Esmaeil PEYGHAN Aydin GEZER Inci GULTEKIN

Abstract Considering the prolongation of a Lie algebroid,the authors introduce Finsler algebroids and present important geometric objects on these spaces.Important endomorphisms like conservative and Barthel,Cartan tensor and some distinguished connections like Berwald,Cartan,Chern-Rund and Hashiguchi are introduced and studied.

Keywords Chern-Rund connection,Distinguished connections,Finsler algebroid,Hashiguchi connection,Lie algebroid

1 Introduction

The notion of Lie algebroids which was introduced by Pradines [12]is a vector bundle such that its sections involve a real Lie algebra.Each section is anchored on a vector field,by means of a linear bundle map named as anchor map,which is further supposed to induce a Lie algebra homomorphism.The Lie algebroid is a good extension of tangent bundle since the homomorphism property of the anchor map grants the basic notions of tangent bundle to the vector bundle.Recently,Lie algebroids are important issues in physics and mechanics since the extension of Lagrangian and Hamiltonian systems to their entity (see [2,5–7,9–10,18,20])and catching the poisson structure (see [11]).

The aim of this paper is to study some concepts of Finsler geometry on Lie algebroid structures.Of course,there are some discussions on Finsler geometry in[9,19].Finsler geometry is a generalization of Riemannian geometry such that interfering of direction and position duplicates the degree of freedom in view of configuration.Variety of tensors in Finsler geometry is more than Riemannian case.A very good reference about Finsler geometry is [1].

The paper is organized as follows.In Section 2,we recall differential,contraction and Lie differential operators,generalized Frlicher-Nijenhuis bracket and vertical and complete lifts on Lie algebroids and we study the relation between these concepts.Also,we present the notion of the prolongation of a Lie algebroid and we recall some concepts on it such as horizontal and vertical endomorphisms,Liouville section,semispray,torsion,tension and almost complex structure.Finally,distinguished connections on the prolongation of a Lie algebroid are introduced and torsion and curvature tensor fields of these connections are considered.In Section 3,we introduce the concept of Finsler algebroid and we study important geometric subjects on this space.Important endomorphisms like Conservative and Barthel,Cartan tensor and some distinguished connections like Berwald,Cartan,Chern-Rund and Hashiguchi are studied by Szilasi and his collaborators from a special point view based on pullback bundle(see[13–17]).In this section we construct them on Finsler algebroids and obtain some results on these concepts.

2 Basic Concepts on Lie Algebroids

LetEbe a vector bundle of ranknover a manifoldMof dimensionmandπ:E→Mbe the vector bundle projection.Denote by Γ(E)theC∞(M)-module of sections ofπ:E→M.A Lie algebroid overMis the triple(E,[.,.]E,ρ)where[·,·]Eis a Lie bracket on Γ(E)andρ:E→T Mis a bundle map,called the anchor map,such that if we also denote byρ:Γ(E) →χ(M) the homomorphism ofC∞(M)-modules induced by the anchor map,then

Moreover,we have the relations

and

Trivial examples of Lie algebroids are real Lie algebras of finite dimension,the tangent bundleT Mof an arbitrary manifoldMand an integrable distribution ofT M.

On Lie algebroid (E,[·,·]E,ρ) we define the differential ofE,dE:Γ(∧kE∗) →Γ(∧k+1E∗),as follows

forµ∈Γ(∧kE∗) andX0,···,Xk∈Γ(E).In particular,iff∈Γ(∧0E∗) =C∞(M) we havedEf(X)=ρ(X)f.Using the above equation it follows that (dE)2=0.

If we take the local coordinates (xi) onMand a local basis {eα} of sections ofE,then we have the corresponding local coordinates(xi,yα)onE,where xi=xi◦πand yα(u)is theα-th coordinate ofu∈Ein the given basis.Such coordinates determine local functionsonMwhich contain the local information of the Lie algebroid structure,and accordingly they are called the structure functions of the Lie algebroid.They are given by

with conditions A sectionωofE∗also defines a functiononEby means of

Ifω=ωαeα,then the linear functionis(x,y)=ωαyα.

ForX∈Γ(∧kE),the contractioniX:Γ(∧pE∗) →Γ(∧p−kE∗) is defined in standard way and the Lie differential operator £EX:Γ(∧pE∗)→Γ(∧p−k+1E∗) is defined by

Note that ifE=T MandX∈Γ(E) =χ(M),thendTMandare the usual differential and the usual Lie derivative with respect toX,respectively.Also,forK∈Γ(∧kE∗⊗E),the contraction

is defined in the natural way.In particular,for simple tensorK=µ⊗X,whereµ∈Γ(∧kE∗),X∈Γ(E),we setiKν=µ∧iXν.The corresponding Lie differential is defined by the formula

and,in particular

The contractioniKcan be extended to an operator

by the formulaiK(µ⊗X)=iK(µ)⊗X.The generalized Frlicher-Nijenhuis bracket is defined for simple tensorsµ⊗X∈Γ(∧kE∗⊗E) andν⊗Y∈Γ(∧lE∗⊗E) by the formula

Moreover,forK∈Γ(∧kE∗⊗E),L∈Γ(∧lE∗⊗E),N∈Γ(∧nE∗⊗E) andX,Y∈Γ(E) we have (see [3–4])

For a functionfonM,one defines its vertical liftf∨onEbyf∨(u)=f(π(u)) foru∈E.Now,letXbe a section ofE.Then,we can consider the vertical lift ofXas the vector field onEgiven byX∨(u) =X(π(u))∨u,u∈E,where∨u:Eπ(u)→Tu(Eπ(u)) is the canonical isomorphism between the vector spacesEπ(u)andTu(Eπ(u)).Let {eα} be a basis of sections ofE.The vertical liftX∨ofX=Xαeα∈Γ(E)has the locally expressionX∨=(Xα◦π)The complete lift of a smooth functionf∈C∞(M) intoC∞(E) is the smooth functionfc:E→R defined byfc(u)=dEf(u)=ρ(u)f.In the local basis,we have

LetXbe a section onE.Then there exists a unique vector fieldXconE,the complete lift ofX,satisfying the following conditions:

(i)Xcisπ-projectable onρ(X),

(ii)Xc(^α)=whereα∈Γ(E∗).It is known thatXchas the following coordinate expression (see [8]):

Also we haveXcfc=(ρ(X)f)cfor allf∈C∞(M).

2.1 The prolongation of a Lie algebroid

Let £πEbe the subset ofE×T Edefined by £πE={(u,z)∈E×T E|ρ(u)=π∗(z)} and denote byπ£:£πE→Ethe mapping given byπ£(u,z)=πE(z),whereπE:T E→Eis the natural projection.Then,(£πE,π£,E)is a vector bundle overEof rank 2n.Indeed,the total space of the prolongation is the total space of the pull-back ofπ∗:T E→T Mby the anchor mapρ.

We introduce the vertical subbundle

whereτ£:£πE→Eis the projection onto the first factor,i.e.,τ£(u,z) =u.Therefore an element ofv£πEis of the form (0,z)∈E×T Esuch thatπ∗(z)=0 which is called vertical.

For local basis{eα}of sections ofEand coordinates(xi,yα)onE,we have local coordinates(xi,yα,kα,zα)on£πEgiven as follows.If(u,z)is an element of£πE,then usingρ(u)=π∗(z),zhas the form

The local basis {Xα,Vα} of sections of £πEassociated to the coordinate system (xi,yα) is given by [6],

The vertical liftXVand the complete liftXCof a sectionX∈Γ(E)as the sections of£πE→Eare given by

with locally coordinate expressions

whereX=Xαeα∈Γ(E).

Here,we consider the anchor mapρ£:£πE→T Edefined byρ£(u,z)=zand the bracket[·,·]£satisfying the relations

forX,Y∈Γ(E).Then,this vector bundle (£πE,π£,E) is a Lie algebroid with structure([·,·]£,ρ£).The Lie brackets of basis {Xα,Vα} are

2.2 A setting for semispray on £πE

A section ofπalong smooth mapf:N→Mis a smooth mapσ:N→Esuch thatπ◦σ=f.The set of sections ofπalongfwill be denoted by Γf(π).Then,there is a canonical isomorphism between Γ(f∗π) and Γf(π) (see [15]).Now we consider pullback bundleπ∗π=(π∗E,pr1,E) of vector bundle (E,π,M),where

andpr1is the projection map onto the first component.The fibres ofπ∗πare then-dimensional real vector spaces {u}×Eπ(u)～=Eπ(u).

We consider the following sequence:

withj(u,z) = (πE(z),Id(u)) = (v,u),z∈TvEandi(u,v) = (0,v∨u),wherev∨u:C∞(E) →R is defined by

F(u+tv).Indeed,we haveThe functionJ=i◦j:£πE→£πEis called the vertical endomorphism(almost tangent structure)of£πE.From the definitions ofi,jandJ,we get

If {Xα,Vα} is the corresponding dual basis of {Xα,Vα},we getJ=Vα⊗Xα.

Letδbe the canonical section alongπgiven byδ(u) = (u,u) ∈π∗Efor eachu∈E.The sectionCgiven byC:=i◦δis called Liouville or Euler section.The Liouville sectionChas the coordinate expression

with respect to {Xα,Vα}.LetXbe a section ofE.Then,we have

Also,the real valued smooth functiononEis homogeneous of degreerif and only if

A sectionSof the vector bundle (£πE,π£,E) is said to be a semispray if it satisfies the conditionJ(S) =C.Moreover,ifSis homogeneous of degree 2,i.e.,[C,S]£=S,we call it spray.A semisprayShas the coordinate expressionS=yαXα+SαVαandSis a spray if and only if

A functionh:£πE→£πEis called a horizontal endomorphism ifh◦h=h,kerh=v£πEandhis smooth on◦£πE= £πE- {0}.Also,v:= Id -his called the vertical projector associated toh.Settingh£πE:=Imhwe have the following splitting for £πE:

Also,from the definition of the horizontal endomorphism we have

It is known thathhas the following locally expression:

Definition 2.1Fork∈N,K∈Γ(∧kE∗⊗E) is called semibasic if

Lethbe a horizontal endomorphism on £πE.Then,are called the tension,weak torsion and strong torsion ofh,respectively,where[·,·]F−N£is the generalized Frlicher-Nijenhuis bracket on£πE.IfH=0,his called homogeneous.Here,H,tandThave the following coordinate expressions:

where

Theorem 2.1(see [8]) Ifh1andh2are horizontal endomorphisms with same associated semisprays and strong torsions,thenh1=h2.

The curvature of a horizontal endomorphismhis defined by Ω = -Nh,whereNhis the Nijenhuis tensor ofhgiven by

The curvature Ω has the following coordinate expression:

where

Let the horizontal endomorphismhbe given on£πE.IfSis an arbitrary semispray of£πE,=hSis also a semispray of £πEwhich does not depend on the choice ofS.Sis called the semispray associated toh.If the horizontal endomorphismhis homogeneous,the semispray associated tohis spray.

LetSbe a semispray on £πE.We consider the maphS:£πE→£πEgiven byhS:=It is known thathSis a horizontal endomorphism on£πEwhich is called horizontal endomorphism generated by semisprayS(see [8]for more details).We have the following theorem.

Theorem 2.2(see [8]) Lethbe a homogeneous horizontal endomorphism on £πEandSbe the semispray associated toh.Then,we have

wheretis the weak torsion ofhandhSis the horizontal endomorphism generated byS.

LetSbe the semispray associated toh.We consider the mapF:£πE→£πEgiven byThen,Fis an almost complex structure on £πE(F2= -Id) which is called the almost complex structure induced byh.Fhas the following coordinate expression:

The following relations hold in [8]:

The map H:=F◦i:E×ME→£πEis called the horizontal map for £πEassociated toh.Also,the map V:=j◦F:£πE→E×MEis called the vertical map for £πEassociated toh.

Lethbe a horizontal endomorphism on £πE.We consider the map

and we call it horizontal lift byh.IfX=Xαeα,we have

The following equations hold in [8]:

Settingδα=ehα= Xα+BβαVβ=h(Xα),it is easy to see that {δα} generates a basis ofh£πEand the frame {δα,Vα} is a local basis of £πEadapted to splitting (2.5) which is called the adapted basis.The dual adapted basis is {Xα,δVα},where

Lie brackets of the adapted basis {δα,Vα} are

2.3 Distinguished connections on Lie algebroids

A linear connection on a Lie algebroid (E,[,]E,ρ) is a map

which satisfies the rules

for any functionf∈C∞(M) andX,Y,Z∈Γ(E).LetDbe a linear connection on £πEandhbe a horizontal endomorphism on £πE.Then,(D,h) is called a distinguished connection (or d-connection) on £πE,if

(i)Dis reducible,i.e.,Dh=0,

(ii)Dis almost complex,i.e.,DF=0,

whereFis the almost complex structure associated byh.It is known that a d-connection has the following coordinate expression:

Let (D,h) be a d-connection.Then

are calledh-covariant derivative andv-covariant derivative,respectively.Moreover,

are calledh-deflection andv-deflection of(D,h),respectively,whereIt is easy to see thath∗(DC) has the following coordinate expression:

Similarly,we can see thatv∗(DC) has the following coordinate expression:

whereδγαis the Kronecker symbol.

Theorem 2.3Let (D,h) be a d-connection on £πE.Then,the torsion tensor fieldTofDis determined by the following,completely:

whereA,B,R1,P1andR1are calledh-horizontal,h-mixed,v-horizontal,v-mixed andvvertical torsion,respectively.

It is easy to check that the components of the torsion tensor field have the following coordinate expressions:

where

Theorem 2.4Let (D,h) be a d-connection on £πE.The curvature tensor fieldKofDis determined by the following,completely:

Here,R,PandQare called horizontal,mixed and vertical curvature,respectively.

By a direct calculation,we can see that the horizontal,mixed and vertical curvature,have the following coordinate expressions:

where

3 Finsler Algebroids

In this section,we introduce Finsler algebroids as a generalization of Finsler manifolds and we present some basic objects such as conservative endomorphism (in particular,Barthel endomorphism) and Cartan tensor on these spaces.

Definition 3.1Finsler algebroid(E,F)is a Lie algebroid£πEprovided with a fundamental Finsler function F:E→R satisfying the conditions:

(i) F is a scalar differentiable function on the manifold=E-{0} and continuous on the null section ofπ:E→M,

(ii) F is a positive function and homogeneous of degree 2,i.e.,££CF =2F,

(iii) the fundamental formω=d£d£JF is nondegenerate,where

For the basis {Xα,Vα} of Γ(£πE) and the dual basis {Xα,Vα} of it,we getd£JF(Vα) = 0 andTherefore,has the following coordinate expression:

Lemma 3.1The fundamental formωof a Finsler algebroid has the following coordinate expression:

ProofUsing (3.1),we have

Thus,we have

Also,it is easy to check thatHence,we have

Setting the above two equations in (3.3) impies (3.2).

From (3.2),we deduce that the fundamental formωis nondegenarate if and only if the symmetric matrixis regular.

Proposition 3.1For the fundamental formωwe have the following identities:

ProofWe have

It is easy to check thatiVγXα= 0 andiVγVα=δαγ.Therefore,from (3.2),we getiVγω=and consequently

Since F is homogeneous of degree 2,we can obtain

Using the above equation in (3.4),we get

Similarly,we can obtain

Hence,we have (ii).It is easy to check thatiCXγ= 0 andiCVγ= yγ.Using (3.2) and (3.5),we get

Definition 3.2Let (E,F) be a Finsler algebroid with fundamental formω.The map

It is easy to check that G is bilinear,symmetric and nondegenerate onSo,we can deduce the following.

Proposition 3.2Lethbe a horizontal endomorphism and G be the vertical metric of Finsler manifold (E,F).The functiongiven by

is a pseudo-Riemannian metric on

The pseudo-Riemannian metricintroduced in the above proposition,is called the prolongation of G alongh.

Using (3.2),we obtain the following coordinate expression:

Also,using (3.6) we can obtain

and consequently

Proposition 3.3For the metrics G,and sectionsX,Yofwe have

ProofUsing (3.7),we get

Since F is homogeneous of degree 2,we can obtainThus,we deduce G(C,C)=2F.Using (iii) of (2.3) and (3.6),we get

Now,letX=XαeαandY=Yβeβbe sections ofThen,we have

Using (3.6) and the above equation,we can obtain

Lethbe a horizontal endomorphism on £πEandbe a pseudo-Riemannian metric given by (3.6).We consider

and we call it the K¨ahler form with respect to

Proposition 3.4We have Kh=ivω.

ProofLet∈Γ(£πE).Then,we have

Using (3.8),the K¨ahler form Khhas the following coordinate expression with respect to{δα,Vα}:

Definition 3.3Let (E,F) be a Finsler algebroid with fundamental formω.Ifφ:E→R is a smooth function,then the section gradφ∈Γ(£πE) characterized by

is called the gradient ofφ.

In the above definition,the nondegeneracy ofωguarantees the existence and unicity of the gradient section.

Ifβis a nonzero 1-form on £πE,we denote byβ♯the section corresponding toω,i.e.,iβ♯ω=β.Also,we can introduce the gradient ofφby gradφ=(d£φ)♯.Since gradφ∈Γ(£πE),we can write it as follows

Using (3.2) and (3.11),we get

which yields

where(Gαβ)is the inverse matric of(Gαβ).Similarly,using(3.2),(3.11)and the above equation we obtain

which gives us

Plugging (3.13) and (3.14) into (3.12) implies the following local expression for gradient:

Proposition 3.5Let (E,F) be a Finsler algebroid andf∈C∞(M).We have

(i) gradf∨∈Γ(v£πE),(ii) [C,gradf∨]£=-gradf∨,(iii)ρ£(gradf∨)(F)=fc.

ProofSincef∨=f◦πis a function with respect to (xi),we haveFrom (3.15),we deduce that gradf∨has the following coordinate expression:

Thus we have (i).The above equation and (2.2) give us

But using (3.5),we can deduceand consequentlySetting this equation in the above equation implies

Therefore,we have (ii).To prove (iii),we use (3.5) and (3.16) as follows

3.1 Conservative endomorphism on Finsler algebroids

Definition 3.4A horizontal endomorphismhon Finsler algebroid (E,F) is called conservative ifd£hF =0.

Using (2.7),it is easy to check thathis conservative if and only if

Proposition 3.6Lethbe a conservative horizontal endomorphism on Finsler algebroid(E,F).We haved£HF =0,whereHis the tension ofh.

ProofUsing (2.8),we can obtaind£HF(Vα)=0 and

Sincehis conservative,differentiating (3.17) with respect to yγwe obtain

Contracting the above equation by yγand using homogeneity of F we get

Setting the above equation in (3.18) and using (3.17) we deduced£HF(Xα) = 0.Therefored£HF =0.

Lemma 3.2Ifωis the fundamental two-form of Finsler algebroid (E,F) andhis a conservative horizontal endomorphism on £πE,then

ProofSincehis conservative,we have (3.19).Then,using (3.2) and (3.19) we get

Also,(2.9) and (2.11) give us

Two above equations yield

Similarly we get

and

Corollary 3.1Ifωis the fundamental two-form of Finsler algebroid (E,F) andhis a torsion free conservative horizontal endomorphism on £πE,then

On any Finsler algebroid there is a sprayS◦:E→£πE,which is uniquely determined onEby the formula

This spray is called the canonical spray of the Finsler algebroid.Using (3.8) and the above equation,the canonical sprayS◦has the coordinate expressionS◦=yαXα+Sα◦Vα,where

and (Gαβ) is the inverse matrix of (Gαβ).

Proposition 3.7LetS◦be the canonical spray andhbe a conservative horizontal endomorphism on Finsler algebroid (E,F) with the associated semisprayS.We have

ProofLeth= (Xα+BβαVβ)⊗Xα,S= yαXα+SαVαandS◦= yαXα+Sα◦Vα,whereSα◦are given by (3.22).Since (iVαω)(Xβ) =and (iVαω)(Vβ) = 0,we haveiVαω=GαβXβ.Therefore,using (3.22) we get

FromS=hS◦,we deduceSα=yγBαγ.Setting this in the above equation gives us

Sincehis conservative,we have (3.17) and (3.19)–(3.20).Using these equations in the above equation and using (2.11) we get

3.2 Barthel endomorphism on Finsler algebroids

LetS◦be the canonical spray on Finsler algebroid (E,F).We consider

In the coordinate expression,we can obtain

From the above equation,we deduceh2◦=h◦and kerh◦=v£πE.Thus,h◦is a horizontal endomorphism on £πEwhich is called Barthel endomorphism.SinceS0is a spray on (E,F),we can deduce that the Barthel endomorphism is homogeneous.

Proposition 3.8Lethbe a conservative and homogeneous horizontal endomorphism andh◦be the Barthel endomorphism on Finsle algebroid (E,F).We have

ProofLetSbe the semispray associated tohandh′be the horizontal endomorphism generated byS.Using Theorem 2.2 we get

Theorem 3.1Barthel endomorphism of Finsler algebroid (E,F) is conservative.

ProofUsing (3.17),it is sufficient to show that

From (i) of (3.25) we derive that

Using (3.22) we obtain

But (ii) of (3.25) implies

Setting the above equation in (3.26),we deducefrom which we have(3.24).

Theorem 3.2Leth1andh2be conservative horizontal endomorphisms on Finsler algebroid(E,F).Ifh1andh2have common strong torsions,thenh1=h2.

ProofWe denote byS1andS2the associated semisprays ofh1andh2,respectively and letT1andT2be the strong torsions ofh1andh2,respectively.From hypothesis we haved£h1F =d£h2F = 0 andT1=T2.Also,from the last equation in the proof of Proposition 3.7,we deduceiS1−S0ω=d£iS1t1F,iS2−S0ω=d£iS2t2F and consequently

wheret1andt2are weak torsions ofh1andh2,respectively.From the definition of strong torsion we have

becaused£H1F =0,whereH1is the tension ofh1.Similarly we obtain=d£T2F.Setting this equation together with the above equation in(3.28),we deduceiS1−S2ω=d£T1F-d£T2F =0.Sinceωis nondegenerate,this equation gives usS1=S2and consequently using Theorem 2.1,we deduceh1=h2.

From the above results,we understand that Barthel endomorphism is homogeneous,conservative and torsion free.Moreover,since Barthel endomorphism is homogeneous and torsion free,we deduce that its strong torsion is zero.Also,from the above theorem we derive that ifhis a homogeneous,conservative and torsion free horizontal endomorphism,then it is coincide with Barthel endomorphism.Hence,we have the following theorem.

Theorem 3.3There exists a unique horizontal endomorphism on Finsler algebroid (E,F)such that it is homogeneous,conservative and torsion free.

3.3 Cartan tensor on Finsler algebroids

Here,we consider the tensor

on Finsler algebroid (E,F) which satisfies

and

Therefore,the first Cartan tensor has the following coordinate expression:

where

From (3.33) and the above equation,we can deduce the following proposition.

Proposition 3.9The first Cartan tensor is semibasic.Moreover,it and its the lowered tensor are symmetric tensors.

Using(3.32)–(3.33),we can obtain the following coordinate expression for the lowered tensor:

where

Proposition 3.10IfSis a semispray on £πE,we haveiSC =iSC♭=0.

ProofLetbe sections ofUsing (3.25),we have

Similarly,we can proveiSC =0.

Now we consider a horizontal endomorphismhon£πEand the prolongationof the vertical metric G alongh.The second Cartan tensor

(belonging toh) is defined by the rules

In a way similar to the first Cartan tensor,using (3.35)–(3.36) we can deduce that the second Cartan tensor has the following coordinate expression:

where

From(3.38),it is easy to see that the second Cartan tensor is semibasic.Moreover,(3.37)–(3.38)give us

where

Proposition 3.11Let (E,F) be a Finsler algebroid.We have

ProofLetX,YandZbe sections ofE.Using the second part of (2.1) we get

Now we prove (3.43).Direct calculation gives us

But,using (3.41) we can see that the above equation is equal toThus,we have (3.43).

Proposition 3.12Let (E,F) be a Finsler algebroid.Ifhis a torsion free and conservative horizontal endomorphism on £πE,the lowered second Cartan tensor is symmetric.

Proof(3.41) implies thatis symmetric with respect to last two variables.Then,it is sufficient to prove thatis symmetric with respect to first two variables.Sincehis conservative,using (3.24) and (i) of (3.25) in (3.41) we obtain

Sincehis torsion free,using (2.11) we haveSetting this equation in the above equation implies

3.4 Distinguished connections on Finsler algebroids

In this section,we study the existence and uniqueness of the distinguished connections Berwald,Cartan,Chern-Rund and Hashiguchi and we present some properties of them.

Theorem 3.4Let (E,F) be a Finsler algebroid andhbe a conservative horizontal endomorphism on £πE.There exists a unique d-connectionon (E,F) such that thev-mixed andh-mixed torsions ofare zero.

ProofThere exists a d-connectionon(E,F)such that thev-mixed andh-mixed torsions of it are zero.If we denote bythev-mixed torsion ofthen we have

Since theh-mixed torsion ofis zero,we have

Relations (3.44)–(3.47)prove the existence and uniqueness of

Using (3.44)–(3.47),the d-connectionhas the following coordinate expression:

Proposition 3.13Let (E,F) be a Finsler algebroid,hbe a conservative horizontal endomorphism on £πEandbe the d-connection given by (3.48).Ifh-deflection andh-horizontal torsions ofare zero,his the Barthel endomorphism.

ProofIt is sufficient to show thathis homogeneous and torsion free.Sinceh-deflection ofis zero,we have

The above equation shows thathis homogeneous.Also,since theh-horizontal torsion ofis zero,we get

From the above equation,we deduce that the weak torsion ofhis zero.

Ifhis the Barthel endomorphism of Finsler algebroid (E,F),the d-connectiongiven in(3.48) is called the Berwald connection of (E,F).

Theorem 3.5Let (E,F) be a Finsler algebroid,hbe a torsion free and conservative horizontal endomorphism on £πEandbe the prolongation of G alongh.There exists a unique d-connectionon (E,F) such thatis metrical,i.e.,= 0 and thev-vertical andhhorizontal torsions ofare zero.

ProofThere exists a d-connectionsuch thatis metrical and thev-vertical andhhorizontal torsions ofare zero.Sinceis metrical,we have

Since theh-horizontal torsion ofis zero,we have

Summing (3.49)–(3.51)and using the above equation give us

Sincehis torsion free,using (2.11) in the above equation we get

Sincehis conservative,we have (3.19).Differentiation of this equation with respect toygives us

Setting two above equations in (3.52),we obtain

Since thev-horizontal torsion ofis zero,we have

If we replaceδα,δβ,δγby Vα,Vβ,Vγin (3.49)–(3.51),summing these equations and using the above equation we get

which gives us

which gives us

Similarly,using (3.55) we get

which gives us

Relations (3.55)–(3.58)prove the existence and uniqueness of

Proposition 3.14Let (E,F) be a Finsler algebroid,hbe a torsion free and conservative horizontal endomorphism on £πEandbe the d-connection given by the above theorem.Ifh-deflection ofis zero,his the Barthel endomorphism.

ProofIt is sufficient to show thathis homogeneous.Sinceh-deflection ofis zero,using (3.25) and (3.58) we obtain

Sincehis conservative,we have (3.19).Using this equation in the above equation we deduce

The above equation shows thathis homogeneous.

Ifhis the Barthel endomorphism of Finsler algebroid (E,F),the d-connectiongiven by(3.55)–(3.58)is called the Cartan connection of (E,F).

Using (2.30)–(2.32)and (3.55)–(3.58),we can obtain

where

Theorem 3.6Let (E,F) be a Finsler algebroid,hbe a torsion free and conservative horizontal endomorphism on £πEandbe the prolongation of G alongh.Then,there exists a unique d-connectionon (E,F) such thatish-metrical,and theh-horizontal torsion ofis zero.Moreover,if theh-deflection ofis zero,his the Barthel endomorphism.

ProofThere exists a d-c onnectionon (E,F) such thatish-metrical,and theh-horizontal torsions ofis zero.Sinceish-metrical andh-horizontal torsion ofis zero,similar to the proof of Theorem 3.5 we can deduce

and consequently

Relations (3.63)–(3.66)prove the existence and uniqueness ofThe proof of the second part of the assertion is similar to Proposition 3.14.

Ifhis the Barthel endomorphism of Finsler algebroid (E,F),the d-connectiongiven by(3.63)–(3.66)is called the Chern-Rund connection of (E,F).

Using (2.30)–(2.32)and (3.63)–(3.66),we can get

where

Theorem 3.7Let (E,F) be a Finsler algebroid,hbe a conservative horizontal endomorphism on £πEandbe the prolongation of G alongh.There exists a unique d-connectionon (E,F) such thatisv-metrical,and thev-vertical andv-mixed torsions ofare zero.

ProofThere exists a d-connectionon(E,F)such thatisv-metrical and thev-vertical andv-mixed torsions ofare zero.Sinceisv-metrical and thev-vertical torsion ofis

zero,similar to the proof of Theorem 3.5 we can deduce

Moreover,since thev-mixed torsion ofis zero,we can obtain

and consequently

Proposition 3.15Let (E,F) be a Finsler algebroid,hbe a conservative horizontal endomorphism on £πEandbe the d-connection given by the above theorem.Ifh-horizontal torsion andh-deflection ofare zero,his the Barthel endomorphism.

ProofThe proof is similar to the proof of Proposition 3.13.

Ifhis the Barthel endomorphism of Finsler algebroid (E,F),the d-connectiongiven by(3.71)–(3.74)is called the Hashiguchi connection of (E,F).

Using (2.30)–(2.32)and (3.71)–(3.74),we can obtain

where

Theorem 3.8Lethbe the Barthel endomorphism on Finsler algebroid (E,F).Then,the Cartan connection

(i) is Chern-Rund connection if

(ii) is Hashiguchi connection if

(iii)is Berwald connection if it is the Chern-Rund connection and the Hashiguchi connection at the same time.