY, X, Y ∈ ΓD, that is horizontal curves γ(s) which satisfy the equation

0 = ∇Sγ(s)γ(s) = prD∇

Qγ(s)γ(s) = (∇M

x x)D.

We used the O’Neil formula iii) and skew-symmetry of the O’Neul tensor A. �

5.2.2 H-geodesics of a Chaplygin metric

Now we consider H-geodesics of the Chaplygin metric gD on Q and study theirrelation with geodesics of its standard extension gQ and the metric gM of the basemanifold.

As above, we fix a trivialization Q = M ×G of the principal bundle π, definedby a section s. Then any fiber Fx = x × G with the metric gF is identified withthe Lie group G with the left invariant metric ( still denoted by gF ) defined bythe metric gg. The metric gQ = gD + gF is invariant with respect to vector fieldsaR and the sub-Riemannian metric gD is invariant with respect to aL and aR fora ∈ g.

The decomposition gQ = gD + gF of the metric in TQ = D + T vQ, defines thedecomposition g−1

Q = g−1D +g−1

F of the contravariant metric g−1Q where the cometrics

g−1D =

m∑i=1

Xi ⊗Xi, g−1F =

∑e∗a ⊗ e∗a

are considered as functions on T ∗Q. Here Xi is an orthonormal frame in D and eaand orthonormal basis of g.

The Hamiltonian hQ of the Riemannian metric Q is the sum hQ = hD + hFof the Hamiltonian hD(ξ) := 1

2g−1D (ξ, ξ) of the sub-Riemannian metric and the

Hamiltonian hF (ξ) =12g

−1F (ξ, ξ) of the fiberwise metric gF .

Now we describe the relations between sub-Riemannian H-geodesics and Rie-mannian geodesics of the metric gM on the base M and left invariant metric gF onthe group G. In the case, when the extension gQ in bi-invariant, i.e. the metric gg

is AdG invariant, they had been proved by R. Montgomery [Mont], Theorem 11.2.5(The main theorem).

Lemma 17 Let gQ = gD + gF be a standard extension of a Chaplygin metric gD.Then the Hamiltonians hQ, hF , hD Poisson commute:

{hF , hD} = {hF , hQ} = 0.

and associated Hamiltonian vector fields hF , hD, hF commute.

Proof: A fundamental field a∗ commutes with the horizontal lift XD of basic vectorfield, see (5), Then

La∗gD)(XD, Y D)

= a∗ · (gD(XD, Y D))− gD([a∗, XD], Y D) + gD(XD, [a∗, Y D])= a∗ · gM (X,Y ) = 0

which shows that the field preserves the sub-Riemannian metric gD. It preservesalso the decomposition TQ = D+T vQ and the dual decomposition T ∗Q = D∗+D0.

21

Let Xi be an orthonormal frame in D and ξi the dual coframe in D∗ = (T vQ)0.Then we mar write

La∗Xj = Akj (q)Xk, La∗ξ

i = Bim(q)ξm.

Since a∗ preserves the metric gD =∑

ξi ⊗ ξi, the matrix Bmi is skew-symmetric.

But0 = La∗ < ξi, Xj >=< La∗ξ

i, Xj > + < ξi,La∗Xj >= Bij +Ai

j .

This shows that the matrix Aij is also skew-symmetric and the field a∗ preserves

the contravariant metric g−1Q =

∑Xi ⊗ Xi. In particular, the Poisson bracket

{a∗, g−1} = {a∗,∑

Xi ⊗Xi} = 0. Then the Leibnitz rule shows that the Hamilto-nians hF = 1

2

∑(e∗i )⊗ (e∗i ) and hD = 1

2gD Poisson commute. �

As a corollary, we get

Theorem 18 i)The sub-Riemannian geodesic flow of the sub-Riemannian metricgD is a composition exp thD = expthQ ◦ exp(−thF ) of the Riemannian geodesicflows of the metric gQ and the fiberwise metric gF .ii) Denote by ga(t) ⊂ G the geodesic of the group G with the left invariant metric gF

as above with initial conditions g(0) = e, g(0) = a ∈ g and by γw(t) the geodesic ofthe standard metric gQ with initial conditions γ(0) = q ∈ Q, γ(0) = w = wv +wh ∈TqQ. Then the curve q(t) = γw(t)ga(t) is a sub-Riemannian geodesic if and only ifit has horizontal velocity q(0) = w + a∗q = wh ∈ Dq that is ϖ(w) = −a.

iii) Horizontal geodesic of gQ are sub-Riemannian geodesics and they projects togeodesics of (M, gM ).iv) Sub-Riemannian geodesics are horizontal lifts of the projection of geodesics γw(t)of gQ to M .

Proof: i) is obvious. ii) The restriction gF |Fx of gF to any fiber Fx = (x,G) isidentified with the left invariant Riemannian metric ( denoted again by gF ) on G, defined by the metric gg.The projection to Q of integral curves exp thF are geodesics of the metric gF , hencealso gQ, since the fibers are totally geodesics. The projection to Q of the integralcurves of gQ are geodesics of gQ. The projection of the composed curve

ξ(t) = exp thF ◦ exp thQ ◦ (ξ), ξ ∈ T ∗q Q

are curves of the form q(t) = Rg(t)γ(t) = γ(t)g(t) where γ(t) = τ(exp)th(ξ) is a

geodesic of gQ and g(t) ⊂ G is a geodesic of the metric gF on G. We may assumethat g(0) = e and g(0) = a ∈ g. If γ(0) = w, then the curve q(t) is a sub-Riemannian geodesic if and only if its velocity vector q(0) = w + a∗q is horizontal ,that is ϖ(w + a∗q) = ϖ(w) + a = 0. This proves ii), which implies iii). iv) followsfrom the remark that the transformation Rg(t) deforms the geodesic γ(t) in verticaldirections. Hence q(t) and γ(t) have the same projection to M ( which are geodesicsif and only if γ(t) is a horizontal geodesic.) �

A sub-Riemannian geodesic q(t) = τ(ξ(t)) is determined by the initial covectorξ(t) = ξ(t)D+λ, where λ ∈ D0 = Ann(D)q and the first term determines the velocityvector v ∈ Dq. We call λ ∈ D0 the codistribution part of the sub-Riemanniangeodesic . The sub-Riemannian geodesics , which are horizontal geodesics of gQ

are characterized as geodesics with trivial codistribution part. As a corollary ofTheorem 18 and theorem 16, we get

22

Theorem 19 Let gD be a Chaplygin sub-Riemannian metric in a principal bundle(π : Q → M,ϖ) and gQ is the standard metric. Then sub-Riemannian S-geodesicscoincide with H-geodesics with trivial codistribution part.

5.2.3 Bi-invariant extension of Chaplygin metric and Yang-Millsdynamics

Assume now that gQ is a bi-invariant extension of the Chaplygin sub-Riemannianmetric gD in the total space Q of a principal bundle π : Q → M with a connec-tion ϖ. In this case , the geodesic Hamiltonian system with Hamiltonian hQ hasnice physical interpretation as dynamical system, which describe the evolution of acharged particle in the base manifold M in the presence of the Yang -Mills field ,defined by the principal connection ϖ : TQ → g , see [W],[Mont], [Mont1]. Moreprecisely, using a trivialisation Q = M × G and local coordinates xi in M , we canwrite the connection form as

ϖ = µ+A = (eaL +Aai dx

i)⊗ ea

where ea is an orthonormal basis of g , eLa ( resp. , eaL) corresponding left invariantfield of frames (resp., coframes) on G and A = Aa

i dxi⊗ ea the Yang-Mills potential.

The horizontal lifts of the coordinates vector fields ∂i := ∂xi are given by ∂Di :=

∂i−Aai e

La . Together with the fundamental field eLa they form a frame in Q = M×G,

which is invariant with respect to the right invariant fields eRa . The sub-Riemannianmetric is given by gD(∂D

i , pDj ) = gM (∂i, ∂j) = gij .

The vertical metric is given by gF (eLa , eLb ) = gg(ea, eb) = δab. The metric gQ = gF +

gD isRG-invariant and the fundamental fields eLa and (locally defined) right invariantfields eRa are Killing vector fields. The contravariant metric is g−1

Q = g−1F +g−1

D where

g−1F =

∑eLa ⊗ eLa ,

g−1D = gij(x)∂D

i ⊗ ∂Dj = gij(∂i −Aa

i eLa )(∂j −Aa

i eLa ).

The cotangent bundle locally is identified with T ∗Q = T ∗M × T ∗G, where (xi, pi)are local coordinates in T ∗M with p = pidx

i and (ga, λa) are local coordinates inT ∗G, where ga are local coordinates in G and T ∗

gG ∋ λ = λaeaL. Note that λa as

linear forms on T ∗gG are identified with eLa . The left invariant vector fields

∂i, ∂pi , ∂λa , eaL

form a frame on T ∗Q = T ∗M × T ∗G. The quadratic in momenta HamiltonianshM , hF , hD, hQ = hF + hD can be written as follows

hM = 12g

ij(x)pipjhF = 1

2

∑eLa e

La

hD = 12g

ij(x)(pi −Aai (x)λa)(pj −Ab

j(x)λb)

Using formula for the Poisson structure Λ = ω−1 on T ∗G, one can easily calculatethe Hamiltonian vector fields and the geodesic equation. But we prefer to use theO’Neil formulas.

23

Lemma 20 The angle between the geodesic γ of gQ and a fundamental field a∗, a ∈g is a constant. In particular, the orthogonal projection prT vQγ(t) of the velocityvector field γ to vertical subbundle is the restriction to γ(t) of some fundamentalvector field a∗ and the velocity vector filed can be written as

γ(t) = a∗(γ(t)) + xD(γ(t))

where xD(γ(t)) is the horizontal lift of the velocity vector filed x(t)) of the projectionx(t) of γ(t) to M .

Remark 21 Physically, the angles αk between γ and basic fundamental fields e∗kcharacterise the charges of a particle with respect to components of the Yang-Millsfield and the condition αk = const is called the conservation of charges. In particu-lar, the evolution of neutral particle is described by horizontal geodesics.

Proof: Let γ(t) be a geodesic and x(t) = prMγ(t) its projection to M . Then x(t) =prTM (γ(t) and is the horizontal part of the velocity vector field is γ(t)h = x(t)D.Hence, we can write

γ(t) = x(t)D + ua(t)e∗a(γ(t)).

Thenddtg

Q(eb, γ(t)) = ub(t)= ∇γg

Q(e∗b , γ(t))= gQ(∇γe

∗b , γ(t)) + gQ(e∗b ,∇γ γ(t))

= 0.

since the covariant derivative ∇·e∗b of a Killing vector field e∗b = eLb is a skew-

symmetric operator. �

The following theorem describes the relation between geodesics of the Rieman-nian metric gM and geodesics of its bi-invariant extension gQ.

Theorem 22 A curve γ(t) ⊂ Q with projection x(t) = prMγ(t) and velocity vectorfield g(t) = a∗(γ(t)) + xD(γ(t)) is a geodesic of gQ if and only if it satisfy theequation

∇Qg(t)γ(t) = (∇M

x x)D + 2A∗xDa

∗ = 0. (7)

Proof: Using O’Neil formulas, we calculate the covariant derivative ∇Qγ γ of the

velocity field γ = a∗(γ(t)) + xD(γ(t)) as follows

nQγ γ = (∇Q

a∗a∗)(γ(t)) + 2∇Q

XDa∗ +∇Q

xD xD = (∇M

x x)D(γ(t)) + 2A∗xDa

∗(γ(t)).

We use the fact that the geodesics of the bi-invariant metric on a group G startingfrom e are 1-parameter subgroups, hence ∇Q

a∗a∗ = ∇F

a∗a∗ = 0 and denote by X an

extension of x to a local vector field. Since the left hand side depends only on γ(t),the result does not depend on such extension. �

Recall that 2A∗XDa

∗ = −F ∗Xλ, λ = gQ ◦ a∗ ∈ D0 where F ∗

X : g∗ → ΓD is thelinear map, dual to the map FXq : Dq → g, associated with the curvature 2-form F .

The equation (7) is equivalent to the equation

∇Mx x = g−1

M λbFbi (x) (8)

where the right hand side is the vector field metrically dual to the 1-form λbFbi (x

and λ ∈ g∗ is a constant covector. �

24

The equation (8) describes the motion of a charged particle in the Yang-Millsfield ϖ with the strength tensor F . In the case, when π : Q → M is a circle bundle, the connection ϖ defines the Maxwell field (if gM has Lorentz signature) and theequation reduces to the Lorentz equation for a charge particle in the electro-magneticfield.

6 Homogeneous sub-Riemannian manifolds

In this section , we consider some classes of homogeneous sub-Riemannian manifolds,for which S-geodesics coincides with H-geodesics.

6.1 Chaplygin system on Lie group associated to a ho-mogeneous Riemannian manifold

Let π : G → M = G/H be the principal bundle associated to a homogeneousRiemannian manifold (M = G/H, gM ) . A reductive decomposition g = h + mdefines a principal connection with connection form ϖ = prhµ

L : TG → h , which

is the projection to h of the left invariant Maurer -Cartan form µL . The horizon-tal distribution D := ker ϖ is the left invariant distribution, obtained by the lefttranslations of the subspace m ⊂ g. The metric gm extends to a left invariant met-ric gD in D. So any Riemannian manifold defines a left invariant sub-RiemannianChaplygin metric (D, gD) in G. Since the stability subgroupH is compact, it admitsa bi-invariant metric gH and sub-Riemannian metric gD admits the bi-invariant ex-tension gG = gH + gD. Note that the metric gG is left invariant and invariant withrespect to the right action of H.

Remark 23 Note that the distribution D ⊂ TG associated to m is bracket generatedif and only if the Lie subalgebra g′ generated by m coincides with g. Since thesubgroup G′ ⊂ G associated with the subalgebra g′ acts on M transitively, changingG to G′ we may assume that m generates the Lie algebra g. In this case, thedistribution D is bracket generated.

As a summary, we get

Proposition 24 Let (M = G/H, gM ) be a homogeneous Riemannian manifold. Areductive decomposition g = h+m defines a sub-Riemannian left invariant Chaply-

gin metric (D, gD) on the Lie group G. Moreover, an AdH-invariant metric gh onthe stability subalgebra h defines a bi-invariant extension gG of the sub-Riemannianmetric to a left invariant and H-right invariant metric on G.

6.2 Sub-Riemaniian homogeneous manifold associatedwith a homogeneous Riemannian manifold with non-simplestabilizer

Now we consider a generalisation of the above construction.Assume that the stabilizer H of a Riemannian homogeneous manifold M = G/H

is an almost direct product H = K ·L of two compact normal subgroups. Then π :Q = G/K → M = G/H is an L-principal bundle with the right action of L. Again

25

a reductive decomposition for the manifold M has the form g = h+m = (k+ l)+m.The projection

ϖG = prlµL : TG → l

of the left invariant Maurer-Cartan form defines a left invariant l-valued 1-form .It is right K-invariant, since K acts trivially on l, π-horizontal and L-equivariant (i.e. R∗

gϖ = Ad−1 ◦ϖ ). Hence it is projected to a G-invariant principal connectionform ϖ : TQ → l. The kernel D = ker ϖ is the invariant distribution associatedwith the subspace m ⊂ m+ l = ToQ. The metric gm = gM |T0M defines an invariantsub-Riemannian Chaplugin metric gD in the distribution D. As above, it admits abi-invariant extension to the metric gQ on the manifold Q = G/K. We get

Proposition 25 A homogeneous Riemannian manifold (M = G/H, gM ) with nonsimple stabilizer H = K ·L defines an invariant sub-Riemannian Chaplygin metric(D, gD) on the total space of the principal L- bundle π : Q = G/K → M = G/H,which admits a bi-invariant extension to an invariant metric gQ on Q.

6.2.1 Homogeneous contact sub-Riemannian manifolds

The above construction may be applied to homogeneous Sasaki manifolds For sim-plicity, consider case of regular compact homogeneous Sasaki manifolds. They aredescribed as total spaces π : Q = G/K → M = G/H of a principal circle bun-dle over a flag manifold M = G/H = AdGZ ⊂ g ( that is the adjoint orbit of acompact semisimple Lie group G), see [A-C-H-K] with Kahler structure (gM , ωM )such that the Kahler form ωM is integer. Then there exists a homogeneous cir-cle bundle π : Q = G/K → M = G/H = G/K · T 1 with a principal connectionϖ : TQ → R = LieT1 such that the ωM = dϖ is the curvature of ϖ. The Chaplyginsub-Riemannian metric on D = ker ϖ, defined by the Kahler metric gM admits abi-invariant extension to the invariant metric gQ on Q, which is the invariant Sasakimetric on Q ( under appropriate normalization).From physical point of view, the principal T 1-bundle π : Q → M with Sasakimetric corresponds to Kaluza-Klein description of electro-magnetic field ( abelianYang-Mills field ) and projections to M of geodesics of Sasaki metric describe theevolution of charges in electro-magnetic field and satisfy the Lorentz equation.

6.3 Symmetric sub-Riemannian manifolds

Strichartz [Str] defined the notion of sub-Riemannian symmetric space as a ho-mogeneous sub-Riemannian manifold (Q = G/H,D, gD) such that H contains aninvolutive element σ (called sub-Riemannian symmetry) which acts on the subspaceDo of the point o = eH ∈ Q as −id.He considered some example of sub-Riemannian symmetric spaces and classified3-dimensional sub-Riemannian symmetric spaces. He stated the problem of exten-sion of this classification to higher dimensions.E. Falbel and C. Gorodski classified symmetric sub-Riemannian manifolds of con-tact type (1995). W. Respondek and A. J. Maciejewski describes all integrablesub-Riemannian metrics on 3-dimensional Lie groups with integrable H-geodesicflow (2008). They are exhausted by sub-Riemannian symmetric spaces.

Below we give a description of bracket generated symmetric sub-Riemannianmanifolds in terms of affine symmetric spaces.

26

6.3.1 Sub-Riemannian symmetric spaces associated with an affinesymmetric space M = G/H

Let σ be an involutive automorphism of a Lie algebra with eigenspace (symmetric)decomposition g = g+ + g−, σ|g±=±id . It is integrated to an involutive decompo-sition σ of the simply connected Lie group G associated to g. Let H be a closedsubgroup such that Gσ

con ⊂ H ⊂ G0, where Gσ is the fixed point subgroup of Gand Gσ

con its connected component. Then M = G/H is an affine symmetric space(associated with the above symmetric decomposition) and any symmetric space isobtained by this construction. The invariant (torsion free) linear connection ∇ in Mis given by ∇XY ∗|o = −1

2 [X,Y ]o, where X,Y ∈ g−, Y∗ the vector field generated

by Y and o = eH ∈ M . The involution is called the central symmetry at the pointeHSince the subalgebra g1 := [g−, g−] + g− generates a transitive subgroup G1 (calledthe group of transvections) such that M = G1/H1 we may assume that G = G1 isthe group of transvections of the symmetric space.Let S = G/H be an affine symmetric space with a symmetric decompositiong = g+ + g−. Chose a compact subgroup K ⊂ H. Then the homogeneous manifoldQ := G/K is the total space of a homogeneous fibrationQ = G/K = G×H(H/K) →S = G/H is a homogeneous fibration with the typical fiber F = H/K over the sym-metric space G/H , defined by the right action of K on H.A reductive decomposition h = k +m+ of the fiber F = H/K defines a reductivedecomposition g = k+(m++g−) of G/K. If G is the group of transvections of S, thesubspace Do := g− generates an invariant bracket generated distribution D ⊂ TQ.An AdK-invariant Euclidean metric g in D0 defines an invariant sub-Riemannianstructure (D, gD) in Q. Moreover , the involution σ induces an involutive auto-morphism σ of (Q = G/K,D, gD) , which is a sub-Riemannian symmetry. Thesub-Riemannian symmetric manifold (Q = G/K,D, gD) is called sub-Riemanniansymmetric space associated to an affine symmetric space M = G/H and a subgroupK ⊂ H.Conversely, let (Q = G/K,D, g) be a sub-Riemannian symmetric space. The sub-Riemannian symmetry σ , which preserves the point o = eK defines a symmetricdecomposition g = g++g− of the Lie algebra of G such that the stability subalgebrak ⊂ g+ and the horizontal subspace at the point o is an AdK-invariant subspaceD0 ⊂ g−. Denote by S = G/G+ the affine symmetrkic space associated with thesymmetric decomposition. Then Q = G/K → S = S = G/G+ is a homogeneousfibration over the affine symmetric space.

We get

Theorem 26 Let (S = G/H, σ) be an affine symmetric space associated to a sym-metric decomposition g = g+ + g− where G is the group of transvections, i.e. g−

generates g. Let K be a compact subgroup of H and g a K-invariant Euclideanmetric in g−. Then the homogeneous manifold Q = G/K admits a structure ofbracket generated sub-Riemannian symmetric space (D, gD, σ), where D is the in-variant distribution defined by D0 = g− with invariant metric , defined by g. Thesub-Riemannian symmetry σ is the automorphism of (Q,D, gD), defined by theinvolutive automorphism σ of G. This symmetry is consistent with the fibrationQ = G/K → M = G/H and induces the central symmetry of M .Any sub-Riemannian symmetric space can be obtained by this construction.

27

6.3.2 Compact sub-Riemannian symmetric space associated to agraded complex semisimple Lie algebra

We describe the structure of bracket generated symmetric sub-Riemannian manifoldon a flag manifold.Let

g =

d∑i=−d

gi

be a graded complex semisimple Lie algebra of depth d ≥ 2 and

g = gev + godd

associated symmetric decomposition.Denote by τ the anti-linear involution which defines the compact real form s.t.gτ = gτ0 +

∑i>0(g−i + gi)

τ We denote by G a complex Lie group with LieG = gand by F = G/P the complex (compact) flag manifold , where P is the parabolicsubgroup generated by p =

∑i≥0 gi. We set h = gτ0 and denote by H the associ-

ated subgroup of the compact Lie group Gτ , which acts transitively on F . Hence,F = Gτ/H.

We choose an adh-invariant metric gm on the space m = (g−1 + g1)τ . The pair

(m, gm) defines an invariant bracket generated sub-Riemannian structure (D, gD)on the flag manifold F = Gτ/H.

Theorem 27 The flag manifold F = Gτ/H with invariant sub-Riemannian struc-ture (D, gD) defined by (m, gm) is a sub-Riemannian symmetric space.

Example Let

g = g−2 + g−1 + g9 + g0 + g1 + g2, dim g±2 = 1

be the contact gradation of a complex simple Lie algebra g , i.e. the eigenspacedecomposition of adHµ where Hµ is the coroot associated to the maximal root µ ofg. Then the symmetric space Gτ/Gτ

ev is the quaternionic Kahler symmetric space( the Wolf space ) and the flag manifold F = Gτ/H is the associated twistor space.The distribution D is the holomorphic contact distribution and gD is unique (up toscaling ) invariant sub-Riemannian metric on D. It is the restriction of the invariantKahler-Einstein metric on F ( for g = sln(C). )

References

[A-B-B] A.A. Agrachev, D. Barilari, U. Boscain, Introduction to Riemannian andsub-Riemannian geometry,(2012).

[A-C-H-K] D.V. Alekseevsky,V.Cortes, K.Hasegawa, Y.Kamishbima, Homogeneouslocally conformally Kahler and Sasaki manifolds, Int. J. Math., v.26, n 5,(2015), 29p.

[A-D] D.V. Alekseevsky, L. David, Tanaka structures (non holonomic G-structures)and Cartan connections, J. Geom. Phys., 91 (2015), 88-100.

[A-D1] D.V. Alekseevsky, L. David, Prolongation of Tanaka structures: an alterna-tive approach, Annali di Mat. Pura ed Appl., 196, n 3, (2017), 1137 -1164.

28

[A-M-S] D.Alekseevsky, A.Medvedev, J.Slovak, Constant curvature models in sub-Riemannian geometry, Journal of Geometry and Physics 138 (2019) , 241256.

[A-S] D.Alekseevsky, A, Spiro, Prolongation of Tanaka structures and regularCR structures , ”Selected topics in Cauchy-Riemannian geometry”, ed. S.Dragomir, v.1, (2001) ,1-38.

[B] V.N. Berestovsky, On the curvature of homogeneous sub-Riemannian mani-folds, European Journal of Mathematics, (2017) 3 , 788807.

[Besse] A.Besse, Einstein Manifolds,

[C] A. Cap, On canonical Cartan connections associated to filtered G-structuresAndreas Cap , arXiv.org:1707.05627.

[C-S] A. Cap, J. Slovak, Parabolic Geometry I, in: Mathematical Survey and Mono-graphs, vol. 154, 2009.

[C-S-Z] A. Cap, J. Slovak, V. Zadnik, On Distinguished Curves in Parabolic Ge-ometries, Transf. Groups, 9,n2, (2004), 143-166.

[C] J.Cortes Monforte, Geometric, Control and Numerical Aspects of Nonholo-nomic Systems, Lect. Notes in Math, 1713 (2002).

[D-Z] B. Doubrov, I. Zelenko, Geometry of curves in generalized flag varieties,Transf. Groups, 18, n2, (2013), 361-383.

[D-G] V. Dragovic and B. Gajic. The Wagner curvature tensor in nonholonomicmechanics, Reg. Chaot. Dyn., 8 (1), (2003), 105-124.

[J] V. Jurdjevic, Optimal Control and Geometry: Integrable Systems, CambridgeUniversity Press, 2016.

[H] M. Herzlich, Parabolic geodesics as parallel curves in parabolic geometries,Intern.J. M., 24 (9) (2013) ,135-167.

[K-N] Sh. Kobayashik, K. Nomuzu , Foundation of Differential Geometry, v.1,2.1963.

[Mont] R. Montgomery, A Tour of Sub-Riemannian Geometry, Their Geodesics andApplications, Mathematical Surveys and Monographs, v.9.

[Mont1] R. Montgomery, A survey of singular curves in sub-Riemannian geometry,J. Dynamical and Control Systems, n1, (1995),1-35.

[M] T. Morimoto , Cartan connection associated with a subriemannian structure,Differential Geometry and its Applications 26 (2008) 75-78.

[M1] T. Morimoto, Geometric structures on filtered manifolds, Hokkaido Math.J.22, n3, (1993), 263347.

[Sh] J. A. Schouten, On nonholonomic connections, Koninklijke akademie vanwetenshappen te Amsterdam, Proceeding of sciences, 31, No. 3,(1928), 291-298.

[Stern] Sh. Sternberg, Lectures in Differential Geometry, Prentice Hall. 1964.

[Str] R.S.Strichartz, Sub-Riemasnnian Geometry, J.Diff.Geom.,24 (1986) 221-263.

[Sy] J. L. Synge, Geodesics in nonholonomic geometry, Math. Ann., (1928) p. 738-751.

[Sy1] J. L. Synge,Classical Dynamics, Springer, 10960.

29

[T] N.Tanaka, On differential systems, graded Lie algebras and pseudogroups. J.Math. Kyoto. Univ. 10,(1970), 182.

[V] Vershik, A. M.: Classical and nonclassical dynamics with constraints, in ”Newin Global Analysis, Voronezh. Gos. Univ., 1984, pp. 2348.

[V-F] A.M. Vershik, L.D. Faddeev, Differential Geometry and Lagrangian Mechan-ics with Constrains, Sov. Phys. Dokl., v.17, n1, (1972), 34-36.

[V-F1] A.M. Vershik, L.D. Faddeev, Lagrangian Mechanics in Invariant Form, Sci.Math. Sov.,v.1, n4, (1981), 339-350.

[V-G] A. M. Vershik, V. Ya. Gershkovich, Nonholonomic dynamical systems. Ge-ometry of distributions and variational problems,Itogi Nauki i Tekhniki.Ser.Sovrem. Probl. Mat. Fund. Napr., 1987, Volume 16, 585.

[V-G1] A. M. Vershik, V. Ya. Gershkovich, Nonholonomic problems and the theoryof distributions, Acta Appl. Math. 12, No.2,(1988), 181-209 .

[V-Gr] A.M. Vershik, O.A.Granichina , Reduction of nonholonomic variationalproblems to isoperimetric ones and connections in principal bundles, Math.Notes 49, No.5, (1991), 467-472.

[Vr] G. Vranceanu, Parallelisme et courlure dans une variete non holonome. Attidel congresso Inter. del Mat. di Bologna, (1928)

[W] Wong S.K.,Field and particle equations for the classical Yang-Mills field andparticle with isoptopic spin, Nuovo Cimento,65a, (1970) 689-693.

[Z] I.Zelenko, Tanakas prolongation procedure for filtered structures of con-stant type, symmetry, integrability and geometry: methods and applications.SIGMA 5(94), 21 ,(2009).

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