Some Curvature Properties of Finsler Warped Product Metrics

Abstract

The class of warped product metrics can often be interpreted as key space models for the general theory of relativity and theory of space-time. In this paper, we first obtain the PDE characterization of Finsler warped product metrics with a vanishing Riemannian curvature. Moreover, we obtain equivalent conditions for locally Minkowski Finsler warped product spaces. Finally, we explicitly construct two types of non-Riemannian examples.

1. Introduction

The Finsler metric is a Riemannian metric without quadratic restrictions [1]. In Riemannian geometry, a necessary and sufficient condition for a Riemannian metric to be projectively flat is that it is of constant section curvature, in which case it is also conformally flat. Unfortunately, these conclusions no longer hold in Finsler geometry. Therefore, in order to study the curvature properties of Finsler geometry, scholars have performed a lot of work. In [2], Kang determined that conformally flat Randers metrics with scalar flag curvature are projectively flat and such metrics were completely classified. Cheng-Yuan [3] showed that conformally flat Randers metrics with isotropic scalar curvature are surely locally Minkowskian. On the other hand, Chen-He-Shen [4] claimed that conformally flat $(\alpha ,\beta )$-metrics with constant flag curvature are Riemannian or locally Minkowskian. However, there are few results for non-trivial conformally flat Finsler metrics. For this reason, to study conformally flat metrics in Finsler geometry is remarkable.

In 1969, in order to study Riemannian manifolds with negative curvature, Bishop-O’Neill proposed a notion of warped product metrics, as a generalization of Riemannian product metrics [5]. Later, Kozma-Peter-Varga [6] and Chen-Shen-Zhao [7] extended the notion of warped products to the case of Finsler manifolds, respectively. Such metrics are named as Finsler warped product metrics. Chen-Shen-Zhao [7] gave formulae of flag curvature and Ricci curvature of Finsler warped product metrics, obtained a characterization of such metrics to be Einstein metrics, and constructed some new Einstein metrics by modifying spherically symmetric Finsler metrics with constant flag curvature. Liu-Mo [8] gave a completely classification of Finsler warped product metrics with vanishing Douglas curvature. In [9], the characterization for Finsler warped product metrics with isotropic Ricci curvature or constant flag curvature were simplified, respectively. For non-Riemannian quantities, Yang-Zhang [10] obtained Finsler warped product metrics with relatively isotropic Landsberg curvature. More recently, Feng-Zhang completely characterized Finsler warped product metrics, which are Berwald metrics or locally Minkowski metrics by a system of partial differential equations [11]. It was also worth pointing out that the Ricci soliton of $\mathcal{CR}$-warped product metrics on a complex manifold was studied in [12].

Based on the warped product notion introduced by Chen-Shen-Zhao [7], Marcal-Shen [13] invented a new class of Finsler metrics with another “warping”, one that keeps pace with static spacetime. They characterized Ricci-flat Finsler warped product metrics of the new type by partial differential equations. Furthermore, they provided two types of non-Riemannian examples. In [14], Chavez–Newton completely classified the new type of Finsler warped product metric of Douglas type. They also showed that Finsler warped product metrics of Berwald type and Landsberg type are equivalent and classified such metrics.

For two Finsler metrics F and $\tilde{F}$, if there is a scalar function $\sigma$ on the manifold such that $\tilde{F}=e^{\sigma }F$, then they are called conformally related. Further a Finsler metric, which is conformally related to a locally Minkowski metric, is called conformally flat. In Finsler geometry, it is an important issue to make investigation and research on geometric structures and properties of conformally flat Finsler metrics.

In this paper, we first recall Finsler warped product metrics of Berwald type. Then, we obtain differential equations of Finsler warped product metrics with vanishing Riemannian curvature. Furthermore, we obtain a classification that Finsler warped product metrics are locally Minkowskian. Finally, we give two types of non-Riemannian examples. Our result is as follows.

Theorem 1.

Let $F=\alpha \sqrt{\varphi (z,\rho )}$ be a Finsler warped product metric on an $(n+1)$-dimensional manifold $M=\mathbb{R}\times {\mathbb{R}}^n$ $(n\ge 2)$, where $\alpha = |\overline{y}|$, $z=\frac{y^0}{|\overline{y}|}$, $\rho = |\overline{x}|$. Then, F is a locally Minkowski metric if and only if one of the following cases holds:

(1) $F=\alpha \sqrt{z^2+c_1{\rho }^{c_2}}$, where $c_1$ $(>0)$ and $c_2$ are constants;

(2) $F=\alpha \sqrt{{\left(\ln \left(c_3\rho \right)\right)}^2{z}^2+c_4{\rho }^{-2}}$, where $c_3$ and $c_4$ are positive constants;

(3) $F=\alpha {\rho }^{c_5}\sqrt{G\left({\rho }^{-c_5}\right|z|})$, where $c_5$ is a constant and $G=G\left(t\right)$ $(t={\rho }^{-c_5}|z\left|\right)$ is a positive differentiable function such that $2G-t{G}^{\prime }>0$ and $2G{G}^{″}-{\left(G^{\prime }\right)}^2>0$.

2. Preliminaries

Set $M=\mathbb{R}\times {\mathbb{R}}^n$ with coordinates on $TM$

$$\begin{array}{cc}& x=\left(x^0,\overline{x}\right),\overline{x}=\left(x^1,\dots ,x^n\right),\hfill \\ & y=\left(y^0,\overline{y}\right),\overline{y}=\left(y^1,\dots ,y^n\right).\hfill \end{array}$$

And, consider a Finsler metric

$$F=\alpha \sqrt{\varphi (z,\rho )},$$

where $\alpha =|\overline{y}|$, $z=\frac{y^0}{|\overline{y}|}$, $\rho =|\overline{x}|$, and $\varphi$ is a suitable function on ${\mathbb{R}}^2$.

Throughout this paper, our index conventions are as follows:

$$\begin{array}{cc}& 0\le A,B,\dots \le n,\hfill \\ & 1\le i,j,\dots \le n.\hfill \end{array}$$

And, note that ${(·)}_z$ and ${(·)}_{\rho }$ mean the derivative of $(·)$ with respect to the first variable z and the second variable $\rho$, respectively. For example, ${\varphi }_z=\frac{\partial \varphi }{\partial z}$, ${\varphi }_{z\rho }=\frac{{\partial }^2\varphi }{\partial z\partial \rho }$, $W_z=\frac{\partial W}{\partial z}$, etc.

For a Finsler warped product metric $F=\alpha \sqrt{\varphi (z,\rho )}$, the fundamental form $g=g_{AB}d{x}^A\otimes d{x}^B$ is given by

$$\left(g_{AB}\right)=\left(\begin{array}{cc}\frac{1}{2}{\varphi }_{zz}& \frac{1}{2}{\Omega }_z\frac{y^j}{\alpha }\\ ine\frac{1}{2}{\Omega }_z\frac{y^i}{\alpha }& \frac{1}{2}\Omega {\delta }_{ij}-\frac{1}{2}z{\Omega }_z\frac{y^i{y}^j}{{\alpha }^2}\end{array}\right),$$

where $\Omega :=2\varphi -z{\varphi }_z$. Then,

$$\det \left(g_{AB}\right)=\frac{1}{2^{n+1}}{\Omega }^{n-1}\Lambda ,$$

where $\Lambda :=2\varphi {\varphi }_{zz}-{{\varphi }_z}^2.$

Henceforth, assume F is non-degenerate. In this case, the inverse of $\left(g_{AB}\right)$ is

$$\left(g^{AB}\right)=\left(\begin{array}{cc}\frac{2}{\Lambda }\left(\Omega -z{\Omega }_z\right)& -\frac{2}{\Lambda }{\Omega }_z\frac{y^j}{\alpha }\\ ine-\frac{2}{\Lambda }{\Omega }_z\frac{y^i}{\alpha }& \frac{2}{\Omega }{\delta }^{ij}+\frac{2{\varphi }_z{\Omega }_z}{\Omega \Lambda }\frac{y^i{y}^j}{{\alpha }^2}\end{array}\right).$$

Lemma 1

([13]). $F=\alpha \sqrt{\varphi (z,\rho )}$ is strongly convex if and only if Ω, $\Lambda >0$.

The spray coefficients $G^A$ are defined by

$$G^A:=\frac{1}{4}{g}^{AC}\left[{\left(F^2\right)}_{y^C{x}^B}{y}^B-{\left(F^2\right)}_{x^C}\right].$$

The Berwald curvature $B=B_{BCD}^A\frac{\partial }{\partial x^A}\otimes d{x}^B\otimes d{x}^C\otimes d{x}^D$ is a family of endomorphisms, defined by

$$\begin{array}{c}\hfill B_{BCD}^A:=\frac{{\partial }^3{G}^A}{\partial y^B\partial y^C\partial y^D}.\end{array}$$

A Finsler metric is called a Berwald metric if $B_{BCD}^A=0$, i.e., the spray coefficients $G^A=G^A(x,y)$ are quadratic in $y\in T_{x}M$ at every point $x\in M$.

Lemma 2

([14]). Let $F=\alpha \sqrt{\varphi (z,\rho )}$ be a Finsler warped product metric on an $(n+1)$-dimensional manifold $M=\mathbb{R}\times {\mathbb{R}}^n$ $(n\ge 2)$, where $\alpha = |\overline{y}|$, $z=\frac{y^0}{|\overline{y}|}$, $\rho = |\overline{x}|$. Then F is a Berwald metric if and only if F is a Landsberg metric. In this case, F is either Riemannian or of the form $F(x,y)=\alpha {\left(h\left(\rho \right)\right)}^{-1}H\left(h\left(\rho \right)\right|z\left|\right)$, where $h\left(\rho \right)$ and $H=H\left(t\right)$ $(t=h(\rho \left)\left|z\right|\right)$ are positive differentiable functions such that $H-t{H}^{\prime }>0$ and $H^{″}>0$.

The Riemanian curvature $R=R_B^{A}d{x}^B\otimes \frac{\partial }{\partial x^A}:T_{x}M\to T_{x}M$ is a family of endomorphisms, defined by

$$\begin{array}{cc}\hfill R_B^A:=& 2{\left(G^A\right)}_{x^B}-{\left(G^A\right)}_{x^C{y}^B}{y}^C+2G^C{\left(G^A\right)}_{y^C{y}^B}-{\left(G^A\right)}_{y^C}{\left(G^C\right)}_{x^B}.\hfill \end{array}$$

For the Riemannian curvature $R_B^A$ of a Finsler warped product $F=\alpha \sqrt{\varphi (z,\rho )}$, we have [13]

$$\begin{array}{cc}\hfill R_0^0=& \left[{\rho }^2(U+zV)W_z-(2{\rho }^{2}W+1)\left(U_z+V+z{V}_z\right)\right]{\alpha }^2\hfill \\ \hfill & +\left[2\left(V+W\right)\left(U_z+V+z{V}_z\right)-{\left(U_z+V+z{V}_z\right)}^2\right.\hfill \\ \hfill & +2U\left(U_{zz}+2V_z+z{V}_{zz}\right)-\left(V_z+W_z\right)\left(U+zV\right)\hfill \\ \hfill & \left.-{\rho }^{-1}\left(U_{z\rho }+V_{\rho }+z{V}_{z\rho }\right)-(U-z{U}_z-z^2{V}_z)V_z\right]{〈\overline{x},\overline{y}〉}^2,\hfill \\ \hfill R_j^0=& z\left[(2{\rho }^{2}W+1)\left(V+U_z+z{V}_z\right)-{\rho }^2{W}_z(U+zV)\right]\alpha y^j\hfill \\ \hfill & +\left[z(U+zV)\left(V_z+W_z\right)+(U-z{U}_z-z^2{V}_z)(5W-U_z)\right.\hfill \\ \hfill & -2zU\left(U_{zz}+2V_z+z{V}_{zz}\right)\left.-{\rho }^{-1}(U_{\rho }-z{U}_{z\rho }-z^2{V}_{z\rho })\right]{〈\overline{x},\overline{y}〉}^2\frac{y^j}{\alpha }\hfill \\ \hfill & +\left[(U+zV)\left(U_z-V+z{V}_z-2W\right)+(V-3W)(U-z{U}_z-z^2{V}_z)\right.\hfill \\ \hfill & \left.+{\rho }^{-1}\left(U_{\rho }+z{V}_{\rho }\right)\right]〈\overline{x},\overline{y}〉\alpha x^j,\hfill \end{array}$$

$$\begin{array}{cc}\hfill R_j^i=& -\left[2W+(2{\rho }^{2}W+1)(V+W)\right]{\alpha }^2{\delta }_j^i\hfill \\ \hfill & +\left[{(V+W)}^2+2U\left(V_z+W_z\right)-{\rho }^{-1}\left(V_{\rho }+W_{\rho }\right)\right]{〈\overline{x},\overline{y}〉}^2{\delta }_j^i\hfill \\ \hfill & +\left[2W\left(2W-z{W}_z\right)+W_z(U-zW)-{\rho }^{-1}{W}_{\rho }\right]{\alpha }^2{x}^i{x}^j+[\left(V+W\right)\hfill \\ \hfill & +z\left(V_z+W_z\right)(2{\rho }^{2}W+1)+({\rho }^2(V+W)+1)\left(2W-z{W}_z\right)]y^i{y}^j\hfill \\ \hfill & -[2zU\left(V_{zz}+W_{zz}\right)+\left(3U-z{U}_z-zV+5zW\right)\left(V_z+W_z\right)\hfill \\ \hfill & -z{\rho }^{-1}\left(V_{z\rho }+W_{z\rho }\right)]{〈\overline{x},\overline{y}〉}^2\frac{y^i{y}^j}{{\alpha }^2}+\left[-{(2W-z{W}_z)}^2-2U\left(W_z-z{W}_{zz}\right)\right.\hfill \\ \hfill & \left.+{\rho }^{-1}\left(2W_{\rho }-z{W}_{z\rho }\right)+W_z(U-z{U}_z-z^2{W}_z)\right]〈\overline{x},\overline{y}〉x^i{y}^j\hfill \\ \hfill & +\left[\left(V_z+W_z\right)\left(U+3zW\right)-{(V+W)}^2+{\rho }^{-1}\left(V_{\rho }+W_{\rho }\right)\right]〈\overline{x},\overline{y}〉x^j{y}^i,\hfill \\ \hfill R_0^i=& \left[{\rho }^2{W}_z(V-W)-(2{\rho }^{2}W+1)V_z\right]\alpha y^i+[\left(2W-V-U_z\right)\left(V_z+W_z\right)\hfill \\ \hfill & +2U\left(V_{zz}+W_{zz}\right)-{\rho }^{-1}\left(V_{z\rho }+W_{z\rho }\right)]{〈\overline{x},\overline{y}〉}^2\frac{y^i}{\alpha }\hfill \\ \hfill & +\left[\left(U_z-W\right)W_z-2U{W}_{zz}+{\rho }^{-1}{W}_{z\rho }\right]〈\overline{x},\overline{y}〉\alpha x^i,\hfill \end{array}$$

where $〈\overline{x},\overline{y}〉:={\sum }_{k=1}^n{x}^k{y}^k$,

$$\begin{array}{c}\hfill U:=\frac{1}{2\rho \Lambda }\left(2\varphi {\varphi }_{z\rho }-{\varphi }_z{\varphi }_{\rho }\right),V:=\frac{1}{2\rho \Lambda }\left({\varphi }_{zz}{\varphi }_{\rho }-{\varphi }_z{\varphi }_{z\rho }\right),W:=\frac{1}{2\rho \Omega }{\varphi }_{\rho }.\end{array}$$

(1)

3. Riemannian Curvature

In this section, we obtain differential equations of Finsler warped product metrics with a vanishing Riemannian curvature.

Lemma 3

([13]). For $n\ge 2$, $A{\alpha }^2+B{〈\overline{x},\overline{y}〉}^2=0$ if and only if $A=B=0$, where A, B are functions of z and ρ.

Lemma 4.

For $n\ge 2$,

$$\begin{array}{c}\hfill A{\alpha }^2{y}^i+B〈\overline{x},\overline{y}〉{\alpha }^2{x}^i+C{〈\overline{x},\overline{y}〉}^2{y}^i=0\end{array}$$

(2)

if and only if $A=B=C=0$, where A, B, C are functions of z and ρ.

Proof. 

“Necessity”. Suppose that (2) holds. Contracting (2) with $y^i$, we have

$$\begin{array}{cc}& A{\alpha }^2+\left(B+C\right){〈\overline{x},\overline{y}〉}^2=0.\hfill \end{array}$$

By Lemma 3, we obtain $A=0$ and $B+C=0$.

Thus, (2) can be simplified as $B\left({\alpha }^2{x}^i-〈\overline{x},\overline{y}〉y^i\right)=0$. Contracting it with $x^i$ yields

$$\begin{array}{c}\hfill B({\rho }^2{\alpha }^2-{〈\overline{x},\overline{y}〉}^2)=0.\end{array}$$

We obtain $B=0$. Thus, $A=B=C=0$.

“Sufficiency”. It is obviously true. □

Lemma 5.

For $n\ge 2$,

$$\begin{array}{c}\hfill A{\alpha }^4{x}^i+B{〈\overline{x},\overline{y}〉}^2{\alpha }^2{x}^i+C〈\overline{x},\overline{y}〉{\alpha }^2{y}^i+D{〈\overline{x},\overline{y}〉}^3{y}^i=0\end{array}$$

(3)

if and only if $A=B=C=D=0$, where A, B, C, D are functions of z and ρ.

Proof. 

“Necessity”. Suppose that (3) holds. Contracting (3) with $y^i$ yields

$$\begin{array}{cc}\hfill A〈\overline{x},\overline{y}〉{\alpha }^4+B{〈\overline{x},\overline{y}〉}^3{\alpha }^2+C〈\overline{x},\overline{y}〉{\alpha }^4+D{〈\overline{x},\overline{y}〉}^3{\alpha }^2& =0,\hfill \end{array}$$

i.e.,

$$\begin{array}{cc}\hfill \left(A+C\right){\alpha }^2+\left(B+D\right){〈\overline{x},\overline{y}〉}^2& =0.\hfill \end{array}$$

By Lemma 3, we obtain $C=-A$ and $D=-B$.

Thus, (3) can be simplified as $A{\alpha }^4{x}^i+B{〈\overline{x},\overline{y}〉}^2{\alpha }^2{x}^i-A〈\overline{x},\overline{y}〉{\alpha }^2{y}^i-B{〈\overline{x},\overline{y}〉}^3{y}^i=0$. Contracting it with $x^i$ yields

$$\begin{array}{cc}\hfill ({\rho }^2{\alpha }^2-{〈\overline{x},\overline{y}〉}^2)(A{\alpha }^2+B{〈\overline{x},\overline{y}〉}^2)& =0.\hfill \end{array}$$

By Lemma 3, we obtain $A=B=0$. Thus, $A=B=C=D=0$.

“Sufficiency”. It is obviously true. □

Lemma 6.

For $n\ge 2$,

$$\begin{array}{cc}\hfill 0=& A{〈\overline{x},\overline{y}〉}^2{y}^i{y}^j+B{\alpha }^2{y}^i{y}^j+C{\alpha }^4{x}^i{x}^j+D〈\overline{x},\overline{y}〉{\alpha }^2{x}^i{y}^j\hfill \\ \hfill & +\tilde{D}〈\overline{x},\overline{y}〉{\alpha }^2{x}^j{y}^i+E{〈\overline{x},\overline{y}〉}^2{\alpha }^2{\delta }^{ij}+F{\alpha }^4{\delta }^{ij}\hfill \end{array}$$

(4)

if and only if $A=0$, $B=-F={\rho }^{2}C$, and $D=\tilde{D}=-E=-C$, where A, B, C, D, $\tilde{D}$, E, F are functions of z and ρ. In particular, for $n>2$, if (4) holds, then $A=B=C=D=\tilde{D}=E=F=0$.

Proof. 

“Necessity”. Suppose that (4) holds. Contracting (4) with $y^i$, we have

$$\begin{array}{cc}& \left(A+D+E\right){〈\overline{x},\overline{y}〉}^2{y}^j+(B+F){\alpha }^2{y}^j+\left(C+\tilde{D}\right)〈\overline{x},\overline{y}〉{\alpha }^2{x}^j=0.\hfill \end{array}$$

By Lemma 4, we obtain $A+D+E=0$, $B+F=0$ and $\tilde{D}=-C$.

Thus, (4) can be simplified as $A{〈\overline{x},\overline{y}〉}^2{y}^i{y}^j+B{\alpha }^2{y}^i{y}^j+C{\alpha }^4{x}^i{x}^j+D〈\overline{x},\overline{y}〉{\alpha }^2{x}^i{y}^j-C〈\overline{x},\overline{y}〉{\alpha }^2{x}^j{y}^i-(A+D){〈\overline{x},\overline{y}〉}^2{\alpha }^2{\delta }^{ij}-B{\alpha }^4{\delta }^{ij}=0$. Contracting it with $x^i$ yields

$$\begin{array}{c}\hfill A{〈\overline{x},\overline{y}〉}^3{y}^j+(B+{\rho }^{2}D)〈\overline{x},\overline{y}〉{\alpha }^2{y}^j-(A+C+D){〈\overline{x},\overline{y}〉}^2{\alpha }^2{x}^j+({\rho }^{2}C-B){\alpha }^4{x}^j=0.\end{array}$$

(5)

By Lemma 5, we obtain $A=0$, $B={\rho }^{2}C$ and $D=-C$. So $A=0$, $B=-F={\rho }^{2}C$ and $D=\tilde{D}=-E=-C$.

Hence, (5) can be rewritten as

$$\begin{array}{c}\hfill {\alpha }^{2}C\left[{\rho }^2{y}^i{y}^j+{\alpha }^2{x}^i{x}^j-〈\overline{x},\overline{y}〉(x^i{y}^j+x^j{y}^i)+{〈\overline{x},\overline{y}〉}^2{\delta }^{ij}-{\rho }^2{\alpha }^2{\delta }^{ij}\right]=0.\end{array}$$

(6)

Now, putting $i=j$ and taking the summation of (6) over the index i, we obtain

$$\begin{array}{c}\hfill (n-2)C{\alpha }^2({〈\overline{x},\overline{y}〉}^2-{\rho }^2{\alpha }^2)=0.\end{array}$$

Thus, when $n=2$, the above equation always holds; when $n>2$, we obtain $C=0$.

“Sufficiency”. When $n>2$, it is obviously true. When $n=2$, we see that the right side of (4) is reduced to the left side of (6). Furthermore, we have that (6) holds for any index i and j $(1\le i$, $j\le 2)$. Thus, (4) holds. This completes the proof of Lemma 6. □

Proposition 1.

Let $F=\alpha \sqrt{\varphi (z,\rho )}$ be a Finsler warped product metric on an $(n+1)$-dimensional manifold $M=\mathbb{R}\times {\mathbb{R}}^n$ $(n\ge 2)$, where $\alpha = |\overline{y}|$, $z=\frac{y^0}{|\overline{y}|}$, $\rho = |\overline{x}|$. Then, F has vanishing Riemannian curvature if and only if $\varphi (z,\rho )$ satisfies Equations (12), (14) and (17)–(23).

Proof. 

Suppose $R_B^A=0$. Then we have $R_0^0=0$, $R_j^i=0$, $R_j^0=0$ and $R_0^i=0$. Since $R_0^0=0$, by Lemma 3, we obtain that

$$\begin{array}{cc}\hfill 0=& {\rho }^2(U+zV)W_z-(2{\rho }^{2}W+1)\left(U_z+V+z{V}_z\right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill 0=& 2\left(V+W\right)\left(U_z+V+z{V}_z\right)-\left(V_z+W_z\right)(U+zV)+2U\left(U_{zz}+2V_z+z{V}_{zz}\right)\hfill \\ \hfill & -{\rho }^{-1}\left(U_{z\rho }+V_{\rho }+z{V}_{z\rho }\right)-{\left(U_z+V+z{V}_z\right)}^2-(U-z{U}_z-z^2{V}_z)V_z.\hfill \end{array}$$

(8)

Similarly, $R_j^i=0$ means that

$$\begin{array}{cc}\hfill 0=& \rho \left[2zU(V_{zz}+W_{zz})+(V_z+W_z)(3U-z{U}_z-zV+5zW)\right]-z(V_{z\rho }+W_{z\rho }),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill 0=& (2{\rho }^{2}W+1)(V_z+W_z)-W_z\left[{\rho }^2(V+W)+1\right],\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill 0=& {\rho }^2\left[2W^2-(3z{W}_z+2V)W+U{W}_z\right]-2\rho W_{\rho }-V-3W,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill 0=& (U+zW)(V_z+W_z),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill 0=& \rho \left[2U{W}_{zz}+(W-U_z)W_z\right]-W_{z\rho },\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill 0=& \rho \left[3W^2-(3z{W}_z+2V)W-2U{V}_z-U{W}_z-V^2\right]+V_{\rho }-W_{\rho }.\hfill \end{array}$$

(14)

$R_j^0=0$ yields that

$$\begin{array}{cc}\hfill 0=& (2{\rho }^{2}W+1)(z{V}_z+U_z+V)-{\rho }^2{W}_z(zV+U),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill 0=& \rho [z(U+zV)\left(V_z+W_z\right)-2zU\left(U_{zz}+2V_z+z{V}_{zz}\right)\hfill \\ \hfill & +(U-z{U}_z-z^2{V}_z)\left(5W-U_z\right)]-U_{\rho }+z{U}_{z\rho }+z^2{V}_{z\rho },\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill 0=& \rho \left[(U+zV)\left(U_z-V+z{V}_z-2W\right)+(V-3W)(U-z{U}_z-z^2{V}_z)\right]\hfill \\ \hfill & +U_{\rho }+z{V}_{\rho }.\hfill \end{array}$$

(17)

And $R_0^i=0$ implies that

$$\begin{array}{cc}\hfill 0=& {\rho }^2{W}_z(V-W)-(2{\rho }^{2}W+1)V_z,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill 0=& \rho \left[\left(2W-V-U_z\right)\left(V_z+W_z\right)+2U\left(V_{zz}+W_{zz}\right)\right]-\left(V_{z\rho }+W_{z\rho }\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill 0=& \rho \left[\left(U_z-W\right)W_z-2U{W}_{zz}\right]+W_{z\rho }.\hfill \end{array}$$

(20)

Hence, $R_B^A=0$ if and only if Equations (7)–(20) hold.

$\left(7\right)-\left(18\right)\times z$ yields

$$\begin{array}{c}\hfill {\rho }^2{W}_z(U+zW)-(2{\rho }^{2}W+1)(U_z+V)=0.\end{array}$$

(21)

$\left(11\right)-\left(21\right)$ yields

$$\begin{array}{c}\hfill (2{\rho }^{2}W+1)(U_z+W)-4W(z{\rho }^2{W}_z+1)-2\rho W_{\rho }=0.\end{array}$$

(22)

Differentiating (17) with respect to the variable z and subtracting the resulting equation from $\left(8\right)\times 3\rho$ yield

$$\begin{array}{c}\hfill (U-z{U}_z-z^2{V}_z)W_z-(U_{zz}+2V_z+z{V}_{zz})(U+zW)+(V_z+W_z)(U+zV)=0.\end{array}$$

(23)

Note that $\left(7\right)+\left(15\right)=0$, $\left(10\right)+\left(18\right)=0$, $\left(13\right)+\left(20\right)=0$, $\left(8\right)\times z+\left(16\right)+\left(17\right)=0$ and $\left(9\right)+\left(12\right)\times 3\rho -\left(19\right)\times z=0$.

Thus, $R_B^A=0$ if and only if Equations (12), (14) and (17)–(23) hold. This completes the proof of Proposition 1. □

Lemma 7.

Let $F=\alpha \sqrt{\varphi (z,\rho )}$ be a Finsler warped product metric on an $(n+1)$-dimensional manifold $M=\mathbb{R}\times {\mathbb{R}}^n$ $(n\ge 2)$, where $\alpha = |\overline{y}|$, $z=\frac{y^0}{|\overline{y}|}$, $\rho = |\overline{x}|$. If $W_z=0$, then $V_z=0$ and $U_{zz}=0$.

Proof. 

Suppose that $W_z=0$, i.e., $W=f\left(\rho \right)$, where $f\left(\rho \right)$ is a differentiable function. By (1), we obtain that

$$\begin{array}{c}\hfill {\varphi }_{\rho }=2\rho f\left(\rho \right)\Omega ,{\varphi }_{z\rho }=2\rho f\left(\rho \right){\Omega }_z.\end{array}$$

Hence, $U=-zf\left(\rho \right)$ and $V=f\left(\rho \right)$. It means that $U_{zz}=0$ and $V_z=0$. This completes the proof of Lemma 7. □

Lemma 8.

Let $F=\alpha \sqrt{\varphi (z,\rho )}$ be a Finsler warped product metric on an $(n+1)$-dimensional manifold $M=\mathbb{R}\times {\mathbb{R}}^n$ $(n\ge 2)$, where $\alpha = |\overline{y}|$, $z=\frac{y^0}{|\overline{y}|}$, $\rho = |\overline{x}|$. Then $U+zW=0$ if and only if $W_z=0$.

Proof. 

By (1), direct computations yields that $W_z=\frac{1}{2\rho \Omega }\left({\varphi }_{z\rho }\Omega -{\varphi }_{\rho }{\Omega }_z\right)$ and $U+zW=\frac{\varphi }{\rho \Lambda \Omega }\left({\varphi }_{z\rho }\Omega -{\varphi }_{\rho }{\Omega }_z\right)$. Thus, $U+zW=0$ is equivalent to $W_z=0$. This completes the proof of Lemma 8. □

Theorem 2.

Let $F=\alpha \sqrt{\varphi (z,\rho )}$ be a Finsler warped product metric on an $(n+1)$-dimensional manifold $M=\mathbb{R}\times {\mathbb{R}}^n$ $(n\ge 2)$, where $\alpha = |\overline{y}|$, $z=\frac{y^0}{|\overline{y}|}$, $\rho = |\overline{x}|$. Then, F has vanishing Riemannian curvature if and only if one of the following cases holds:

(1) $F=\alpha {\rho }^c\sqrt{G\left({\rho }^{-c}\left|z\right|\right)}$, where c is a constant and $G=G\left(t\right)$ $(t={\rho }^{-c}\left|z\right|)$ is a positive differentiable function such that $2G-t{G}^{\prime }>0$ and $2G{G}^{″}-{\left(G^{\prime }\right)}^2>0$;

(2) $\varphi (z,\rho )$ satisfies the following equations: $W_z\ne 0$, $V+W=-{\rho }^{-2}$ and

$$\begin{array}{cc}\hfill 0=& {\rho }^2{W}_z(U+zW)-(2{\rho }^{2}W+1)(U_z-{\rho }^{-2}-W),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill 0=& (2{\rho }^{2}W+1)(U_z+W)-4W(z{\rho }^2{W}_z+1)-2\rho W_{\rho },\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill 0=& 2{\rho }^{2}U(U_z-5W)-3z{U}_z+9zW+4z{\rho }^2{W}^2+2\rho U_{\rho }+4z{\rho }^{-2},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill 0=& 3W_z-U_{zz}.\hfill \end{array}$$

(27)

Proof. 

“Necessity”. Suppose that F has vanishing Riemannian curvature. By Proposition 1, we have Equations (12), (14) and (17)–(23) hold. By (12), namely, $(U+zW)(V_z+W_z)=0$, we divide the problem into three cases:

Case (i): $U+zW=0$ and $V_z+W_z\ne 0$. Since $U+zW=0$, by Lemmas 7 and 8, we have $W_z=0$ and $V_z=0$. It is a contradiction to $V_z+W_z\ne 0$.

Case (ii): $U+zW=0$ and $V_z+W_z=0$. By Lemmas 7 and 8, we have $W_z=0$ and $V_z=0$. Thus, Equations (18)–(20) and (23) are automatically satisfied. Furthermore, the remaining equations can be simplified as:

$$\begin{array}{cc}\hfill 0=& (2{\rho }^{2}W+1)(V-W),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill 0=& 2W+\rho W_{\rho },\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill 0=& \rho (V-W)(V+3W)-V_{\rho }+W_{\rho }.\hfill \end{array}$$

(30)

By (29), there exists a constant c such that

$$W=\frac{c}{2}{\rho }^{-2}.$$

Then, substituting it into $W:=\frac{1}{2\rho \Omega }{\varphi }_{\rho }$ yields ${\varphi }_{\rho }=c{\rho }^{-1}(2\varphi -z{\varphi }_z).$ Furthermore, it can be rewritten as

$$\begin{array}{c}\hfill {\left({\rho }^{-2c}\varphi \right)}_{\rho }+c{\rho }^{-1}z{\left({\rho }^{-2c}\varphi \right)}_z=0.\end{array}$$

Thus, there exists a positive differentiable function $G:\mathbb{R}\to \mathbb{R}$ such that

$$\begin{array}{c}\hfill \varphi (z,\rho )={\rho }^{2c}G\left({\rho }^{-c}\left|z\right|\right).\end{array}$$

(31)

Plugging (31) into (1) yields $V=W$. Thus, Equations (28)–(30) hold.

Finally, let us discuss strong convexity of the metric. For (31), we have

$$\begin{array}{cc}\hfill \Omega :=& 2\varphi -z{\varphi }_z={\rho }^{2c}(2G-{\rho }^{-c}\left|z\right|G^{\prime }),\hfill \\ \hfill \Lambda :=& 2\varphi {\varphi }_{zz}-{{\varphi }_z}^2={\rho }^{2c}\left[2G{G}^{″}-{\left(G^{\prime }\right)}^2\right].\hfill \end{array}$$

Hence, by Lemma 1, we ascertain that $F=\alpha {\rho }^c\sqrt{G\left({\rho }^{-c}\left|z\right|\right)}$ is a regular Finsler metric if and only if $2G-t{G}^{\prime }>0$ and $2G{G}^{″}-{\left(G^{\prime }\right)}^2>0$, where $G=G\left(t\right)$ and $t={\rho }^{-c}\left|z\right|$.

Case (iii): $U+zW\ne 0$ and $V_z+W_z=0$. Namely, $W_z\ne 0$ and $V_z+W_z=0$ by Lemma 8. Thus, Equation (19) is automatically satisfied. In this case, we have $V+W=-{\rho }^{-2}$ by (18). Then, the remaining equations can be simplified as:

$$\begin{array}{cc}\hfill 0=& {\rho }^2{W}_z(U+zW)-(2{\rho }^{2}W+1)(U_z-{\rho }^{-2}-W),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill 0=& \rho \left[4W(W-z{W}_z)+W_z(U+zW)\right]+{\rho }^{-3}-2W_{\rho },\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill 0=& (2{\rho }^{2}W+1)(U_z+W)-4W(z{\rho }^2{W}_z+1)-2\rho W_{\rho },\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill 0=& (3U-z{U}_z+z^2{W}_z+2zW)W_z-(U_{zz}-z{W}_{zz})(U+zW),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill 0=& \rho \left[U(U_z-z{W}_z-5W)+zW(3U_z-3W_z+W)\right]+U_{\rho }+z{\rho }^{-3}-z{W}_{\rho },\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill 0=& \rho \left[(U_z-W)W_z-2U{W}_{zz}\right]+W_{z\rho }.\hfill \end{array}$$

(37)

Note that $\left(32\right)+\left(34\right)-\left(33\right)\times \rho =0$ and $\left(32\right)\times z-\left(35\right)\times {\rho }^2=0$. $\left(32\right)\times 2z-\left(34\right)\times z+\left(36\right)\times 2\rho$ yields

$$\begin{array}{c}\hfill 2{\rho }^{2}U(U_z-5W)-3z{U}_z+9zW+4z{\rho }^2{W}^2+2\rho U_{\rho }+4z{\rho }^{-2}=0.\end{array}$$

(38)

Differentiating (32) and (34) with respect to the variable z, respectively, we have

$$\begin{array}{cc}\hfill 0=& {\rho }^2{W}_{zz}(U+zW)+{\rho }^2{W}_z(3W-U_z+z{W}_z+2{\rho }^{-2})-(2{\rho }^{2}W+1)(U_{zz}-W_z),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill 0=& 2{\rho }^2{W}_z(U_z-W-2z{W}_z-2{\rho }^{-2})-4z{\rho }^{2}W{W}_{zz}-2\rho W_{z\rho }+(2{\rho }^{2}W+1)(U_{zz}+W_z).\hfill \end{array}$$

(40)

Thus $\left(39\right)\times 12+\left(40\right)+\left(37\right)\times 6\rho$ yields $(2{\rho }^{2}W+1)(3W_z-U_{zz})=0$. Since $W_z\ne 0$, we obtain $2{\rho }^{2}W+1\ne 0$. Thus

$$\begin{array}{c}\hfill 3W_z-U_{zz}=0.\end{array}$$

(41)

Hence $R_B^A=0$ implies that Equations (32), (34), (38) and (41) hold.

“Sufficiency”. By simple calculations, we obtain $R_B^A=0$. This completes the proof of Theorem 2. □

4. Locally Minkowski Finsler Warped Product Spaces

In this section, we first recall necessary and sufficient conditions for locally Minkowski Finsler warped product spaces. Moreover, we are going to prove Theorem 1. Finally, we construct two non-Riemannian classes of locally Minkowski Finsler warped product spaces.

Lemma 9

([1]). Let $(M,F)$ be a Finsler manifold. Let $R_{BCD}^A$ and $P_{BCD}^A$ be, respectively, the hh and hv curvatures of the Chern connection. Then, the following three conditions are equivalent:

(1) $(M,F)$ is locally Minkowskian;

(2) $R_{BCD}^A=0$ and $P_{BCD}^A=0$;

(3) $R_{AC}:=l^B{R}_{BACD}{l}^D=0$ and $P_{BACD}=0$, where $l^B:=\frac{y^B}{F}$.

Remark 1.

$R_{BCD}^A=0$ if and only if $R_C^A=0$. $P_{BCD}^A=0$ if and only if ${\Gamma }_{BC}^A={\Gamma }_{BC}^A\left(x\right)$, i.e., F is a Berwald metric.

Remark 2.

Suppose that there is a homothetic transformation between two Finsler warped product metrics F and $\tilde{F}$. If F is locally Minkowskian, then so is $\tilde{F}$.

Remark 3.

Let $F=\alpha \sqrt{\varphi (z,\rho )}$ be a Finsler warped product metric. Then, the following three conditions are equivalent:

(1) F is a Riemannian metric;

(2) ${\varphi }_z-z{\varphi }_{zz}=0$;

(3) $\varphi (z,\rho )=f\left(\rho \right)z^2+g\left(\rho \right)$, where $f\left(\rho \right)$ and $g\left(\rho \right)$ are positive differentiable functions.

Proof of Theorem 1.

“Necessity”. Suppose that F is a locally Minkowski metric. By Lemma 9, we have that $R_B^A=0$ and $B_{BCD}^A=0$. By Lemma 2, $B_{BCD}^A=0$ means that F is either Riemannian or of the form $F(x,y)=\alpha {\left(h\left(\rho \right)\right)}^{-1}H\left(h\left(\rho \right)\right|z\left|\right)$. Hence, we divide the problem into two cases:

Case (i): Suppose that F is Riemannian.

Meanwhile, by Theorem 2, $R_B^A=0$ implies that $F=\alpha {\rho }^c\sqrt{G\left({\rho }^{-c}\left|z\right|\right)}$ or $\varphi (z,\rho )$ satisfies the case (2) of Theorem 2. Thus, we divide the problem into two cases:

Case (i-i): Suppose that $F=\alpha {\rho }^c\sqrt{G\left({\rho }^{-c}\left|z\right|\right)}$, i.e., $\varphi (z,\rho )={\rho }^{2c}G\left({\rho }^{-c}\left|z\right|\right)$.

Since F is Riemannian, we obtain that ${\varphi }_z-z{\varphi }_{zz}={\rho }^c{G}^{\prime }-\left|z\right|G^{″}=0$ holds by Remark 3. Its solution gives

$$G\left(t\right)=\widehat{c}(t^2+c_1),$$

where $t={\rho }^{-c}\left|z\right|$, $\widehat{c}$ $(>0)$ and $c_1$ are constants. Thus, we have that $\varphi (z,\rho )={\rho }^{2c}G\left({\rho }^{-c}\left|z\right|\right)=\widehat{c}(z^2+c_1{\rho }^{c_2})$ for some constants $\widehat{c}$, $c_1$ $(>0)$ and $c_2$. In this case, by Remark 2, $\varphi (z,\rho )$ can be rewritten as $\varphi (z,\rho )=z^2+c_1{\rho }^{c_2}$ for some constants $c_1$ $(>0)$ and $c_2$.

Case (i-ii): Suppose that $\varphi (z,\rho )$ satisfies case (2) of Theorem 2. For $V+W=-{\rho }^{-2}$, we obtain $\rho g^{\prime }+2g=0$ by Remark 3. Its solution is $g\left(\rho \right)=\tilde{c}{\rho }^{-2}$ for some constant $\tilde{c}$ $(>0)$. Plugging $\varphi (z,\rho )=f\left(\rho \right)z^2+\tilde{c}{\rho }^{-2}$ into (25) yields

$$\rho f{f}^{″}-\frac{1}{2}\rho {\left(f^{\prime }\right)}^2+f{f}^{\prime }=0.$$

Its solution is $f\left(\rho \right)=\overline{c}{\left[\ln \left(c_3\rho \right)\right]}^2$ for positive constants $\overline{c}$ and $c_3$. The remaining equations in case (2) of Theorem 2 are automatically satisfied. Hence, we ascertain that $\varphi (z,\rho )={\left[\ln \left(c_3\rho \right)\right]}^2{z}^2+c_4{\rho }^{-2}$, where $c_3$ and $c_4$ are positive constants.

Case (ii): Suppose that $F(x,y)=\alpha {\left(h\left(\rho \right)\right)}^{-1}H\left(h\left(\rho \right)\right|z\left|\right)$, i.e., $\varphi (z,\rho )$ = ${\left[h{\left(\rho \right)}^{-1}H\left(h\left(\rho \right)\right|z\left|\right)\right]}^2$. Similarly, by Theorem 2, we divide the problem into two cases:

Case (ii-i): Suppose that $F=\alpha {\rho }^c\sqrt{G\left({\rho }^{-c}\left|z\right|\right)}$, i.e., $\varphi (z,\rho )={\rho }^{2c}G\left({\rho }^{-c}\left|z\right|\right)$. Comparing it with $\varphi (z,\rho )={\left[h{\left(\rho \right)}^{-1}H\left(h\left(\rho \right)\right|z\left|\right)\right]}^2$ yields that $h\left(\rho \right)={\rho }^{-c}$ and $G\left(t\right)=H^2\left(t\right)$ $(t={\rho }^{-c}\left|z\right|)$. Thus, we obtain that $\varphi (z,\rho )={\rho }^{2c_5}G({\rho }^{-c_5}\left|z\right|)$, where $c_5$ is a constant and $G=G\left(t\right)$ $(t={\rho }^{-c_5}|z\left|\right)$ is a positive differentiable function such that $2G-t{G}^{\prime }>0$ and $2G{G}^{″}-{\left(G^{\prime }\right)}^2>0$.

Case (ii-ii): Suppose that $\varphi (z,\rho )$ satisfies case (2) of Theorem 2.

For $\varphi (z,\rho )={\left[h{\left(\rho \right)}^{-1}H\left(h\left(\rho \right)\right|z\left|\right)\right]}^2$, we have that $W=-\frac{h{\left(\rho \right)}^{\prime }}{2\rho h\left(\rho \right)}$ by (1). Thus, $W_z=0$. It is a contradiction to $W_z\ne 0$ by the case (2) of Theorem 2.

“Sufficiency”. By a direct calculation, it is easy to check that F is a locally Minkowski metric for three cases of Theorem 1. □

At the end of this section, let us construct some locally Minkowski metrics for Finsler warped product metrics by Theorem 1.

Example 1.

Let

$$G\left(t\right)=\exp (-\sum _{i=1}^s{t}^{\frac{1}{2i}}),$$

where s is a positive integer. Then we have

$$\begin{array}{cc}\hfill 2G-t{G}^{\prime }=& G\left(t\right)\left(2+\sum _{i=1}^s\frac{1}{2i}{t}^{\frac{1}{2i}}\right)>0,\hfill \\ \hfill 2G{G}^{″}-{\left(G^{\prime }\right)}^2=& G^2\left(t\right)\left[{\left(\sum _{i=1}^s\frac{1}{2i}{t}^{\frac{1}{2i}-1}\right)}^2+\sum _{i=1}^s\left(1-\frac{1}{2i}\right)\frac{1}{i}{t}^{\frac{1}{2i}-2}\right]>0.\hfill \end{array}$$

Let $\varphi (z,\rho )={\rho }^{2c_5}G({\rho }^{-c_5}\left|z\right|)$. By the case (3) of Theorem 1, the regular Finsler warped product metric

$$F=\alpha \sqrt{\varphi (z,\rho )}=\alpha {\rho }^{2c_5}\exp \left[-\frac{1}{2}\sum _{i=1}^s({\rho }^{-c_5}{\left|z\right|)}^{\frac{1}{2i}}\right]$$

is locally Minkowskian.

Example 2.

Let

$$G\left(t\right)=\sum _{i=0}^s{t}^{-2i},$$

where s is a positive integer. Then we have

$$\begin{array}{cc}\hfill 2G-t{G}^{\prime }=& 2\sum _{i=0}^s(i+1)t^{-2i}>0,\hfill \\ \hfill 2G{G}^{″}-{\left(G^{\prime }\right)}^2=& 4\sum _{i=0}^s{t}^{-2i}\sum _{i=0}^{s}i(i+1)t^{-2(i+1)}>0.\hfill \end{array}$$

Let $\varphi (z,\rho )={\rho }^{2c_5}G({\rho }^{-c_5}\left|z\right|)$. By the case (3) of Theorem 1, the regular Finsler warped product metric

$$F=\alpha \sqrt{\varphi (z,\rho )}=\alpha {\rho }^{2c_5}{\left[\sum _{i=0}^s({\rho }^{-c_5}{\left|z\right|)}^{-2i}\right]}^{\frac{1}{2}}$$

is locally Minkowskian.

5. Conclusions

In this paper, locally Minkowski Finsler warped product spaces are considered. Firstly, we recall Finsler warped product metrics of Berwald type and obtain differential equations of Finsler warped product metrics with a vanishing Riemannian curvature. Based on these, we obtain a classification that Finsler warped product metrics are locally Minkowskian in Theorem 1. We still have a lot of things to do, such as the classification of Finsler warped product metrics with vanishing Riemannian curvature, the application of our results by incorporating elements from Ricci soliton theory [15,16,17], etc. These can be considered as a future aspect of this paper.