Conformal deformations preserving the Finslerian $R$-Einstein criterion

TL;DR

This paper investigates how conformal deformations can preserve the Finslerian $R$-Einstein criterion, providing a characterization of local conformal invariance between such metrics on manifolds.

Contribution

It introduces conditions under which conformal deformations preserve the Finslerian $R$-Einstein property and characterizes local conformal invariance between these metrics.

Findings

Conformal deformations preserving the $R$-Einstein criterion are explicitly described.

A characterization of local conformal invariance between Finslerian $R$-Einstein metrics is provided.

The results apply to $C^4$-manifolds with Finslerian metrics.

Abstract

Given a Finslerian metric $F$ on a $C^4$-manifold, conformal deformations of $F$ preserving the $R$-Einstein criterion are presented. In particular, locally conformal invariance between two Finslerian $R$-Einstein metrics is characterized.

Equations

$$g_{ij}(x,y):=\frac{1}{2}\frac{{\partial }^2{F}^2(x,y)}{\partial y^i\partial y^j}$$

$$T\mathring{T}M=\mathcal{H}\oplus \mathcal{V}.$$

$$\mathcal{H}:=ker\theta ,$$

$$\theta =\frac{\partial }{\partial x^i}\otimes \frac{1}{F}(d{y}^i+N_j^{i}d{x}^j).$$

$$\theta {|}_{(x,y)}:T_{(x,y)}\mathring{T}M⟶T_{x}M,$$

$${\pi }_{\ast }(\text{v}(\xi )){|}_{(x,y)}=0_{(x,y)}\text{ and }\theta (\text{v}(\xi )){|}_{(x,y)}={\xi }_{(x,y)}.$$

$${\pi }_{\ast }{|}_{(x,y)}:T_{(x,y)}\mathring{T}M⟶T_{x}M,$$

$${\pi }_{\ast }(\text{h}(\xi )){|}_{(x,y)}={\xi }_{(x,y)}\text{ and }\theta (\text{h}(\xi )){|}_{(x,y)}=0_{(x,y)}.$$

$$T:\underset{p_1-times}{\underbrace{{\pi }^{\ast }TM\otimes ...\otimes {\pi }^{\ast }TM}}\otimes \underset{p_2-times}{\underbrace{T\mathring{T}M\otimes ...\otimes T\mathring{T}M}}⟶\stackrel{q}{⨂}{\pi }^{\ast }TM,$$

$$T=T_{i_1...i_{p_1}{j}_1...j_{p_2}}^{k_1...k_q}{\partial }_{k_1}\otimes ...\otimes {\partial }_{k_q}\otimes d{x}^{i_1}\otimes ...\otimes d{x}^{i_{p_1}}\otimes {\epsilon }^{j_1}\otimes ...\otimes {\epsilon }^{j_{p_2}}$$

$${\nabla }_X{\pi }_{\ast }Y-{\nabla }_Y{\pi }_{\ast }X={\pi }_{\ast }[X,Y],$$

$$X(g(\xi ,\eta ))=g({\nabla }_X\xi ,\eta )+g(\xi ,{\nabla }_X\eta )+2\mathcal{A}(\theta (X),\xi ,\eta ),$$

$$\varphi :\begin{array}{ccc}\chi (\mathring{T}M)\times \chi (\mathring{T}M)\times \Gamma ({\pi }^{\ast }TM)& \to & \Gamma ({\pi }^{\ast }TM)\\ (X,Y,\xi )& \mapsto & \varphi (X,Y)\xi ={\overline{\nabla }}_X{\overline{\nabla }}_Y\xi -{\overline{\nabla }}_Y{\overline{\nabla }}_X\xi -{\overline{\nabla }}_{[X,Y]}\xi .\end{array}$$

$${\overline{\nabla }}_X={\overline{\nabla }}_{\hat{X}}+{\overline{\nabla }}_{\check{X}},$$

$$\Phi (\xi ,\eta ,X,Y)$$

$${\text{Ric}}_F^H(\xi ,X):=\sum _{a=1}^n\text{R}(\xi ,e_a,X,{\hat{e}}_a)$$

$${\text{Scal}}_F^H:=\sum _{a=1}^n{\text{Ric}}_F^H(e_a,{\hat{e}}_a)=\sum _{a,b=1}^n\text{R}(e_a,e_b,{\hat{e}}_a,{\hat{e}}_b).$$

$${\text{Ric}}_F^H=\frac{1}{n}{\text{Scal}}_F^H\underline{g}.$$

$${\tilde{\text{E}}}_{\tilde{F}}^H={\text{E}}_F^H-(n-2)\left(H_u-du\circ du\right)-\frac{(n-2)}{n}\left({\Delta }^{H}u+||▽u|{|}_g^2\right)\underline{g}+{\Psi }_u$$

$${\Psi }_u(\xi ,X)$$

$$\mathcal{B}={\mathcal{B}}_j^i{\partial }_i\otimes d{x}^j$$

$${\mathcal{B}}_j^i=\frac{1}{2F}\left({\nabla }_{r}u\right)\frac{\partial (F^2{g}^{ir}-2y^i{y}^r)}{\partial y^j}.$$

$$0$$

$$I_{11}$$

$$I_{12}$$

$$I_{13}$$

$$I_{14}$$

$$I_{15}$$

$$I_{16}$$

$${\nabla }_j{\nabla }_{i}u$$

$$\stackrel{1}{p}$$

$$\stackrel{2}{p}$$

$$F:\mathring{T}\stackrel{1}{M}\times \mathring{T}\stackrel{2}{M}⟶{\mathbb{R}}^{+}$$

Full text

1. Introduction

Let $F$ be a Finslerian metric on an $n$-dimensional manifold $M$. One of the most interesting problems is to study the conformal invariance of some important geometric quantities associated with $F$ [3]. For example, we have the Liouville transformation that is a conformal deformation which preserves the Finslerian Ricci tensor [18]. It is known that Einstein metrics play an important role in conformal geometry (see [5, 7, 12]). However, little work has been done on the conformal deformations between Einstein metrics of type Finslerian. In 2013, Zhang Xiao-ling [22] has shown that conformal deformations between Einstein metrics of type Randers must be homothetic.

The main objective of the present work is to study the conformal deformations preserving the $R$-Einstein criterion. In particular, we classify locally conformal deformations between two Finslerian $R$-Einstein metrics.

The sections 2 and 3 review the main notions on global Finslerian $R$-Einstein spaces introduced in [10]. In the Section 4, we prove our main results given by the Proposition 4.3 and, established in the Theorem 4.7 and the Corollary 4.8. This is concluded by an example given in Section 5.

2. Preliminaries

Throughout this work, all manifolds and mappings are supposed to be differentiable of classe $C^{4}$. Let $M$ be an $n-$dimensional manifold. We denote by $T_{x}M$ the tangent space at $x\in M$ and by $TM:=\bigcup_{x\in M}T_{x}M$ the tangent bundle of $M$. Set $\mathring{T}M=TM\backslash\{0\}$ and $\pi:TM\longrightarrow M:\pi(x,y)\longmapsto x$ the natural projection. Let $(x^{i})_{i=1,...,n}$ be a local coordinate on an open subset $U$ of $M$ and $(x^{i},y^{i})_{i=1,...,n}$ be the local coordinate on $\pi^{-1}(U)\subset TM$. We have the coordinate bases $\{\frac{\partial}{\partial x^{i}}\}_{i=1,...,n}$ and $\{dx^{i}\}_{i=1,...,n}$ respectively, for the tangent bundle $TM$ and cotangent bundle $T^{*}M$. We use Einstein summation convention.

Definition 2.1**.**

A Finslerian metric on $M$ is a function $F:TM\longrightarrow[0,\infty)$ with the following properties:

• (1)

$F$* is $C^{\infty}$ on the entire slit tangent bundle $\mathring{T}M$,*

• (2)

$F$* is positively $1$-homogeneous on the fibers of $TM$, that is *

$\forall c>0,~{}F(x,cy)=cF(x,y),$**

• (3)

the Hessian matrix $(g_{ij}(x,y))_{1\leq i,j\leq n}$ with elements

[TABLE]

is positive definite at every point $(x,y)$ of $\mathring{T}M$.

Remark 2.2**.**

$F(x,y)\neq 0\text{ for all }x\in M\text{ and for every }y\in T_{x}M\backslash\{0\}.$**

Consider the tangent mapping $\pi_{*}$ of the submersion $\pi:\mathring{T}M\longrightarrow M$. The vertical subspace of $T\mathring{T}M$ is defined by $\mathcal{V}:=ker({\pi_{*}})$ which is locally spanned by the set $\{F\frac{\partial}{\partial y^{i}},1\leq i\leq n\}$, on each $\pi^{-1}(U)\subset\mathring{T}M$.

An horizontal subspace $\mathcal{H}$ of $T\mathring{T}M$ is by definition any complementary to $\mathcal{V}$. The bundles $\mathcal{H}$ and $\mathcal{V}$ give a smooth splitting

[TABLE]

An Ehresmann connection is a selection of a horizontal subspace $\mathcal{H}$ of $T\mathring{T}M$. As explain in [11], $\mathcal{H}$ can be canonically defined from the geodesics equation.

Definition 2.3**.**

Let $\pi:\mathring{T}M\longrightarrow M$ be the submersion.

• (1)

An Finslerian Ehresmann connection of $\pi$ is the subbundle $\mathcal{H}$ of $T\mathring{T}M$ given by

[TABLE]

where $\theta:T\mathring{T}M\longrightarrow\pi^{*}TM$ is the bundle morphism defined by

[TABLE]

• (2)

The form $\theta:T\mathring{T}M\longrightarrow\pi^{*}TM$ induces a linear map

[TABLE]

*for each point $(x,y)\in\mathring{T}M$; where $x=\pi(x,y)$. *

The vertical lift of a section $\xi$ of $\pi^{*}TM$ is a unique section $\textbf{v}(\xi)$ of $T\mathring{T}M$ such that for every $(x,y)\in\mathring{T}M$,

[TABLE]

• (3)

The differential projection $\pi_{*}:T\mathring{T}M\longrightarrow\pi^{*}TM$ induces a linear map

[TABLE]

*for each point $(x,y)\in\mathring{T}M$; where $x=\pi(x,y)$. *

The horizontal lift of a section $\xi$ of $\pi^{*}TM$ is a unique section $\textbf{h}(\xi)$ of $T\mathring{T}M$ such that for every $(x,y)\in\mathring{T}M$,

[TABLE]

Definition 2.4**.**

A tensor field $T$ of type $(p_{1},p_{2};q)$ on $(M,F)$ is a mapping

[TABLE]

\big{(}p_{1},p_{2}* and q\in\mathbb{N}\big{)} which is $C^{\infty}(\mathring{T}M,\mathbb{R})$ -linear in each argument.*

Remark 2.5**.**

In a local chart,

[TABLE]

where $\{\partial_{k_{r}}:=\frac{\partial}{\partial x^{k_{r}}}\}_{r=1,...,q}$ and $\{\varepsilon^{j_{s}}\}_{s=1,...,p_{2}}$ are respectively the basis sections for $\pi^{*}TM$ and $T\mathring{T}M$.

Example 2.6**.**

• (1)

A vector field $X$ on $\mathring{T}M$ is of type $(0,1;0)$.

• (2)

*A section $\xi$ of $\pi^{*}TM$ is of type $(1,0;0)$. *

The following lemma defines the Chern connection on $\pi^{*}TM$.

Lemma 2.7**.**

[11]** Let $(M,F)$ be a Finslerian manifold and $g$ its fundamental tensor. There exist a unique linear connection $\nabla$ on the bundle $\pi^{*}TM$ such that, for all $X,Y\in\chi(\mathring{T}M)$ and for every $\xi,\eta\in\Gamma(\pi^{*}TM)$, one has the following properties:

• (i)

Symmetry:

[TABLE]

• (ii)

Almost $g$-compatibility:

[TABLE]

where $\mathcal{A}:=\frac{F}{2}\frac{\partial g_{ij}}{\partial y^{k}}dx^{i}\otimes dx^{j}\otimes dx^{k}$ is the Cartan tensor.

One has $\nabla_{\frac{\delta}{\delta x^{j}}}\frac{\partial}{\partial x^{k}}=\Gamma_{jk}^{i}\frac{\partial}{\partial x^{i}},\text{ }\Gamma_{jk}^{i}:=\frac{1}{2}g^{il}\left(\frac{\delta g_{jl}}{\delta x^{k}}+\frac{\delta g_{lk}}{\delta x^{j}}-\frac{\delta g_{jk}}{\delta x^{l}}\right)$where

$\left\{\frac{\delta}{\delta x^{i}}:=\frac{\partial}{\partial x^{i}}-N_{i}^{j}\frac{\partial}{\partial y^{j}}=\textbf{h}(\frac{\partial}{\partial x^{i}})\right\}_{i=1,...,n}\text{ with }N_{j}^{i}=\Gamma_{jk}^{i}y^{k}.$

3. Finslerian $R$-Einstein metrics

3.1. First curvatures $R$ associated with the Chern

Definition 3.1**.**

The full curvature of a linear connection $\overline{\nabla}$ on the vector bundle $\pi^{*}TM$ over the manifold $\mathring{T}M$ is the application

[TABLE]

By the relation (2.2), we have

[TABLE]

where $X=\hat{X}+\check{X}$ with $\hat{X}\in\Gamma(\mathcal{H})$ and $\check{X}\in\Gamma(\mathcal{V})$.

One can define the full curvature of $\overline{\nabla}$ by the following formula:

[TABLE]

where $\textbf{R}(\xi,\eta,X,Y)=g(\phi(\hat{X},\hat{Y})\xi,\eta),$ $\textbf{P}(\xi,\eta,X,Y)=g(\phi(\hat{X},\check{Y})\xi,\eta)+g(\phi(\check{X},\hat{Y})\xi,\eta)$ and $\textbf{Q}(\xi,\eta,X,Y)=g(\phi(\check{X},\check{Y})\xi,\eta)$ are respectively the first (horizontal) curvature, mixed curvature and vertical curvature.

In particular, if $\overline{\nabla}$ is the Chern connection, the Q-curvature vanishes.

3.2. $R$-Einstein metric

With respect to the Chern connection, we have:

Definition 3.2**.**

• (1)

The horizontal Ricci tensor $\textbf{Ric}_{F}^{H}$ and the horizontal scalar curvature $\textbf{Scal}_{F}^{H}$ of $(M,F)$ are respectively defined by

[TABLE]

and

[TABLE]

• (2)

A Finslerian metric $F$ on an $n$-dimensional manifold is $R$-Einstein if

[TABLE]

Remark 3.3**.**

If $F$ satisfies (3.2) for a constant function $\textbf{Scal}_{F}^{H}$ (respectively for $\textbf{Scal}_{F}^{H}\equiv 0$) then $F$ is said to be horizontally Ricci-constant (respectively, $F$ is called horizontally Ricci-flat metric).

Lemma 3.4**.**

[10]** If $F$ is horizontally an Einstein metric on a connected manifold of dimension $n\geq 3$ then its horizontal scalar curvature $\textbf{Scal}_{F}^{H}$ is constant.

4. Finslerian $R$-Einstein conformal invariance

Definition 4.1**.**

Let $F$ and $\widetilde{F}$ be two conformal Finslerian metrics on a manifold $M$, with $\widetilde{F}=e^{u}F$ . A geometric object $\mathcal{O}$ associated with $F$ is said to be conformally invariant (respectively conformally $u$-invariant) if the object $\widetilde{\mathcal{O}}$ associated with $\widetilde{F}$ satisfies $\widetilde{\mathcal{O}}=\mathcal{O}$. (respectively, $\widetilde{\mathcal{O}}=e^{u}\mathcal{O}$).

Proposition 4.2**.**

[18]** Let $F$ and $\widetilde{F}$ be two Finslerian metrics on an $n$-dimensional manifold $M$. If $F$ is conformal to $\widetilde{F}$, with $\widetilde{F}=e^{u}F$, then the trace-free horizontal Ricci tensors $\textbf{E}_{F}^{H}$ and $\widetilde{\textbf{E}}_{\widetilde{F}}^{H}$, associated with $F$ and $\widetilde{F}$ respectively, are related by

[TABLE]

where ${\varPsi}_{u}$ is the $(1,1;0)$-tensor on $(M,F)$ given by

[TABLE]

*for every $\xi\in\Gamma(\pi^{*}TM)$ and $X\in\chi(\mathring{T}M)$ with $\Theta_{ij}=\Theta(\hat{\partial}_{i},\hat{\partial}_{j})$ and $\mathcal{B}$ is the application which maps $\pi^{*}TM$ to $\pi^{*}TM$ given by *

[TABLE]

with

[TABLE]

By the Proposition 4.2, if $F$ and $\widetilde{F}$ are conformal then the $R$-Einstein criterions $\textbf{E}_{F}^{H}$ and $\widetilde{\textbf{E}}_{\widetilde{F}}^{H}$ satisfy the relation (4.1). In a local chart, if $\textbf{E}_{F}^{H}=\widetilde{\textbf{E}}_{\widetilde{F}}^{H}$ then we have

[TABLE]

By (4.3), we have $\mathcal{B}(\hat{\partial}_{l})=\mathcal{B}_{l}^{s_{1}}\partial_{s_{1}}$ where $\mathcal{B}_{l}^{s_{1}}=\frac{1}{2F}\left(\nabla_{r}u\right)\frac{\partial(F^{2}g^{{s_{1}}r}-2y^{s_{1}}y^{r})}{\partial y^{l}}$ and $\mathcal{B}(\textbf{h}(\triangledown u))=\nabla^{l}u\mathcal{B}_{l}^{s_{1}}\partial_{s_{1}}.$ Thus, from (4.5), we have

[TABLE]

[TABLE]

Hence, putting the expressions of $I_{11},I_{12},I_{13},I_{14},I_{15}$ and $I_{16}$ in the right-hand side of (4.5) we obtain the equation in (4.6) given in the following Proposition.

Proposition 4.3**.**

Let $F$ and $\widetilde{F}$ be two conformal Finslerian metrics on an $n$-dimensional manifold $M$, with $\widetilde{F}=e^{u}F$. Then $(M,F)$ and $(M,\widetilde{F})$ have locally a same $R$-Einstein criterion if and only if

[TABLE]

4.1. Warped product of Finslerian metrics

Let $\stackrel{{\scriptstyle{1}}}{{M}}$ and $\stackrel{{\scriptstyle{2}}}{{M}}$ be two $C^{\infty}$ manifolds. For every $(x_{1},x_{2})\in\stackrel{{\scriptstyle{1}}}{{M}}\times\stackrel{{\scriptstyle{2}}}{{M}}$, we have the following properties.

• (1)

The projections

[TABLE]

are $C^{\infty}$ submersions.

• (2)

$dim(\stackrel{{\scriptstyle{1}}}{{M}}\times\stackrel{{\scriptstyle{2}}}{{M}})=dim\stackrel{{\scriptstyle{1}}}{{M}}+dim\stackrel{{\scriptstyle{2}}}{{M}}$.

The warped product manifold of two Finslerian manifolds is defined as follows.

Definition 4.4**.**

Let $(\stackrel{{\scriptstyle{1}}}{{M}},\stackrel{{\scriptstyle{1}}}{{F}})$ and $(\stackrel{{\scriptstyle{2}}}{{M}},\stackrel{{\scriptstyle{2}}}{{F}})$ be two Finslerian manifolds. Let $f$ be a positive $C^{\infty}$ function on $\stackrel{{\scriptstyle{1}}}{{M}}$. The warped product of $(\stackrel{{\scriptstyle{1}}}{{M}},\stackrel{{\scriptstyle{1}}}{{F}})$ and $(\stackrel{{\scriptstyle{2}}}{{M}},\stackrel{{\scriptstyle{2}}}{{F}})$ is a manifold $M=\stackrel{{\scriptstyle{1}}}{{M}}\times_{f}\stackrel{{\scriptstyle{2}}}{{M}}$ equipped with the Finslerian metric

[TABLE]

such that for any vector tangent $y\in T_{x}M$, with $x=(x_{1},x_{2})\in M$ and $y=(y_{1},y_{2})$,

[TABLE]

where $\stackrel{{\scriptstyle{1}}}{{p}}$ and $\stackrel{{\scriptstyle{2}}}{{p}}$ are respectively the projections of $\stackrel{{\scriptstyle{1}}}{{M}}\times\stackrel{{\scriptstyle{2}}}{{M}}$ onto $\stackrel{{\scriptstyle{1}}}{{M}}$ and $\stackrel{{\scriptstyle{2}}}{{M}}$.

The function $F$ defined in (4.7) and (4.8) is a Finslerian manifold. More precisely,

• (i)

$F$ is $C^{\infty}$ on $\mathring{T}\stackrel{{\scriptstyle{1}}}{{M}}\times\mathring{T}\stackrel{{\scriptstyle{2}}}{{M}}$ since $\stackrel{{\scriptstyle{1}}}{{F}}$ and $\stackrel{{\scriptstyle{2}}}{{F}}$ are respectively $C^{\infty}$ on $\mathring{T}\stackrel{{\scriptstyle{1}}}{{M}}$ and $\mathring{T}\stackrel{{\scriptstyle{2}}}{{M}}$.

• (ii)

$F$ is homogeneous of degree $1$ in $y=(y_{1},y_{2})\in T_{x}M$.

• (iii)

If $n_{1}$ and $n_{2}$ are respectively the dimensions of $(\stackrel{{\scriptstyle{1}}}{{M}},\stackrel{{\scriptstyle{1}}}{{F}})$ and $(\stackrel{{\scriptstyle{2}}}{{M}},\stackrel{{\scriptstyle{2}}}{{F}})$, each element of the Hessian matrix $(g_{ij}(x,y))_{1\leq i,j\leq n_{1}+n_{2}}$ of $\frac{1}{2}F^{2}$, has the following form:

[TABLE]

for every point $(x,y)=(x_{1},x_{2},y_{1},y_{2})\in\mathring{T}\stackrel{{\scriptstyle{1}}}{{M}}\times\mathring{T}\stackrel{{\scriptstyle{2}}}{{M}}$. Thus,

[TABLE]

where $\stackrel{{\scriptstyle{1}}}{{g}}_{ij}(x_{1},y_{1}):=\frac{1}{2}\frac{\partial^{2}\stackrel{{\scriptstyle{1}}}{{F^{2}}}(x_{1},y_{1})}{\partial y_{1}^{i}\partial y_{1}^{j}}$ and $\stackrel{{\scriptstyle{2}}}{{g}}_{ij}(x_{2},y_{2}):=\frac{1}{2}f^{2}(x_{1})\frac{\partial^{2}\stackrel{{\scriptstyle{2}}}{{F^{2}}}(x_{2},y_{2})}{\partial y_{2}^{i}\partial y_{2}^{j}}.$ So the Hessian matrix of $F$ is positive definite at every point $(x_{1},x_{2},y_{1},y_{2})\in\mathring{T}\stackrel{{\scriptstyle{1}}}{{M}}\times\mathring{T}\stackrel{{\scriptstyle{2}}}{{M}}$ because the Hessian matrices of $\stackrel{{\scriptstyle{1}}}{{F}}$ and $\stackrel{{\scriptstyle{2}}}{{F}}$ are.

4.2. Curvatures associated with warped product Finslerian metrics

Proposition 4.5**.**

Let $(\stackrel{{\scriptstyle{1}}}{{M}},\stackrel{{\scriptstyle{1}}}{{F}})$ and $(\stackrel{{\scriptstyle{2}}}{{M}},\stackrel{{\scriptstyle{2}}}{{F}})$ be two Finslerian manifolds. On a warped product manifold $M=\stackrel{{\scriptstyle{1}}}{{M}}\times_{f}\stackrel{{\scriptstyle{2}}}{{M}}$, if $\stackrel{{\scriptstyle{1}}}{{\xi}}\in\Gamma(\stackrel{{\scriptstyle{1}}}{{\pi^{*}}}T\stackrel{{\scriptstyle{1}}}{{M}})$, $\stackrel{{\scriptstyle{2}}}{{\xi}}\in\Gamma(\stackrel{{\scriptstyle{2}}}{{\pi^{*}}}T\stackrel{{\scriptstyle{2}}}{{M}})$ and $\stackrel{{\scriptstyle{1}}}{{X}}\in\chi(\mathring{T}\stackrel{{\scriptstyle{1}}}{{M}})$ then

• (i)

$\nabla_{\stackrel{{\scriptstyle{1}}}{{X}}}\stackrel{{\scriptstyle{1}}}{{\xi}}=\stackrel{{\scriptstyle{1}}}{{\nabla}}_{\stackrel{{\scriptstyle{1}}}{{X}}}\stackrel{{\scriptstyle{1}}}{{\xi}}$* where $\stackrel{{\scriptstyle{1}}}{{\nabla}}$ is the Chern connection associated with $(\stackrel{{\scriptstyle{1}}}{{M}},\stackrel{{\scriptstyle{1}}}{{F}})$.*

• (ii)

$\nabla_{\stackrel{{\scriptstyle{1}}}{{X}}}\stackrel{{\scriptstyle{2}}}{{\xi}}=\frac{1}{f}\stackrel{{\scriptstyle{1}}}{{X}}(f)\stackrel{{\scriptstyle{2}}}{{\xi}}$.

As a direct consequence, we have

Corollary 4.6**.**

Let $(\stackrel{{\scriptstyle{1}}}{{M}},\stackrel{{\scriptstyle{1}}}{{F}})$ and $(\stackrel{{\scriptstyle{2}}}{{M}},\stackrel{{\scriptstyle{2}}}{{F}})$ be two Finslerian manifolds. On a warped product manifold $M=\stackrel{{\scriptstyle{1}}}{{M}}\times_{f}\stackrel{{\scriptstyle{2}}}{{M}}$, if $\stackrel{{\scriptstyle{1}}}{{\xi}},\stackrel{{\scriptstyle{1}}}{{\eta}}\in\Gamma(\stackrel{{\scriptstyle{1}}}{{\pi^{*}}}T\stackrel{{\scriptstyle{1}}}{{M}})$, $\stackrel{{\scriptstyle{1}}}{{X}},\stackrel{{\scriptstyle{1}}}{{Y}}\in\chi(\mathring{T}\stackrel{{\scriptstyle{1}}}{{M}})$ and $\stackrel{{\scriptstyle{2}}}{{X}}\in\chi(\mathring{T}\stackrel{{\scriptstyle{2}}}{{M}})$ then

• (i)

$\textbf{R}(\stackrel{{\scriptstyle{1}}}{{\xi}},\stackrel{{\scriptstyle{1}}}{{\eta}},\stackrel{{\scriptstyle{1}}}{{X}},\stackrel{{\scriptstyle{1}}}{{Y}})=\stackrel{{\scriptstyle{1}}}{{\textbf{R}}}(\stackrel{{\scriptstyle{1}}}{{\xi}},\stackrel{{\scriptstyle{1}}}{{\eta}},\stackrel{{\scriptstyle{1}}}{{X}},\stackrel{{\scriptstyle{1}}}{{Y}})$.

• (ii)

$\textbf{R}(\stackrel{{\scriptstyle{1}}}{{\xi}},\stackrel{{\scriptstyle{1}}}{{\eta}},\stackrel{{\scriptstyle{2}}}{{X}},\stackrel{{\scriptstyle{1}}}{{Y}})=0$.

4.3. Main results

We prove the following.

Theorem 4.7**.**

Let $F$ and $\widetilde{F}$ be two Finslerian metrics on a manifold $M$ of dimension $n\geq 3$. A conformal deformation $\widetilde{F}$ of $F$, with $\widetilde{F}=\varphi^{-1}F$, preserves the $R$-Einstein criterion if and only if :

• (1)

$\varphi$* is constant if $M$ is (locally) closed with $\widetilde{\textbf{Ric}}_{\widetilde{F}}^{H}=\textbf{Ric}_{F}^{H}$.*

• (2)

$\varphi$* is everywhere non-constant in a neighborhood $U$ of a point $x\in M$ if $(M,\varphi^{-1}F)$ is a Finslerian cylinder of the form \Big{(}(0,\varepsilon)\times\stackrel{{\scriptstyle{2}}}{{M}},\sqrt{t^{2}+(\varphi^{{}^{\prime}}(t))^{2}\stackrel{{\scriptstyle{2}}}{{F}}^{2}}\Big{)}, with $\varphi$ depending only on $t\in(0,\varepsilon)$.*

Proof.

Let $\widetilde{F}:=e^{u}F$ with $e^{u}=\varphi^{-1}$ be a conformal deformation of $F$. We can show that, for the conformal factor $\varphi$, the equation (4.6) takes the form $\nabla_{j}\nabla_{i}\varphi=fg_{ij}$ for some $f\in C^{\infty}(\mathring{T}M,\mathbb{R})$. Precisely, we have

[TABLE]

where f:=\frac{1}{n}\Big{[}\nabla^{d}\nabla_{d}{\varphi}-\frac{(n-1)}{2(n-2)F}\left(\nabla_{r}\varphi\nabla^{q}\varphi\right)\frac{\partial(F^{2}g^{rs}-2y^{r}y^{s})}{\partial y^{q}}g^{kl}\mathcal{A}_{skl}\Big{]}.

$(1)$ If $\widetilde{F}=\varphi^{-1}F$ on $U\subseteq M$ closed and $\widetilde{\textbf{Ric}}_{\widetilde{F}}^{H}=\textbf{Ric}_{F}^{H}$ then $\widetilde{\textbf{E}}_{\widetilde{F}}^{H}=\textbf{E}_{F}^{H}$. As shown in [18], $\varphi$ is constant.

Conversely, set $\varphi=e^{-u}$. If $\varphi$ is constant then from the Proposition 4.2, $\widetilde{\textbf{E}}_{\widetilde{F}}^{H}=\textbf{E}_{F}^{H}$.

$(2)$ Define $\varphi:(0,\varepsilon)\times\stackrel{{\scriptstyle{2}}}{{M}}\longrightarrow(0,\infty)$ by $\varphi(t,p)=\varphi(t)$. Then

[TABLE]

and

[TABLE]

The relation (4.14) shows that $\nabla_{t}\nabla_{t}\varphi=\ddot{\varphi}(t)g_{tt}$.

Now we show that $\nabla_{\alpha}\nabla_{\beta}\varphi=f\stackrel{{\scriptstyle{2}}}{{g}}_{\alpha\beta}$ for $\alpha,\beta=1,...,n-1$. We have

[TABLE]

Since $g_{\alpha n}=0=\frac{\delta g_{nl}}{\delta x^{\alpha}}$, we obtain from relation (4.15)

[TABLE]

That is $\nabla_{\alpha}\nabla_{\beta}\varphi=\ddot{\varphi}\stackrel{{\scriptstyle{2}}}{{g}}_{\alpha\beta},\alpha,\beta=1,...,n$.

Conversely, if $\varphi$ is everywhere non-constant on $(M,F)$ and if $(M,\widetilde{F})$ is a Finslerian cylinder of the form \Big{(}(0,\varepsilon)\times\stackrel{{\scriptstyle{2}}}{{M}},\sqrt{t^{2}+(\varphi^{{}^{\prime}}(t))^{2}\stackrel{{\scriptstyle{2}}}{{F}}^{2}}\Big{)} then the equation (4.12) holds. It follows from the Proposition 4.2 and by setting $e^{u}=\varphi^{-1}$ that $\widetilde{\textbf{E}}_{\widetilde{F}}^{H}=\textbf{E}_{F}^{H}$. ∎

Thus, we claim.

Corollary 4.8**.**

A Finslerian $R$-Einstein space $(M,F)$ is locally conformal to an other $R$-Einstein space $(M,\widetilde{F})$, with $\widetilde{F}=\varphi^{-1}F$, if and only if

• (1)

the conformal factor is constant if $M$ is (locally) closed with $\widetilde{\textbf{Ric}}_{\widetilde{F}}^{H}=\textbf{Ric}_{F}^{H}$.

• (2)

the conformal factor $\varphi$ is everywhere non-constant in a neighborhood $U$ of a point $x\in M$ if $(M,\varphi^{-1}F)$ has the form \Big{(}(0,\varepsilon)\times\stackrel{{\scriptstyle{2}}}{{M}},\sqrt{t^{2}+(\varphi^{{}^{\prime}}(t))^{2}\stackrel{{\scriptstyle{2}}}{{F}}^{2}}\Big{)}, with $\varphi$ depending only on $t\in(0,\varepsilon)$.

Remark 4.9**.**

The last corollary is a particular case of the Theorem 4.7.

5. Example

Let $\varphi:(0,\pi)\longrightarrow(0,\infty)$ be a $C^{\infty}$ map such that $\varphi(t)=cost+c$ with $c\in(1,\infty)$ and, $\mathbb{S}^{3}$ and $\mathbb{S}^{2}$ the unit spheres. Consider the warped product $M=(0,\pi)\times_{\varphi^{\prime}}\mathbb{S}^{2}$ and the map $i:(M,F)\longrightarrow(\mathbb{S}^{3},F_{0})$ defined by $i(t,z^{1},z^{2})=(cost,z^{1}sint,z^{2}sint)$. We can show that $i$ is a diffeomorphism and a local isometry. For $y=(y^{1},y^{2},y^{3})\in\mathbb{S}^{3}$ and $z=(z^{1},z^{2})\in\mathbb{S}^{2}$, we have $y=(cost,z^{1}sint,z^{2}sint)$. Then,

[TABLE]

The fundamental tensor associated with $F_{0}$ is

[TABLE]

Hence, by the formula (4.8), $F_{0}(y)=\sqrt{t^{2}+sin^{2}t\stackrel{{\scriptstyle{2}}}{{F}}^{2}}$ where $\stackrel{{\scriptstyle{2}}}{{F}}$ is the Finslerian metric on $\mathbb{S}^{2}$.