On a class of conformally invariant semi basic vector one forms

TL;DR

This paper introduces a new class of conformally invariant semi-basic vector one-forms on Finsler manifolds, characterizes associated conservative connections, and explores their geometric properties and correspondences.

Contribution

It defines conservative semibasic vector 1-forms, characterizes conservative L-Ehresmann connections, and establishes a correspondence with torsion-free semibasic vector forms, advancing Finsler geometry theory.

Findings

Characterization of conservative L-Ehresmann connections

Establishment of a correspondence between torsion-free forms and vertical vector fields

Construction of semisprays generating the Ehresmann connections

Abstract

In this paper, we define conservative semibasic vector $1-$forms on the tangent bundle of a Finsler manifold. Using these vector $1-$forms, we characterize conservative $L-$Ehresmann connections with respect to the energy function. Then we find a correspondence between torsion-free semibasic vector $1-$forms and the subset of vertical vector fields. Taking into account this correspondence, we construct a class of semisprays that generates the Ehresmann connections mentioned above.

Equations

$${\mathbf{h}}_L:={\mathbf{h}}_0+L+[J,(d_{L}E{)}^{\#}],$$

$${\mathbf{t}}_L=[J,L].$$

$$\overset{\,\,V}{\mbox{\small{$\mathcal{S}$}}}=\mbox{\small{$\mathcal{S}$}}_{0}+2\,V+2\,(d_{\,[J,V]}E)^{\#},$$

$$i_K\upharpoonright C^{\infty }(TM)=0;i_K\circ \alpha =\alpha \circ K,\text{if}\alpha \in {\Omega }^1(TM);$$

$$d_K:=[i_K,d]=i_K\circ d-(-1{)}^{k-1}d\circ i_K.$$

$$d_K\phi =d\phi \circ K;\phi \in C^{\infty }(TM),K\in {\Psi }^k(TM).$$

$$d_{[K,L]}=[d_K,d_L].$$

$$[K,Y]X=[KX,Y]-K[X,Y],$$

$$i_{[K,Y]}=i_Y\circ d_K+d_K\circ i_Y-{\mathcal{L}}_{KY},$$

$$i_Y\circ i_K=i_K\circ i_Y+i_{KY}.$$

$$ImJ=KerJ={\mathfrak{X}}^v(TM),[J,J]=0,[J,C]=J.$$

$$\mathbf{h}=\frac{1}{2}\,(1_{\mathfrak{X}(TM)}+[J,\mbox{\small{$\mathcal{S}$}}]).$$

$$i_J\omega =0,i_C\omega =d_{J}E,{\mathcal{L}}_C\omega =\omega .$$

$$i_{{\beta }^{\#}}\omega =\beta .$$

$$\mbox{\small{$\mathcal{S}$}}_{0}:=-(dE)^{\#}\quad\textrm{over}\quad\overset{\circ}{T}M,\quad\mbox{\small{$\mathcal{S}$}}_{0}(0):=0$$

$$d_{{\mathbf{h}}_0}E=0.$$

$$\mathbf{h}={\mathbf{h}}_0+\frac{1}{2}{\mathbf{t}}^{\circ }+\frac{1}{2}[J,(d_{{\mathbf{t}}^{\circ }}E{)}^{\#}].$$

$$\overline{\mathbf{h}}:={\mathbf{h}}_0+f^{c}J-E[J,\text{textsl}grad{f}^v]-d_{J}E\otimes \text{textsl}grad{f}^v,$$

$$L:=f^{c}J-E[J,\text{textsl}grad{f}^v]-d_{J}E\otimes \text{textsl}grad{f}^v.$$

$$\phi :=exp\circ f^v.$$

$$\forall X\in \mathfrak{X}(TM):(d_K\phi )X\stackrel{(\text{ref}d3)}{=}(d\phi \circ K)X=(KX)\phi =0,$$

$$d_L\stackrel{\sim }{E}=d_L\phi E=(d_L\phi )E+\phi d_{L}E=\phi d_{L}E.$$

$${\mathbf{h}}_L:={\mathbf{h}}_0+L+[J,(d_{L}E{)}^{\#}],$$

$${\beta }^{\#}E$$

$$\displaystyle=\omega(\beta^{\#},\mbox{\small{$\mathcal{S}$}}_{0})=\big{(}i_{\beta^{\#}}\omega\big{)}(\mbox{\small{$\mathcal{S}$}}_{0})$$

$$\displaystyle=\beta\,\mbox{\small{$\mathcal{S}$}}_{0}=i_{\,\mbox{\scriptsize{$\mathcal{S}$}}_{0}}\beta$$

$$={\beta }^{\circ }.$$

$$U:=(d_{L}E{)}^{\#}.$$

$$d_{{\mathbf{h}}_L}E=d_{{\mathbf{h}}_0}E+d_{L}E+d_{[J,U]}E$$

$$d_{[J,U]}E$$

$$\stackrel{(\text{ref}d2)}{=}-i_{U}d{d}_{J}E+d_J{i}_{U}dE$$

$$=-i_U\omega +d_J{i}_{U}dE$$

$$\stackrel{(\text{ref}w)}{=}-d_{L}E+d_J{i}_{U}dE.$$

$$d_{{\mathbf{h}}_L}E$$

$$=d_J(UE)\stackrel{\text{Lem}.\text{ref}lem}{=}{d}_J((d_{L}E{)}^{\circ })$$

Full text

Keywords: Finsler manifold, Ehresmann connection, Semibasic vector 1-form.111 2010 Mathematics subject Classification: 53C07, 53C60.

1 Introduction

The difference of two Ehresmann connections over a manifold $M$ is a semibasic vector $1-$forms on $TM$. Motivated by this fact, it is natural to define a conservative semibasic vector $1-$form on a Finsler manifold with respect to an energy function induced by a Finsler metric. A semibasic vector $1-$form is called conservative on a Finsler manifold $(M,E)$ if it can be expressed as a difference of two conservative Ehresmann connections on $M$. The conservativity property of a semibasic vector $1-$form $L$ on $TM$ implies the conservativity of an $L-$Ehresmann connection, which is defined by

[TABLE]

where $\mathbf{h}_{\,0}$ is the Berwald connection. It is well known that this class of Ehresmann connections contains the Wagner connection of the Finsler manifold $(M,E)$ and among them the conservative ones are conformally closed. For more details see [2] and [3].

In Section 3, using conservative semibasic vector $1-$forms, we characterize conservative $L-$Ehresmann connections with respect to the energy function $E$.

The weak torsion of an $L-$Ehresmann connection is in the following form

[TABLE]

Thus, it is natural to define torsion-freeness of a semibasic vector $1-$form $L$ on the tangent manifold $TM$ by requiring the condition $[J,L]=0.$ In Section 4, we establish a correspondence between the subset of vertical vector fields $\mathfrak{X}^{v}(TM)$ and the set of torsion – free semibasic vector $1-$forms on $TM$.

Taking into account this correspondence, in Subsection 4.2, we introduce a set of semisprays as follows

[TABLE]

where $V$ is a vertical vector field on $TM$. If $V$ is a two-homogeneous vertical vector field, then the semispray \overset{\,\,V}{\mbox{\small{\mathcal{S}}}} is a spray and we can find projectively related relation between two sprays in this form.

In [10], Vincze presents the theory of conservative semisparays on a Finsler manifold $(M,E)$ with respect to the energy function $E$. Then he defines a conservative vertical vector field on a Finsler manifold. In this paper, using conservativity and torsion-freeness of the $L-$Ehresmann connections, we find a class of conservative vertical vector fields on a Finsler manifold.

2 Preliminaries

We work on an $n-$dimensional connected smooth manifold $M$ whose topology is Hausdorff and has a countable base. $C^{\infty}(M)$ denotes the ring of smooth real-valued functions on $M$, $\mathfrak{X}(M)$ and $\Omega^{k}(M)$ stand for the $C^{\infty}(M)-$ module of (smooth) vector fields and differential $k-$forms on $M$.

$TM$ is the tangent manifold of $M$, and $\overset{\circ}{T}M$ is the open submanifold of the non-zero tangent vectors to $M$. The vertical and the complete lift of a smooth function $f$ on $M$ into $TM$ are denoted by $f^{v}$ and $f^{c}$, respectively.

The $C^{\infty}(TM)-$module of vertical vector fields on $TM$ will be denoted by $\mathfrak{X}^{v}(TM)$. $X^{v}$ stands for the vertical lift of a vector field $X$ on $M$ and $C\in\mathfrak{X}^{v}(TM)$ is the Liouville vector field.

By a vector $k-$form on $TM$ we mean a skew symmetric $C^{\infty}(TM)-$ multilinear map $K:(\mathfrak{X}(TM))^{k}\rightarrow\mathfrak{X}(TM)$ if $k\in\{1,\ldots,2n\}$, and a vector field on $TM$, if $k=0$. In particular, a vector $1-$form on $TM$ is just a tensor field of type $(1,1)$. The $C^{\infty}(TM)-$module of vector $k-$forms on $TM$ will be denoted by $\Psi^{k}(TM)$.

By the Frölicher-Nijenhuis theory of vector forms to any vector $k-$form $K\in\Psi^{k}(TM)$ two graded derivations of $\Omega(TM)$ are associated, denoted by $i_{K}$ and $d_{K}$, which the former is of degree $k-1$, and the later is of degree $k$, and the following rules are prescribed:

[TABLE]

Then, in particular,

[TABLE]

To any vector forms $K\in\Psi^{k}(TM)$, $L\in\Psi^{\ell}(TM)$ there is a unique vector $(k+l)-$form $[K,L]\in\Psi^{k+l}(TM)$, the Frölicher-Nijenhuis bracket of $K$ and $\L$ such that

[TABLE]

In particular, if $K\in\Psi^{1}(TM)$, $Y\in\Psi^{\circ}(TM)$ then $[K,Y]\in\Psi^{1}(TM)$, and for any vector field $X\in\mathfrak{X}(TM)$, we have,

[TABLE]

There is a unique vector $1-$form $J$ on $TM$ such that

[TABLE]

$J$ is called the vertical endomorphism. A differential form $\alpha\in\Omega^{k}(TM)$ is semibasic, if $i_{J\xi}\alpha=0;$ a vector form $K\in\Psi^{k}(TM)$ is semibasic, if $i_{J\xi}K=0$ and $J\circ K=0$ $(k\geq 1,\ \xi\in\mathfrak{X}(TM))$.

A vector field $\mathcal{S}$ on $TM$ of class $C^{1}$, smooth on $\overset{\circ}{T}M$ is said to be a semispray over $M$ if J\mbox{\small{\mathcal{S}}}=C. A semispray $\mathcal{S}$ is called a spray if it is homogeneous of degree 2, i.e., [C,\mbox{\small{\mathcal{S}}}]=\mbox{\small{\mathcal{S}}}.

A vector 1–form $\mathbf{h}\in\Psi^{1}(TM)$, smooth –in general– only over $\overset{\circ}{T}M$ is said to be a Ehresmann connection over $M$ if it is a projector (i.e., $\mathbf{h}^{2}=\mathbf{h}$) and $\ker\mathbf{h}=\mathfrak{X}^{v}(TM)$, or, equivalently, if $J\circ\mathbf{h}=J$ and $\mathbf{h}\circ J=0$. An Ehresmann connection $\mathbf{h}$ is called homogeneous if its tension vanishes, i.e., $\mathbf{H}=[C,\mathbf{h}]=0$. The weak torsion of $\mathbf{h}$ is the vector 2-form $\mathbf{t}:=[J,\mathbf{h}]$.

A fundamental result due to M. Crampin and J. Grifone states that any semispray $\mathcal{S}$ generates a Ehresmann connection of zero weak torsion by the formula

[TABLE]

Let a function $E:TM\rightarrow\mathbb{R}$ be given. Assume:

• (i)

$E(v)>0$ for all $v\in\overset{\circ}{T}M,\,\,\,E(0)=0$;

• (ii)

$E$ is of class $C^{1}$ on $TM$, smooth on $\overset{\circ}{T}M$;

• (iii)

$E$ is (positive–) homogeneous of degree 2, i.e., $CE=2E$;

• (iv)

The fundamental 2-form $\omega:=d\,d_{J}E$ is nondegenerate.

Then $(M,E)$ is said to be a Finsler manifold with the energy function $E$. Notice that $\omega$ is semibasic and we have the relations

[TABLE]

Due to the nondegeneracy of $\omega$, for any $1-$form $\beta\in\Omega^{1}(TM)$ there is unique vector field $\beta^{\#}$ on $TM$ (smooth, in general, only on $\overset{\circ}{T}M$) such that

[TABLE]

This map $\#:\beta\rightarrow\beta^{\#}$ is called the (Finslerian) sharp operator. In particular, the gradient of a function $f\in C^{\infty}(TM)$ is the vector field $\textsl{grad\;}\,f:=(df)^{\#}$.

Following Grifone [1], by the potential of a semibasic $k-$form $K$ on $TM$ we mean the $(k-1)-$form K^{\circ}:=i_{\mbox{\small{\mathcal{S}}}}K, where $\mathcal{S}$ is any semispray over $M\,\,$ $(k\geqq 1)$. Clearly, $K^{\circ}$ is independent of the choice of $\mathcal{S}$.

To conclude this section, we recall the fundamental lemma of Finsler geometry due to J. Grifone [1]. Let $(M,E)$ be a Finsler manifold. If

[TABLE]

then \mbox{\small{\mathcal{S}}}_{\,0} is a spray over $M$, called the canonical spray of $(M,E)$. \mbox{\small{\mathcal{S}}}_{0} generates a homogeneous Ehresmann connection $\mathbf{h}_{\,0}$ according to (9), called the Berwald connection of $(M,E)$. $\mathbf{h}_{\,0}$ is conservative in the sense that

[TABLE]

3 Conservative Semibasic Vector 1-Form

The difference of two Ehresmann connections is a semibasic vector $1-$form, thus it is natural to define the conservativity of semibasic vector $1-$form as follows.

Definition 1**.**

Suppose that $(M,E)$ is a Finsler manifold. A semibasic vector $1-$form $L$ on $TM$ is said to be conservative with respect to energy function E, if it is a difference of two conservative Ehresmann connections.

It is easy to see that a semibasic vector $1-$form $L$ on $TM$ is conservative if and only if $d_{L}E=0$. Therefore, the set of all conservative semibasic vector $1-$forms on $TM$ is a $C^{\infty}(TM)-$module.

Authors have shown in [5] that any homogeneous, conservative Ehresmann connection $\mathbf{h}$ over $M$ can be expressed as follows:

[TABLE]

Hence every homogeneous, conservative Ehresmann connection $\mathbf{h}$ over $M$ generates a conservative semibasic vector $1-$form $L=\mathbf{t}^{\circ}+[J,(d_{\,\mathbf{t}^{\circ}}E)^{\#}]$.

Another example of conservative semibasic vector $1-$form, is difference of the Berwald and the Wagner Ehresmann connections. Let $f$ be a smooth function on $M$. It is known [9] the Wagner connection of a Finsler manifold $(M,E)$ have the form,

[TABLE]

Then the following semibasic vector $1-$form is conservative on Finsler manifold $(M,E)$:

[TABLE]

3.1 Conformal Change of Energy Function

Assume that $(M,E)$ is a Finsler manifold. Let $f$ be a smooth function on $M$ and define a positive function on $TM$ by

[TABLE]

If $\overset{\sim}{E}=\varphi E$, then $(M,\overset{\sim}{E})$ is also a Finsler manifold (see [7]). We say that $(M,\overset{\sim}{E})$ has been obtained by a conformal change of $E$ given by the scale function $\varphi$.

It is known that a smooth function $\varphi$ on $TM$ is a vertical lift if and only if $d_{J}\,\varphi=0$, where $J$ is the canonical vertical endomorphism on $TM$ (see [7]).

Let $K$ be a semibasic vector $1-$form on $TM$. If a smooth function $\varphi$ on $TM$ is a vertical lift, then

[TABLE]

since $KX$ is a vertical vector field. Thus $d_{K}\,\varphi=0$.

Therefore for the scale function $\varphi:=exp\circ f^{v}$, we have $d_{K}\,\varphi=0$.

Proposition 1**.**

The conservativity property of a semibasic vector $1-$form on $TM$ (with respect to energy function of Finsler manifold) is invariant under any conformal change of energy function.

Proof.

Let us consider the conformal change $\overset{\sim}{E}=\varphi E\ (\varphi:=exp\circ f^{v})$ of a Finsler manifold $(M,E)$. Then

[TABLE]

This relation implies that conservativity of a semibasic vector $1-$form on $TM$ is invariant under any conformal change of energy function. ∎

3.2 L-Ehresmann Connection on a Finsler Manifold

Let $(M,E)$ be a Finsler manifold and $L$ be a semibasic vector $1-$form on $TM$. The Ehresmann connection

[TABLE]

is called $L-$Ehresmann connection on Finsler manifold $(M,E)$. For more details of $L-$Ehresmann connection, we refer to [3].

Next proposition shows that the conservativity of semibasic vector $1-$form $L$ on $TM$ implies the conservativity of $L-$Ehresmann connection.

Proposition 2**.**

Let $(M,E)$ be a Finsler manifold. Suppose that $L$ is a conservative semibasic vector $1-$form on $TM$ with respect to energy function $E$. Then the $L-$Ehresmann connection is conservative.

Proof.

By conservativty of $L$ we obtain $\mathbf{h}_{L}=\mathbf{h}_{\,0}+L$. This proves what we want. ∎

We obtain a condition for conservativity of $L-$Ehresmann connection in the following theorem. First we need to prove next lemma.

Lemma 1**.**

If $\beta$ is a semibasic $1-$form on $TM$, then $\beta^{\#}E=\beta^{\circ}.$

Proof.

Using definition of canonical spray of Finsler manifold, we get

[TABLE]

This completes the proof. ∎

Theorem 1**.**

Let $(M,E)$ be a Finsler manifold and $L$ be a semibasic vector $1-$form on $TM$. Then $L-$Ehresmann connection is conservative if and only if $L^{\circ}E$ is a vertical lift.

Proof.

For simplicity, let us put

[TABLE]

By definition of $L-$Ehresmann connection we get

[TABLE]

The first term of right hand side vanishes by conservativity of the Berwald connection, and for the third one, we have

[TABLE]

Thus, we have

[TABLE]

In ${\dagger}$ we have used following relations for an arbitrary semispray $\mathcal{S}$,

[TABLE]

The relation

[TABLE]

implies that $L-$Ehresmann connection is conservative if and only if $L^{\circ}E$ is a vertical lift. ∎

Corollary 1**.**

Let $(M,E)$ be a Finsler manifold. Suppose that $K$ is a semibasic vector $1-$form on $TM$ and $f$ is a smooth function on $M$. Define $L:=\frac{1}{K^{\circ}E}f^{v}K$. Then $L-$Ehresmann connection is conservative, provided that the function $K^{\circ}E$ nowhere vanishes.

Corollary 2**.**

A Wagner connection is conservative.

Proof.

We know that a Wagner connection is an $L-$Ehresmann connection by $L=\frac{1}{2}\,(f^{c}J-df^{v}\otimes C)$ (see [3]). Since

[TABLE]

thus $L^{\circ}E=0$. Therefore by Theorem 1 we obtain the conservativity of Wagner connections. ∎

Authors have shown in [3] that conservativity property of $L-$Ehresmann connection conformally invariant. Here, we give a simplified proof for this fact by the help of Theorem 1.

Proposition 3**.**

The set of all conservative $L-$Ehresmann connections on a Finsler manifold is conformally closed, i.e., a conservative $L-$Ehresmann connection remains a conservative $L-$Ehresmann connection by any conformal change of energy function.

Proof.

Let us consider conformal change $\overset{\sim}{E}=\varphi E$. Suppose that $\mathbf{h}_{L}$ is conservative on $(M,E)$, then we have

[TABLE]

Since $\varphi$ and $L^{\circ}E$ are vertical lifts thus $\overset{\sim}{\mathbf{h}}_{L}$ is conservative on $(M,\overset{\sim}{E})$ by Theorem 1. ∎

Now, we state and prove a property of $L-$Ehresmann connection for next uses.

Proposition 4**.**

Let $L$ be a semibasic vector $1-$form on $TM$. Then

[TABLE]

where $\Theta$ is an operator on semibasic vector $1-$forms on $TM$ defined by

[TABLE]

It results

[TABLE]

where $\mathbf{H}_{L}$ is the tension of $\mathbf{h}_{L}$.

Proof.

The following holds

[TABLE]

Using graded Jacobi identity, we obtain

[TABLE]

Thus

[TABLE]

Then we get the proof. ∎

4 Torsion-free Semibasic Vector 1-Form

By Proposition 1 of [3], the weak torsion of $L-$Ehresmann connection is

[TABLE]

Due to this result, we define a torsion-free semibasic vector $1-$form as follows.

Definition 2**.**

A Semibasic vector $1-$form $L$ on $TM$ is called torsion-free if

[TABLE]

Thus a semibasic vector $1-$form $L$ on $TM$ is torsion–free if and only if the $L-$Ehresmann connection is torsion–free.

Let $V$ be a vertical vector field on $TM$. It is easy to see that the semibasic $1-$form $L:=[J,V]$ is torsion-free. The following proposition states that the converse is also true. Hence, we get a characterization of torsion-free semibasic vector $1-$forms.

Proposition 5**.**

Let $L$ be a torsion–free semibasic vector $1-$form on $TM$. Then there is a vertical vector field $V_{L}$ on $TM$ such that

[TABLE]

Moreover, the set of all vertical vector fileds on $TM$ satisfying (12) is given by

[TABLE]

Proof.

Since $L$ is torsion-free, there is a semispray $\mathcal{S}$ on $M$ such that

[TABLE]

On the other hand

[TABLE]

Then

[TABLE]

Therefore, it suffices to define

[TABLE]

Since for every vector field $X$ on $M$ we have $[J,X^{v}]=0$, the vertical vector field $V_{L}+X^{v}$ satisfies (12). Conversely, suppose that the relation (12) is true for two vertical vector fields. Then by Lemma 1.15 of [6], the difference of these vertical vector fields is a vertical lift. This completes the proof. ∎

Remark 1. Let $L$ be a torsion-free semibasic vector $1-$form on $TM$ and homogeneous of degree $r\neq-1$. By a remark of [1] we have $L={1\over r+1}\,[J,L^{\circ}]$. Hence, the set of all vertical vector fields satisfying (12) is given by

[TABLE]

Lemma 2**.**

Let $L$ be a torsion–free semibasic vector $1-$form on $TM$. If $V_{L}$ is homogeneous of degree 2, then $L-$Ehresmann connection is also homogeneous.

Proof.

By graded Jacobi identity and using Proposition 5 we obtain

[TABLE]

Thus $L$ is homogeneous of degree 1, therefore Proposition 4, implies the homogeneity of $L-$Ehresmann connection. ∎

Proposition 6**.**

Let $L$ be a torsion-free semibasic vector $1-$form on $TM$. Then $d_{\,\mathbf{h}_{L}}\omega$ vanishes, where $\omega$ is the fundamental form of Finsler manifold $(M,E)$.

Proof.

It is sufficient to show that

[TABLE]

where $\mathbf{t}_{L}$ is the weak torsion of $L-$Ehresmann connection. First

[TABLE]

For simplicity let $P:=[J,(d_{L}E)^{\#}]$. Then, we have

[TABLE]

Since $[J,P]$ vanishes by graded Jacobi identity. On the other hand in the proof of Theorem 1, we obtain

[TABLE]

therefore

[TABLE]

Since $d_{J}^{\,2}\overset{(\ref{e})}{=}{1\over 2}d_{\,[J,J]}\overset{(\ref{ee4})}{=}0$. Hence

[TABLE]

The proof follows from the torsion–freeness of $L$. ∎

4.1 Some Projectively Related Sprays

In this section we construct a class of semisprays that generates the torsion-free $L-$Ehresmann connections.

Proposition 7**.**

Let $V$ be a vertical vector field on $TM$, then the Ehresmann connection generated by semispray

[TABLE]

is just the $[J,V]-$Ehresmann connection. Furthermore, if $V$ is a vertical vector field homogeneous of degree 2, then the semispray \overset{V}{\mbox{\small{\mathcal{S}}}} is a spray.

Proof.

By a direct computation, we get

[TABLE]

Suppose $V$ is a $2-$homogeneous vertical vector field. By Jacobi identity, we have

[TABLE]

Therefore $[J,V]$ is homogeneous of degree 1. Using Lemma 2 of [3], we obtain that $V+2\,(d_{\,[J,V]}E)^{\#}$ is homogeneous of degree 2. Hence the semispray \overset{V}{\mbox{\small{\mathcal{S}}}} is a spray. ∎

By Proposition 7, we get the following.

Corollary 3**.**

Suppose that $L$ is a torsion–free semibasic vector $1-$form on $TM$. Then \overset{\ V_{L}}{\mbox{\small{\mathcal{S}}}} generates $L-$Ehresmann connection.

Two sprays \mbox{\small{\mathcal{S}}}_{1} and \mbox{\small{\mathcal{S}}}_{2} over $M$ are said to be (pointwise) projectively related if there is a smooth function $\lambda$ on $\overset{\circ}{T}M$ such that

[TABLE]

Then the projective factor $\lambda$ is necessarily $1-$homogeneous, i.e., $C\lambda=\lambda$. For more details see [8].

Proposition 8**.**

Let $V$ and $U$ be two vertical vector fields homogeneous of degree 2 on $TM$. Suppose that \overset{V}{\mbox{\small{\mathcal{S}}}} and \overset{U}{\mbox{\small{\mathcal{S}}}} are projectively related sprays on $M$, with the projective factor $\lambda$. Then

[TABLE]

Proof.

Suppose that \overset{V}{\mbox{\small{\mathcal{S}}}} and \overset{U}{\mbox{\small{\mathcal{S}}}} are projectively related sprays on $M$, with the projective factor $\lambda$. Then

[TABLE]

Therefore by (13) we get

[TABLE]

Put $Y:=2\,(V-U)$ then we obtain

[TABLE]

We have

[TABLE]

on the other hand

[TABLE]

Hence, for any semispray $\mathcal{S}$,

[TABLE]

and

[TABLE]

and

[TABLE]

thus $3\,YE=2\lambda E.$ Therefore

[TABLE]

as we want. ∎

5 Some Conservative Vector Fields

In [10], Vincze defines a conservative vector field on a Finsler manifold. He proves that a vertical vector field $V$ is conservative on a Finsler manifold $(M,E)$ if and only if

[TABLE]

In this section, we introduce a class of conservative vertical vector fields.

Theorem 2**.**

Let $(M,E)$ be a Finsler manifold and $V$ be a vertical vector field on $TM$. If $[J,V]-$Ehresmann connection is conservative then $U:=V+(d_{[J,V]}E)^{\#}$ is a conservative vector field.

Proof.

By assumption we obtain

[TABLE]

Suppose that $[J,V]-$Ehresmann connection is conservative. By conservativity of $\mathbf{h}_{\,0}$ and using Cartan magic formula we have

[TABLE]

which implies that $U$ is a conservative vector field. ∎

Corollary 4**.**

Let $(M,E)$ be a Finsler manifold and $L$ be a torsion-free semibasic vector $1-$form on $TM$. If $L-$Ehresmann connection is conservative then

[TABLE]

is conservative in Vincze’s sense.