1 Introduction

In early days of Einstein’s relativity theory, Kasner studied the possibility of two fields both obeying Einstein’s equation of gravitation having the same light rays. He proved that if an Einstein manifold is conformal to a flat space, then it is isometric to a flat space. The strong ties between conformal gradient fields and conformal transformations between Einstein spaces made the study of these special vector fields important. This goes back to Brinkman’s investigation in 1920’s [5], where he showed that the conformal characteristic function of such a transformation has conformal gradient. In Riemannian geometry, conformal gradient fields are essential solutions of the following differential equation

$$\begin{aligned} \nabla ^{2}f=\left( \frac{\Delta f}{n}\right) g. \end{aligned}$$
(1)

This equation has been studied by Fialkow, Yano, Obata and others. Equation (1) helps to prove that for the fuliation \({\mathcal {F}}\) of M whose leaves are the connected components of the fibers of f, the transverse submanifolds are perpendicular to leaves and the metric tensor has warped product representation in foliated chart [11]. This raises the question of how this result can be generalized to Finsler geometry, c.f. [4, 7].

A generalization of warped product structures to Finsler geometry was initiated by Asanov in order to find a reversible Finsler metric on \({\mathbb {R}}\times M\), when M is a Riemannian manifold (cf. [2]), and appeared in full content for reversible Finsler functions in [8]. Warped product manifolds in Finsler geometry are defined by a condition on Finsler function which is proved to impose much the same structure on Finsler metric tensor as in the Riemannian case. Although this warped product of Finsler functions is not necessarily smooth on vectors of the form \((v_{1},0), (0,v_{2})\in TM\times TN\) and therefore is not a Finsler function on \(M\times N\) by usual definition, it is shown in [6] that it can be used to define a Hamiltonian structure on \(TM_{0}\times TN_{0}\).

As the first step in answering the question, some basic results for Finsler manifolds obtaining a conformal gradient field are stated in this note after reiterating the global and local definitions of conformal gradient fields. This chain of statements concludes with the following main theorem.

Theorem 1.1

Let \(f:M\longrightarrow {\mathbb {R}}\) be a \(C^{\infty }\) function on the Finsler manifold (M, g), \( {{\,\mathrm{grad}\,}}f\) a conformal vector field on M and p an ordinary point of \( {{\,\mathrm{grad}\,}}f\). Then the norm of \( {{\,\mathrm{grad}\,}}f\) is locally constant on the \(f-\)hypersurface of f which contains p.

Afterwards, It will be shown that how the geodesics of this warped product structure, as the extremal points of related energy functional, are related to those of constructing Finsler manifolds in Theorem 1.2.

Theorem 1.2

Let \(\alpha :[0,1]\longrightarrow M\) and \(\beta :[0,1]\longrightarrow N\) be regular curves on \((M,F_{1})\) and \((N,F_{2})\), which are extremal points of energy functional related to \(L_{1}\) and \(L_{2}\) respectively. Then \((\alpha ,\beta ):[0,1]\longrightarrow (M\times N,F= F_{1}\times _{f}F_{2})\) is an extremal point of the energy functional related to \(L=1/2 F^{2}\), if and only if f is a constant function on M.

2 Finsler structure

Let \(\pi :TM_{0}\longrightarrow M\) be slit tangent bundle, where (M, F) is a \(C^{\infty }\) connected differentiable Finsler manifold as defined in [3]. More precisely, \(F:TM\longrightarrow {\mathbb {R}}\) is a positive function that is \(C^{\infty }\) away from zero section, positive homogeneous of degree one on fibers and strongly convex on \(TM_{0}\).

The canonical linear mapping is defined by \(\varrho (v)=(u,\pi _{*u}(v))\) for each \(u\in TM_{0}\) and \(v\in T_{u}TM_{0}\). For each \(u\in TM_{0}\), \(\ker \pi _{*u}=({\mathcal {V}}TM_{0})_{u}\) and therefore \(\varrho \) defines a bundle isomorphism from \({\mathcal {H}}TM_{0}\) to \(\pi ^{*}TM\), which will be also denoted by \(\varrho \) through this note.

Let \(\nabla \) be a covariant derivative in \(\pi ^{*}TM\) and \( l \) radial vector field, which is a section of \(\pi ^{*}TM\) globally defined on \(TM_{0}\) with local expression \( l :={\dot{x}}^{i}\frac{\partial }{\partial x^{i}}\) in every coordinate system. \(\nabla \) is said to be regular if linear mapping defined by \(\mu ({\tilde{X}}) :=\nabla _{{\tilde{X}}} l \) vanishes on \({\mathcal {H}}TM_{0}\). This regularity condition means that \(\mu \) defines a bundle isomorphism from \({\mathcal {V}}TM_{0}\) to \(\pi ^{*}TM\), which will be denoted by \(\mu \) as well.

The torsion and curvature of the regular connection \(\nabla \) are given by

$$\begin{aligned} \tau ({\tilde{X}},{\tilde{Y}})&=\nabla _{{\tilde{X}}}\varrho ({\tilde{Y}}) -\nabla _{{\tilde{Y}}}\varrho ({\tilde{X}})-\varrho [{\tilde{X}},{\tilde{Y}}],&\\ \Omega ({\tilde{X}},{\tilde{Y}})Z&= \nabla _{{\tilde{X}}}\nabla _{{\tilde{Y}}}Z-\nabla _{{\tilde{Y}}} \nabla _{{\tilde{X}}}Z-\nabla _{[{\tilde{X}},{\tilde{Y}}]}Z,&\end{aligned}$$

where \({\tilde{X}}, {\tilde{Y}}\in {\mathfrak {X}}(TM_{0})\) and \(Z\in \Gamma (\pi ^{*}TM)\).

The torsion gives two torsion tensors S and T, which are defined for each \(X, Y\in \Gamma (\pi ^{*}TM)\) by \(S(X, Y)=\tau (\varrho ^{-1}(X), \varrho ^{-1}(Y))\) and \( T(X, Y)=\tau (\mu ^{-1}(X),\varrho ^{-1}(Y))\).

Let g be the Riemannian metric defined on the bundle \(\pi ^{*}TM\) by \(g_{ij}(x,{\dot{x}})=\frac{1}{2}(\dfrac{\partial ^{2} F}{\partial {\dot{x}}^{i}\partial {\dot{x}}^{j}})\). Cartan connection is the unique metric compatible regular connection, for which \(S=0\) and \(g\big (T(X,Y),Z)\big )=g\big (T(X,Z),Y)\). Chern connection is the unique torsion free regular connection \({\tilde{\nabla }}\), for which \(({\tilde{\nabla }}_{_{\mu ^{-1}(Z)}}g)(X,Y)=2C(Z,X,Y)\), where C is Cartan torsion tensor with the component functions \(C_{ijk}:=1/2 \frac{\partial g_{ij}}{{\dot{x}}^{k}}\), cf. [1].

Let \(\omega ^{j}_{i}\) be the Chern connection 1-forms related to the local basis sections \(\{\dfrac{\partial }{\partial x^{i}}, i=1 , \ldots ,n\}\), defined by \({\tilde{\nabla }}\dfrac{\partial }{\partial x^{i}}=\omega ^{j}_{i}\dfrac{\partial }{\partial x^{j}}\) and \({\tilde{\nabla }}dx^{i}=-\omega _{j}^{i}dx^{j}\). Then \(\omega ^{j}_{i}=\Gamma ^{j}_{ ik}dx^{k}\), where \(\Gamma ^{i}_{\,jk}=1/2 g^{il}(\dfrac{\delta g_{lj}}{\delta x^{k}}+\dfrac{\delta g_{lk}}{\delta x^{j}}-\dfrac{\delta g_{jk}}{\delta x^{i}})\) and \(\{\frac{\delta }{\delta x^{i}}:=\frac{\partial }{\partial x^{i}}-G^{j}_{i}\frac{\partial }{\partial {\dot{x}}^{j}}, i=0,..,n\}\) is the smooth local basis of \(\Gamma ({\mathcal {H}}TM_{0})\). The 1-forms of Cartan connection are given by \(\omega _{j}^{i}+ C_{j\,\,k}^{\,i}\delta {\dot{x}}^{k}\), where \(C_{ijk}:=1/2 \frac{\partial g_{ij}}{{\dot{x}}^{k}}\) are components of Cartan tensor and \(\delta {\dot{x}}^{i}:=d{\dot{x}}^{i}+G^{i}_{j}dx^{j}\).

3 Conformal gradient

In Riemannian geometry it is well known that, if the \(C^{\infty }\) function f has conformal gradient, then trajectories of \( {{\,\mathrm{grad}\,}}f\) are geodesic arcs except at stationary points of f. A geodesic curve containing such an arc is called f-curve. In a neighborhood of each ordinary point of f the family of f-curves form the normal congruence of f- hypersurfaces, which are connected components of the level sets of f defined by \(f=constant \). This fact helps to choose a local coordinate system known as adapted coordinate system in a neighborhood of every ordinary point, in which the Riemannian metric has a warped product structure. The first coordinate of adapted coordinate system is equal to the arc length of f-curves and the rest \(n-1\) coordinates belong to f-hypersurfaces.

It will be shown in Example 1 that the same construction will not result in a warped product structure for a Finsler manifold obtaining the conformal gradient field \( {{\,\mathrm{grad}\,}}f\). However, it will be proven that the trajectories of \( {{\,\mathrm{grad}\,}}f\) are still geodesic arcs in a neighborhood of each ordinary point and the length of \( {{\,\mathrm{grad}\,}}f\) is locally constant on f-hypersurfaces.

Let (M, g) be a Finsler manifold, V a smooth vector field on M and \({\hat{V}}\in {\mathfrak {X}}(TM)\) its complete lift as defined in [12]. If \(\left\{ \phi _{t}\right\} _{t\in I}\) be the local 1-parameter group of diffeomorphisms corresponding to the local flow of V, then \(\left\{ \phi _{t_{*}}\right\} _{t\in I}\) is the local \(1-\)parameter group of diffeomorphisms generated by \({\hat{V}}\). It can be shown that \(\varrho ({\hat{V}})=V\) and \(\mu ({\hat{V}})=\nabla _{\hat{ l }}V\), where \(\hat{ l }:=\varrho ^{-1}( l )\) is a horizontal vector field on \(TM_{0}\) with local expression \(\hat{ l }={\dot{x}}^{i}\frac{\delta }{\delta x^{i}}\).

Lemma 3.1

For a (0, k)-tensor W in Finsler sense, Lie derivative of W along \({\hat{V}}\) is obtained by the following relation

$$\begin{aligned} ({\mathcal {L}}_{_{{\hat{V}}}}W)(X_{1},X_{2} , \ldots ,X_{k})&=(\nabla _{{\hat{V}}}W)(X_{1},X_{2} , \ldots ,X_{k})\nonumber \\&\quad +\Sigma _{i}W(X_{1} , \ldots ,X_{i-1},\nabla _{\varrho ^{-1}(X_{i})}V, X_{i+1} , \ldots ,X_{k})\nonumber \\&\quad +\Sigma _{i}W\left( X_{1} , \ldots ,X_{i-1},T\left( \mu \left( {\hat{V}}\right) ,X_{i}\right) , X_{i+1} , \ldots ,X_{k}\right) . \end{aligned}$$
(2)

Proof

Let W be a Finsler tensor field of type (0, k). This means that W is a k-linear mapping from \((\Gamma (\pi ^{*}TM))^{k}\) to \(C^{\infty }(TM_{0})\). Then for \(X_{1},X_{2} , \ldots ,X_{k}\in \Gamma (\pi ^{*}TM)\)

$$\begin{aligned} ({\mathcal {L}}_{_{{\hat{V}}}}W)(X_{1},X_{2} , \ldots ,X_{k})&={\hat{V}}(W(X_{1},X_{2} , \ldots ,X_{k}))\nonumber \\&\quad -\Sigma _{i}W(X_{1} , \ldots ,X_{i-1},\varrho [{\hat{V}},{\tilde{X}}_{i}], X_{i+1} , \ldots ,X_{k}),&\end{aligned}$$
(3)

where \(\varrho ({\tilde{X}}_{i})=X_{i}\). Since \(\varrho ({\mathcal {H}}{\hat{V}})=\varrho ({\hat{V}})=V\), the component functions of \({\mathcal {H}}{\hat{V}}\) are constant on each fiber of \(TM_{0}\) and therefore \(\varrho [{\mathcal {H}}{\hat{V}},{\mathcal {V}}{\tilde{X}}_{i}]=0\). On the other hand, \([{\mathcal {V}}{\hat{V}},{\mathcal {V}}{\tilde{X}}_{i}]\) is a vertical vector filed, which implies that \(\varrho [{\mathcal {V}}{\hat{V}},{\mathcal {V}}{\tilde{X}}_{i}]=0\). Therefore, \(\varrho [{\hat{V}},{\tilde{X}}_{i}]=\varrho [{\hat{V}},\varrho ^{-1}(X_{i})]\).

According to the fact that \(S(V,X_{i})=0\), one obtains the following relation for \(\varrho [{\mathcal {H}}{\hat{V}},\varrho ^{-1}(X_{i})]=\varrho [\varrho ^{-1}(V),\varrho ^{-1}(X_{i})]\)

$$\begin{aligned} \varrho [{\mathcal {H}}{\hat{V}},\varrho ^{-1}(X_{i})]= \nabla _{{\mathcal {H}}{\hat{V}}}X_{i}-\nabla _{\varrho ^{-1}(X_{i})}V. \end{aligned}$$
(4)

Since \(\mu ({\mathcal {V}}{\hat{V}})=\mu ({\hat{V}})=\nabla _{\hat{ l }}V\), then \( \tau ({\mathcal {V}}{\hat{V}},\varrho ^{-1}(X_{i}))=T(\nabla _{\hat{ l }}V, X_{i})\) and

$$\begin{aligned} \varrho [{\mathcal {V}}{\hat{V}},\varrho ^{-1}(X_{i})]= \nabla _{{\mathcal {V}}{\hat{V}}}X_{i}-T(\nabla _{\hat{ l }}V, X_{i}) \end{aligned}$$
(5)

The result is obtained by replacing (4) and (5) into (3). \(\square \)

For the Finsler metric tensor g, using properties of Cartan connection and (2), it can be shown that

$$\begin{aligned} ({\mathcal {L}}_{_{{\hat{V}}}}g)(X,Y)=g(\nabla _{\varrho ^{-1} (X)}V,Y)+g(X,\nabla _{\varrho ^{-1}(Y)}V)+2g(T(\nabla _{\hat{ l }}V,X),Y). \end{aligned}$$
(6)

This is written by the following equation in local coordinate system, cf. [7, 12].

$$\begin{aligned} {\mathcal {L}}_{_{{\hat{V}}}}g_{ij}=\nabla _{i}V_{j}+\nabla _{j}V_{i} +2(\nabla _{0}V^{l})C_{lij}, \end{aligned}$$
(7)

where \(\nabla _{i}=\nabla _{\frac{\delta }{\delta x^{i}}}\), \(\nabla _{0}={\dot{x}}^{i}\nabla _{i}=\nabla _{\hat{ l }}\) and \((g_{ij})\) and its inverse matrix \((g^{ij})\) are used to lower or raise the indices. For example \(V_{j}=g_{ij}V^{i}\) are components of the unique tensor filed \(V^{\#}:\Gamma (\pi ^{*}TM)\longrightarrow C^{\infty }(TM_{0})\) associated to \(V\in \Gamma (\pi ^{*}TM)\) by the following relation

$$\begin{aligned} V^{\#}(X)=g(V,X),\, \forall X\in \Gamma (\pi ^{*}TM) \end{aligned}$$

It is worth mentioning that components of the torsion tensor T in Cartan connection are equal to \(C_{ij}^{k}=g^{kl}C_{ijl}\).

V is said to be conformal if \({\mathcal {L}}_{_{{\hat{V}}}}g_{ij}=2\rho g_{ij}\), where \(\rho \) is a real valued function on M called conformal characteristic function of V. When \(\rho \) is constant or zero, the conformal vector filed V is said to be homothetic or Killing respectively.

There is another characterization of a conformal vector field on Finsler manifold which is based on considering Chern connection as a family of maps.

Let \(y\in T_xM_0\), one can define a nondegenerate bilinear form \(g_y\) on \(T_xM\) by

$$\begin{aligned} g_y(u,v):=\frac{1}{2}\frac{\partial ^{2}}{\partial s\partial t}|_{s=t=0}F^{2}(y+su+tv), \;\forall u,v\in T_xM. \end{aligned}$$

In a local coordinate system \( g_y(u,v)=g_{ij}(x,y)u^iv^j\), where \(u=u^i\frac{\partial }{\partial x^i}\) and \(v=v^i\frac{\partial }{\partial x^i}\). If \(Y\in {\mathfrak {X}}(M)\) is a nowhere zero vector field, then \(g_Y\) defines a Riemannian metric on M as follows

$$\begin{aligned} g_Y(U,V)(x):=g_{Y(x)}(U(x),V(x)),\; \forall x\in M,\;U,V\in {\mathfrak {X}}(M). \end{aligned}$$

Similarly on can define the 3-linear function \(C_y\) by

$$\begin{aligned} C_y(u,v,w):=\frac{1}{4}\frac{\partial ^{3}}{\partial s\partial t\partial r}|_{s=t=r=0}F^{2}(y+su+tv+rw), \;\forall u,v,w\in T_xM, \end{aligned}$$

which has the local expression \(C_y(u,v,w)=C_{ijk}(x,y)u^iv^jw^k\), where \(C_{ijk}(x,y)\) are component functions of Cartan torsion.

So the fundamental tensor (or Finsler metric tensor) of (M, F) is the family \(\{g_y:=y\in T_xM_0, x\in M\}\) of nondegenerate symmetric bilinear forms and the Cartan torsion is the family \(\{C_y:=y\in T_xM_0, x\in M\}\) of symmetric covariant tensors of order three.

The Landsberg curvature of (M, F) is defined as the family \(\{L_y:=y\in T_xM_0,\ x\in M\}\) of symmetric covariant tensors of order three, where for each \(y\in T_xM_0\) the coefficient functions of \(L_y\), i.e. \(L_{ijk}(x,y)\) is defined by

$$\begin{aligned} L_{ijk}(x,y):=-\frac{1}{2}{\dot{x}}^m(y)g_{ml}(x,y)\frac{\partial ^2 G^l_i}{\partial {\dot{x}}^j \partial {\dot{x}}^k}(x,y). \end{aligned}$$

Definition 3.2

[9] A Finsler connection \(\nabla \) on (M, F) is a family

$$\begin{aligned} \{\nabla ^y:T_xM\times {\mathfrak {X}}(M)\longrightarrow T_xM, \; y\in T_xM_0, x\in M\} \end{aligned}$$

of maps with the following properties:

  1. (a)

    \(\nabla ^{\lambda y}_{u}V=\nabla ^y_{u}V,\; \forall \lambda >0,\; u\in T_xM,\; V\in {\mathfrak {X}}(M)\),

  2. (b)

    \(\nabla ^{y}_{u}fV+U=u(f)V+f\nabla ^{y}_{u}V+\nabla ^{ y}_{u}U, \forall u\in T_xM,\; f\in c^{\infty }(M),\; U,V\in {\mathfrak {X}}(M)\),

  3. (c)

    \(\nabla ^{y}_{su+v}V=s\nabla ^{y}_{u}V+\nabla ^{y}_{v}V,\;\forall u,v\in T_xM,\; s\in {{\mathbb {R}}},\; V\in {\mathfrak {X}}(M)\),

  4. (d)

    \(\nabla ^{ Y}_{U}V-\nabla ^{ Y}_{V}U=[U,V],\; \forall Y,U,V\in {\mathfrak {X}}(M)\), where Y is a nowhere zero vector filed.

  5. (e)

    For U, V, Y as above, the vector field \(\nabla ^{ Y}_{U}V\) is smooth.

Let \(\nabla \) be a connection on (M, F) with coefficient functions \(\Gamma _{ij}^{k}(x,y)\) in the local coordinate system \((x^i,{\dot{x}}^i)\) defined by

$$\begin{aligned} \nabla ^{y}_{\frac{\partial }{\partial x^i}}\dfrac{\partial }{\partial x^j}=\Gamma _{ij}^{k}(x,y)\dfrac{\partial }{\partial x^k}. \end{aligned}$$

Then for any \(u=u^i\dfrac{\partial }{\partial x^i}|_{x}\in T_xM\) and \(V\in {\mathfrak {X}}(M)\) with local expression \(V^i\dfrac{\partial }{\partial x^i}\) one has

$$\begin{aligned} \nabla ^{y}_{u}V=\{u(V^k)+u^iV^j\Gamma _{ij}^{k}(x,y)\}\dfrac{\partial }{\partial x^k}|_{x}. \end{aligned}$$

Proposition 3.3

(cf. [10]) Chern connection \({\tilde{\nabla }}\) is the unique Finsler connection with the following characteristic property

$$\begin{aligned} w(g_Y(U,V))=g_Y({\tilde{\nabla }}^Y_wU,V)+g_Y(U, {\tilde{\nabla }}^Y_wV)+2C_Y(U,V,{\tilde{\nabla }}^Y_wY). \end{aligned}$$

for all \(x\in M\), \(w\in T_xM\), \(U,V,\in {\mathfrak {X}}(M)\) and nowhere zero vector field \(Y\in {\mathfrak {X}}(M)\).

The coefficients of Chern connection in the local coordinate system \((x^i,{\dot{x}}^i)\) are equal to

$$\begin{aligned} \Gamma _{ij}^{k}(x,y)=G^k_{ij}(x,y)-L^{k}_{ij}(x,y), \end{aligned}$$

where \(G_{ij}^{k}:=\frac{\partial G^K_i}{\partial {\dot{x}}^j }(x,y)\) are coefficients of Berwald connection and \(L^{k}_{ij}(x,y):=g^{kl}(x,y)L_{lij}\).

If \(K\in {\mathfrak {X}}(M)\) is a nowhere zero vector field and \(X,Y,V\in {\mathfrak {X}}(M)\) are arbitrary vector fields, according to the formula for computing the Lie derivative of a Riemannian metric and Proposition 3.3, it can be shown that

$$\begin{aligned} ({\mathcal {L}}_Vg_K)(X,Y)=g_K({\tilde{\nabla }}^K_XV,Y) +g_Y(X,{\tilde{\nabla }}^K_YV)+2C_Y({\tilde{\nabla }}^K_{K}V,X,Y). \end{aligned}$$

Moreover, if \(V\in {\mathfrak {X}}(M)\) is conformal, Then

$$\begin{aligned} g_{K}({\tilde{\nabla }}^{K}_{X}V,Y)+g_{K}(X,{\tilde{\nabla }}^{K}_{Y}V) +2C_{K}(X,Y,{\tilde{\nabla }}^{K}_{K}V)=\rho g_K(X,Y), \end{aligned}$$
(8)

where \(\rho \in C^{\infty }(M)\) is the conformal characteristic function of V, the vector fields X and Y are arbitrary and K is any vector field non-zero everywhere.

Definition 3.4

Suppose \(f:M\longrightarrow {\mathbb {R}}\) is a \(C^{\infty }\) function on M. At each point \(p\in M\) with \(df_{p}\ne 0\), \( {{\,\mathrm{grad}\,}}f(p) \) is defined as a vector of \(T_{p}M\), for which

$$\begin{aligned} g_{ {{\,\mathrm{grad}\,}}f(p)}(u, {{\,\mathrm{grad}\,}}f(p))=df_{p}(u), \, \forall u\in T_{p}M. \end{aligned}$$

If \(df_{p}=0\), then \( {{\,\mathrm{grad}\,}}f(p) \) is defined to be zero.

In local coordinate system \( {{\,\mathrm{grad}\,}}f(p)=g^{ij}(p, {{\,\mathrm{grad}\,}}f(p))f_{i}\frac{\partial }{\partial x^{j}}\), where \(f_{i}:=\frac{\partial f}{\partial x^{i}}\) are components of df.

Example 1

Consider the Finsler metric on \({\mathbb {R}}^{2}\) defined by the Minkowski norm

$$\begin{aligned} F((x,y),(u,v)):=\sqrt{\sqrt{u^{4}+v^{4}}+\lambda (u^{2}+v^{2})}, \end{aligned}$$

where \(\lambda \) is any non-negative constant and (x, y) indicates the coordinate on manifold and (u, v) coordinates on tangent space. A straightforward computation gives

$$\begin{aligned} (g_{ij})= \left( \begin{array}{c@{\quad }c} \lambda +\dfrac{u^{2}(u^{4}+3v^{4})}{(u^{4}+v^{4})^{\frac{3}{2}}} &{} \dfrac{-2u^{3}v^{3}}{(u^{4}+v^{4})^{\frac{3}{2}}} \\ \dfrac{-2u^{3}v^{3}}{(u^{4}+v^{4})^{\frac{3}{2}}} &{} \lambda +\dfrac{v^{2}(v^{4}+3u^{4})}{(u^{4}+v^{4})^{\frac{3}{2}}} \end{array} \right) . \end{aligned}$$
(9)

Let \(f:{\mathbb {R}}^{2}\longrightarrow {\mathbb {R}}\) be a \(C^{\infty }\) function on the Finsler manifold (M, g) defined by \(f(x,y):=ax+b\) for constants a and b. Then at each point p(x, y), \( {{\,\mathrm{grad}\,}}f(p)=(\frac{a}{\lambda +1},0),\) which is horizontal vector at each point \(p\in {\mathbb {R}}^{2}\). Moreover, level sets of f are lines parallel to y-axis.

For the Finsler manifold (M, g) and \(f\in C^{\infty }(M)\), the following Lemma shows that if \( {{\,\mathrm{grad}\,}}f\) is conformal, then its trajectories are geodesic arcs except at stationary points of f .

Lemma 3.5

Let \(f:M\longrightarrow {\mathbb {R}}\) be a \(C^{\infty }\) function on the Finsler manifold (M, g) and \( {{\,\mathrm{grad}\,}}f\) a conformal vector field on M. Then the integral curves of \( {{\,\mathrm{grad}\,}}f\) are geodesics of the Finsler structure.

Proof

Let \(c:I\longrightarrow M\) be the integral curve of \(V:= {{\,\mathrm{grad}\,}}f\). To see whether c is geodesic or not we only need to restrict the vectors and derivatives to the direction of c, which is nothing but V. Therefore, (15) becomes

$$\begin{aligned} {\mathcal {L}}_{_{{\hat{V}}}}g_{ij}=2(\nabla _{i}f_{j}+(V^{k} \nabla _{k}V^{l})C_{lij}(x,V(x))=2\rho (x)g_{ij}(x,V(x)). \end{aligned}$$

Contracting this equation with \(V^{i}g^{jk}(x,V)\) we have \(V^{i}\nabla _{i}V^{k}=2\rho (x)V^{k}\). So c is a geodesic of Finsler structure and moreover, \(V^{k}\nabla _{k}V^{l}C_{lij}(x,V(x))=0\) \(\square \)

Proposition 3.6

Let \(f:M\longrightarrow {\mathbb {R}}\) be a \(C^{\infty }\) function on the Finsler manifold (M, g) that satisfies the equation \(\nabla _{H{\tilde{X}}} {{\,\mathrm{grad}\,}}f=0\), for each \({\tilde{X}}\in \chi (TM_{0})\). Then in a neighborhood of each ordinary point of \( {{\,\mathrm{grad}\,}}f\), f-hypersurfaces are locally isometric.

Proof

Let p be an ordinary point of \( {{\,\mathrm{grad}\,}}f\) and W be a neighborhood of p with compact closure and with \( {{\,\mathrm{grad}\,}}f(q)\ne 0\) for all \(q\in W\). Hence for each \(q\in W\), \(c_{q}:=f(q)\) is a regular value of f. Let V be the restriction of \( {{\,\mathrm{grad}\,}}f\) to W and \(\phi _{t}\) the local 1-parameter group of diffeomorphisms generated by V. Denoting by \({\bar{M}}_{q}\), \(q\in W\), the connected component of \(f^{-1}(c_{q})\cap W\) containing q, \(\phi _{t}\) maps each \({\bar{M}}_{q}\) to another one. According to (11), both \({\mathcal {L}}_{_{{\hat{V}}}}g\) and \({\mathcal {L}}_{_{{\hat{V}}}}F\) vanish, since V is parallel along horizontal vector fields. This means that \(\frac{d}{dt}|_{t=0}(\phi _{t_{*}})^{*}F=0\), i.e.

$$\begin{aligned} F\circ \phi _{t_{*}}(u)=F(u), \; \forall q\in W,\, u\in T{\bar{M}}_{q}. \end{aligned}$$

\(\square \)

If the condition of Proposition 3.6 satisfies, then \( {{\,\mathrm{grad}\,}}f\) is a Killing vector filed by (11).

Proof of Theorem 1.1

Let p be an ordinary point of \( {{\,\mathrm{grad}\,}}f\) and W be a neighborhood of p with \( {{\,\mathrm{grad}\,}}f(q)\ne 0\) for all \(q\in W\). Hence for each \(q\in W\), \(c_{q}:=f(q)\) is a regular value of f. Let V be the restriction of \( {{\,\mathrm{grad}\,}}f\) to W. Since V is a non-vanishing vector field, one can choose \(K=V\) in (8) to obtain

$$\begin{aligned} g_{V}({\tilde{\nabla }}^{V}_{X}V,Y)+g_{V}(X, {\tilde{\nabla }}^{V}_{Y}V)+2C_{V}(X,Y,{\tilde{\nabla }}^{V}_{V}V)=\rho g_V(X,Y), \end{aligned}$$

for each smooth vector fields X, Y on W and hence for \(X=V\) and Y tangent to the f-hypersurface passing through p.

Then using the fact that \(C_{V}(V,X,Y,)=0\) by zero homogeneity of the fundamental tensor, and the definition of \( {{\,\mathrm{grad}\,}}(f)\) one has

$$\begin{aligned} g_{V}({\tilde{\nabla }}^{V}_V V,Y)+g_{V}(V,{\tilde{\nabla }}^{V}_{Y}V)=0 \end{aligned}$$
(10)

Let \( {{\,\mathrm{Hess}\,}}f\) be the Hessian of f with respect to the torsion free affine connection \({\tilde{\nabla }}^{V}\) on the Riemannian manifold \((W,g_V)\), which is defined by

$$\begin{aligned} {{\,\mathrm{Hess}\,}}f(X,Y):=({\tilde{\nabla }}^{V}df)(X,Y) \end{aligned}$$

for all \(X, Y\in {\mathfrak {X}}(W)\). Then one has

$$\begin{aligned} {{\,\mathrm{Hess}\,}}f(X,Y)&=({\tilde{\nabla }}^{V}df)(X,Y)&\\&=XY(f)-({\tilde{\nabla }}^{V}_XY)f \\&=YXf-({\tilde{\nabla }}^{V}_YX)f\;\text {(by torsion freeness assumption)}&\\&= {{\,\mathrm{Hess}\,}}f(Y,X). \end{aligned}$$

On the other hand, according to definitions of \( {{\,\mathrm{grad}\,}}(f)\) and Chern connection, one obtains

$$\begin{aligned} {{\,\mathrm{Hess}\,}}f(X,Y)=g_{ {{\,\mathrm{grad}\,}}(f)}(X,{\tilde{\nabla }}^{ {{\,\mathrm{grad}\,}}(f)}_{Y} {{\,\mathrm{grad}\,}}(f))=g_{ {{\,\mathrm{grad}\,}}(f)}(Y,{\tilde{\nabla }}^{ {{\,\mathrm{grad}\,}}(f)}_{X} {{\,\mathrm{grad}\,}}(f)). \end{aligned}$$

Therefore, Eq. (10) becomes

$$\begin{aligned} g_{V}(V,{\tilde{\nabla }}^{V}_{Y}V)=0. \end{aligned}$$

Equivalently,

$$\begin{aligned} Y(F^2(V))&=Y(g_V(V,V))&\\&=2g_V({\tilde{\nabla }}^{V}_YV,V)+2C_V(V,V,{\tilde{\nabla }}^{V}_YV)=0, \end{aligned}$$

which concludes the result. \(\square \)

4 Warped product Finsler manifolds and their geodesics

Warped product metrics are defined on product of two Riemannian (psuedo-Riemannian) manifolds by special modification of usual product of their metrics and can be considered as the higher dimensional generalizations of surfaces of revolution. They play an essential role in the study of conformal vector fields on Einstein spaces, which indicates the importance of a proper generalization of warped product structures to Finsler geometry. In this section, after briefly presenting the Hamiltonian structure of warped product Finsler manifolds from [6], It will be shown that how the geodesics of this warped product structure, as the extremal points of related energy functional, are related to those of constructing Finsler manifolds.

4.1 Finsler manifolds from Hamiltonian point of view

Let \(\pi :TM_{0}\longrightarrow M\) be slit tangent bundle, where (M, F) is a \(C^{\infty }\) connected differentiable Finsler manifold. Then \(L:=1/2 F^{2}\) is a lagrangian on tangent bundle that is smooth on \(TM_{0}\), and \(J:TTM_{0}\longrightarrow TTM_{0}\) be the almost tangent structure, defined for each \(v\in T_{u}TM_{0}\) by \(J(v):=\frac{d}{dt}|_{t=0}(u+t\pi _{*}(v))\). Let \(\Theta _{L}:=dL\circ J\) and \(\omega _{L}:=-d\Theta _{L}\) be Poincaré–Cartan forms defined on \(TM_{0}\). It is easy to check that \(\omega _{L}\) is a non-degenerate 2-form according to the strong convexity criterion for F. Therefore, \((TM_{0},\omega _{L})\) becomes a symplectic manifold and the triple \(\left( TM_{0}, \omega _{L}, L\right) \) is a Hamiltonian formalism. The Hamiltonian vector field related to this formalism is the vector field \(X_{L}\in \chi (TM_{0})\), which satisfies the relation \(\omega _{L}(X_{L}, Y)=Y(L)\) for all \(Y\in \chi (TM_{0})\).

Let \((U, x^{i})\) be a local chart of M, then \((\pi , {\dot{x}}^{i}:=dx^{i})\) is the corresponding bundle chart for \(TM_{0}\) and \((\pi ^{-1}(U), {\bar{x}}^{i}:=\pi \circ x^{i}, {\dot{x}}^{i})\) a local chart of \(TM_{0}\) as manifold. By direct computations one obtains

$$\begin{aligned} J&=d{\bar{x}}^{i}\otimes \frac{\partial }{\partial {\dot{x}}^i},\\ \Theta _{L}&= \frac{\partial L}{\partial {\dot{x}}^i}d{\bar{x}}^{i},&\omega _{L}&=\dfrac{\partial ^{2}L}{\partial {\dot{x}}^{i}\partial {\dot{x}}^{j}}d{\bar{x}}^{i}\wedge d{\dot{x}}^{j}+\frac{\partial ^{2}L}{\partial {\dot{x}}^{i}\partial {\bar{x}}^{j}}d{\bar{x}}^{i}\wedge d{\bar{x}}^{j},\\ X_{L}&= {\dot{x}}^{i}\frac{\partial }{\partial {\bar{x}}^i}-2 G^{i}\frac{\partial }{\partial {\dot{x}}^i},&G^{i}&=\frac{1}{4}g^{ij}\left( {\dot{x}}^{k}\frac{\partial ^{2} F^{2}}{\partial {\dot{x}}^{j}\partial {\bar{x}}^{k}}-\frac{\partial F^{2}}{\partial {\bar{x}}^{j}}\right) , \end{aligned}$$

where \((g^{ij})\) is the inverse matrix of \((g_{ij}):=(\frac{\partial ^{2} L}{\partial {\dot{x}}^{i}\partial {\dot{x}}^{j}})\).

The regular curve \(\alpha :[0,1]\longrightarrow M\) is an integral curve of \(X_{L}\) iff it is an extremal curve of the length integral \({\mathcal {L}}(\alpha )=\int ^{1}_{0}F({\dot{\alpha }}(t))dt\), iff it is the extremal value of the energy functional \({\mathcal {E}}(\alpha )=\int ^{1}_{0}L({\dot{\alpha }}(t))dt\).

4.2 Warped product

Let \((M,F_{1})\) and \((N,F_{2})\) be two Finsler manifolds with respective Finsler metrics \({\dot{g}}\) and \(\ddot{g}\), and f be a smooth real function defined on M, which is called the warping function. Then the following expression defines a function on \(TM\times TN\) which is smooth on \(TM_{0}\times TN_{0}\).

$$\begin{aligned} F(v_{1}, v_{2}):=\sqrt{F_{1}^{2}(v_{1})+f^{2}(\tau _{M}(v_{1}))F_{2}^{2}(v_{2})} \,,\, (v_{1}, v_{2})\in TM\times TN. \end{aligned}$$

F is positive homogeneous of degree one on each fiber \(T_{p}M_{0}\times T_{q}N_{0}\), \((p,q) \in M\times N\), and its related Hessian matrix is of the form

$$\begin{aligned} (g_{ij})= \left( \begin{array}{cc} {\dot{g}}_{ij}&{} 0 \\ 0 &{} f^{2}(x_{1})\ddot{g}_{ij} \end{array} \right) , \end{aligned}$$

which defines a Riemannian metric on \(\pi _{1}^{*}TM\times \pi _{2}^{*}TN\), where \(\pi _{1}:TM_{0}\longrightarrow M\) and \(\pi _{2}:TN_{0}\longrightarrow N\) are the natural projections. For the metric tensor g with components \(g_{ij}\), (M, g) is called the warped product of \((M_{1},{\dot{g}})\) and \((M_{2},\ddot{g})\) and is denoted by \(g={\dot{g}}\times _{f}\ddot{g}\) or \(F=F_{1}\times _{f}F_{2}\).

The almost tangent structure of \(TM_{0}\times TN_{0}\) is therefore defined by

$$\begin{aligned} J:T(TM_{0}\times TN_{0})&\longrightarrow T(TM_{0}\times TN_{0}) \\ v\in T_{u}(TM_{0}\times TN_{0})&\longmapsto \frac{d}{dt}|_{t=0}(u+t(\pi _{1}\times \pi _{2})_{*}(v)). \end{aligned}$$

It can be seen that \(J(v)=(J_{1}(p_{1*}(v)),J_{2}(p_{2*}(v)))\), where \(p_{1}:TM_{0}\times TN_{0}\longrightarrow TM_{0}\) and \(p_{2}:TM_{0}\times TN_{0}\longrightarrow TN_{0}\) are projections on the first and second components and \(J_{1}\) and \(J_{2}\) are almost tangent structures of \(TM_{0}\) and \(TN_{0}\). Considering \(L:=\frac{1}{2}F^{2}\) as a smooth function on \(TM_{0}\times TN_{0}\), one can define Poincaré–Cartan forms by \(\Theta _{L}=dL\circ J\) and \(\omega _{L}=-d\Theta _{L}\).

$$\begin{aligned} \Theta _{L}(v)&=\Theta _{L_{1}}(p_{1*}(v))+ f^{2}\Theta _{L_{2}}(p_{2*}(v)), \end{aligned}$$
(11)
$$\begin{aligned} \omega _{L}(v,w)&{=}\omega _{L_{1}}(p_{1*}(v),p_{1*}(w))+f^{2}\omega _{L_{2}}(p_{2*}(v),p_{2*}(w))-df^{2}\wedge \Theta _{L_{2}}(p_{1*}(v),p_{2*}(w))\nonumber \\&\quad -df^{2}\wedge \Theta _{L_{2}}(p_{2*}(v),p_{1*}(w)). \end{aligned}$$
(12)

\((TM_{0}\times TN_{0}, \omega _{L})\) is a symplectic manifold and the triple \((TM_{0}\times TN_{0},\omega _{L},L)\) is a Hamiltonian formalism. The Hamiltonian vector field \(X_{L}\) related to this formalism is equal to \(X_{1}+X_{2}\), where \(X_{1}\) (resp. \(X_{2}\)) is a vector field on \(TM_{0}\times TN_{0}\) with values in \(TTM_{0}\) (resp. \(TTN_{0}\)), i.e. \(X_{1}\in \Gamma (TTM_{0})\) and \(X_{2}\in \Gamma (TTN_{0})\).

Proposition 4.1

Let \(X_{L}=X_{1}+X_{2}\) be the Hamiltonian vector field related to the Hamiltonian formalism \((TM_{0}\times TN_{0},\omega _{L},H)\). Then

  1. (i)

    \(X_{1}=X_{L_{1}}+Y\), where \(X_{L_{1}}\) is the Hamiltonian vector field related to the Hamiltonian formalism \((TM_{0},\omega _{L_{1}},L_{1})\) and \(Y\in \Gamma ({\mathcal {V}}TM_{0})\)

  2. (ii)

    \(X_{2}=X_{L_{2}}+Z\), where \(X_{L_{2}}\) is the Hamiltonian vector field related to the Hamiltonian formalism \((TN_{0},\omega _{L_{2}},L_{2})\) and \(Z\in \Gamma ({\mathcal {V}}TN_{0})\)

Proof

Let \((U, x^{i})\) be a local chart on M, \((\pi _{1}, {\dot{x}}^{i})\) the corresponding bundle chart for \(TM_{0}\) and \((\pi _{1}^{-1}(U), x^{i}, {\dot{x}}^{i})\) a local chart on \(TM_{0}\) as manifold. Similarly, let \((V, \xi ^{\mu })\) be a local chart on N, \((\pi _{2}, {\dot{\xi }}^{\mu })\) the corresponding bundle chart for \(TN_{0}\) and \((\pi _{2}^{-1}(V), \xi ^{\mu }, {\dot{\xi }}^{\mu })\) a local chart on \(TN_{0}\) as manifold. For \(Y\in \chi (TM_{0})\) and \(Z\in \chi (TN_{0})\), according to the second relation in (11), one has

$$\begin{aligned}&\omega _{L}(X_{L},Y)=\omega _{L_{1}}(X_{1}, Y)+Y(f^{2})\Theta _{L_{2}}(X_{2}),&\\&\omega _{L}(X_{L},Z)=f^{2}\omega _{L_{2}}(X_{2}, Z)-X_{1}(f^{2})\Theta _{L_{2}}(Z).&\end{aligned}$$

On the other hand, by the definition of \(X_{L}\), \(\omega _{L_{1}}\) and \(\omega _{L_{2}}\) one obtains

$$\begin{aligned} \omega _{L}(X_{L},Y)&=Y(L_{1})+L_{2}Y(f^{2})= \omega _{L_{1}}(X_{L_{1}},Y)+L_{2}Y(f^{2}), \end{aligned}$$
(13)
$$\begin{aligned} \omega _{L}(X_{L},Z)&=f^{2}Z(L_{2})=f^{2}\omega _{L_{2}}(X_{L_{2}},Z). \end{aligned}$$
(14)

By putting \(Y=\frac{\partial }{\partial {\dot{x}}^{k}}\), \(Y=\frac{\partial }{\partial x^{k}}\), \(Z=\frac{\partial }{\partial {\dot{\xi }}^{\nu }}\) and \(Z=\frac{\partial }{\partial \xi ^{\nu }}\) one by one and after some computation one can see that \(X_{1}\) and \(X_{2}\) has the following expressions on \(\pi _{1}^{-1}(U)\times \pi _{2}^{-1}(V)\)

$$\begin{aligned} X_{1}&={\dot{x}}^{i}\frac{\partial }{\partial x^{i}}-2G^{i}\frac{\partial }{\partial {\dot{x}}^{i}},&X_{2}={\dot{\xi }}^{\mu }\frac{\partial }{\partial \xi ^{\mu }}-2G^{\mu }\frac{\partial }{\partial {\dot{\xi }}^{\mu }}, \end{aligned}$$

where \(G^{i}={\dot{G}}^{i}-\frac{1}{2}L_{2}(f^{2})^{i}\), \(G^{\mu }={\ddot{G}}^{\mu }+\frac{1}{2}{\dot{x}}^{i}\frac{\partial \ln f^{2}}{\partial x^{i}}\dot{\xi ^{\mu }}\), and \({\dot{G}}^{i}\) and \({\ddot{G}}^{\mu }\) are spray coefficients corresponding to the restricted spray \(X_{L_{1}}\) and \(X_{L_{2}}\) respectively. Therefore, \(X_{1}=X_{L_{1}}+L_{2}(f^{2})^{i}\frac{\partial }{\partial {\dot{x}}^{i}}\) and \(X_{2}=X_{L_{2}}-{\dot{x}}^{i}\frac{\partial \ln f^{2}}{\partial x^{i}}\dot{\xi ^{\mu }}\) \(\square \)

Corollary 4.2

The functions \(G^{i}\) and \(G^{\mu }\), \(i=1 , \ldots ,m\) and \(\mu =1 , \ldots ,n\), are positive homogeneous of degree 2 on each fiber of \(\pi _{1}^{-1}(U)\times \pi _{2}^{-1}(V)\). Thus \(X_{L}\) defines a restricted spray on \((TM_{0}\times TN_{0}, \omega _{L})\) in the sense that its integral curves are of the form \(({\dot{\alpha }},{\dot{\beta }})\), where \(\alpha \) and \(\beta \) are regular curves on M and N, and for every re-parametrization \((\alpha (\lambda t),\beta (\lambda t))\) with \(\lambda >0\), \(({\dot{\alpha }}(\lambda t),{\dot{\beta }}(\lambda t))\) will still be an integral curve of \(X_{L}\).

Proof of Theorem 1.2

\((\alpha , \beta )\) is an extremal point of the energy functional, if and only if it satisfies the following Euler–Lagrange equations

$$\begin{aligned} E_{i}(L)&:=\frac{\partial L}{\partial x^{i}}-\frac{d}{dt}\frac{\partial L}{\partial \dot{x^{i}}}=0,\nonumber \\ E_{\mu }(L)&:=\frac{\partial L}{\partial \xi ^{\mu }}-\frac{d}{dt}\frac{\partial L}{\partial \dot{\xi ^{\mu }}}=0. \end{aligned}$$
(15)

By direct calculation one obtains

$$\begin{aligned} \frac{\partial L}{\partial x^{i}}&= \frac{\partial L_{1}}{\partial x^{i}}+L_{2}\frac{\partial f^{2}}{\partial x^{i}},\quad \;\frac{\partial L}{\partial \xi ^{\mu }}=f^{2}\frac{\partial L}{\partial \xi ^{\mu }},\nonumber \\ \frac{d}{dt}\frac{\partial L}{\partial \dot{x^{i}}}&=\frac{d}{dt}\frac{\partial L_{1}}{\partial \dot{x^{i}}},\quad \quad \ \quad \; \frac{d}{dt}\frac{\partial L}{\partial \dot{\xi ^{\mu }}}=f^{2}\frac{d}{dt}\frac{\partial L_{2}}{\partial \dot{\xi ^{\mu }}}+{\dot{x}}^{i}\frac{\partial f^{2}}{\partial x^{i}}\frac{\partial L_{2}}{\partial \dot{\xi ^{\mu }}}. \end{aligned}$$
(16)

Substituting (16) in (15), gives the following equations for extremal points of the energy functional related to L

$$\begin{aligned}&E_{i}(L_{1})+L_{2}\frac{\partial f^{2}}{\partial x^{i}}=0,&\\&f^{2}E_{\mu }(L_{2})-{\dot{x}}^{i}\frac{\partial f^{2}}{\partial x^{i}}\frac{\partial L_{2}}{\partial \dot{\xi ^{\mu }}}=0.&\end{aligned}$$

If \(\alpha \) satisfies \(E_{i}(L_{1})=0\) and \(\beta \) satisfies \(E_{\mu }(L_{2})=0\), the above Euler–Lagrange equations implies that the necessary and sufficient condition for \((\alpha ,\beta )\) to be an extremal point of the energy functional is that f be a constant function on M, considering the fact that \(\beta \) is a regular curve and therefore \(L_{2}(\beta , \beta ')\ne 0\). \(\square \)

Corollary 4.3

Let \(\alpha :[0,1]\longrightarrow M\) and \(\beta :[0,1]\longrightarrow N\) be regular curves on \((M,F_{1})\) and \((N,F_{2})\), where \(\alpha \) is an extremal point of energy functional related to \(L_{1}\) and \((\alpha ,\beta ):[0,1]\longrightarrow (M\times N,F= F_{1}\times _{f}F_{2})\) is an extremal point of the energy functional related to L. Then \(\beta \) is an an extremal point of energy functional related to \(L_{2}\)