Kropina Metrics with Isotropic Scalar Curvature

Abstract

In this paper, we study Kropina metrics with isotropic scalar curvature. First, we obtain the expressions of Ricci curvature tensor and scalar curvature. Then, we characterize the Kropina metrics with isotropic scalar curvature on by tensor analysis.

1. Introduction

The curvature properties of metrics play very important roles in Riemannian and Finsler geometry. Riemannian curvature and Ricci curvature are the most important Riemannian geometric quantities in Finsler geometry. In 1988, the concept of Ricci curvature was first proposed by Akbar-Zadeh, and its tensor form can be naturally obtained [1]. In recent years, many scholars have conducted a great deal of research on them. Cheng-Shen-Tian proved that the polynomial $(\alpha ,\beta )$-metric is an Einstein metric if and only if it is Ricci-flat [2]. Zhang-Shen gave the expression of Ricci curvature of Kropina metric. Furthermore, they proved that a non-Riemannian Kropina metric with a constant Killing form $\beta$ is an Einstein metric if and only if $\alpha$ is also an Einstein metric [3]. By using navigation date $(h,W)$, they proved that n (≥2)-dimensional Kropina metric is an Einstein metric if and only if Riemann metric h is an Einstein metric and W is a Killing vector field with respect to h. Xia gave the expression for the Riemannian curvature of Kropina metrics and proved that a Kropina metric is an Einstein metric if and only if it has non-negative constant flag curvature [4]. Cheng-Ma-Shen studied and characterized projective Ricci-flat Kropina metrics and obtained its equivalent characterization Equation [5].

Unlike the notion of Riemannian curvature, there is no unified definition of scalar curvature in Finsler geometry, although several geometers have offered several versions of the definition of the Ricci curvature tensor [1,6,7,8]. In 2015, Li–Shen introduced a new definition of the Ricci curvature tensor [6]. This tensor is symmetric. They proved that a Finsler metric F has isotropic Ricci curvature tensor if and only if it has isotropic Ricci curvature and $\chi$-curvature tensor satisfies ${\chi }_i=f_{ij}\left(x\right)y^j$, where $f_{ij}+f_{ji}=0$. It was further proven that for Randers metrics, they are isotropic Ricci curvature tensors if and only if they are of isotropic Ricci curvature.

In Finsler geometry, there are several versions of the definition of scalar curvature. We used Akbar-Zadeh’s definition [1] of the scalar curvature, based on Li–Shen’s definition of the (symmetric) Ricci curvature tensor [6]. For a Finsler metric F on an n-dimensional manifold M, the scalar curvature R of F is defined as $R:=g^{ij}Ri{c}_{ij}$. Tayebi studied general fourth-root metrics [9]. They characterized general fourth-root metrics with isotropic scalar curvature and also for general fourth-root metrics with isotropic scalar curvature under conformal variation. Finally, they characterized Bryant metric with isotropic scalar curvature. Chen–Xia studied a conformally flat $(\alpha ,\beta )$-metric with weakly isotropic scalar curvature [10]. They proved that if conformally flat polynomial $(\alpha ,\beta )$-metrics have weakly isotropic scalar curvature R, then R vanishes.

In this paper, we obtain a characterization of Kropina metrics with isotropic scalar curvature and have the following results.

Theorem 1. 

Let F be a Kropina metric on an n (≥3)-dimensional manifold M. Then, F is of isotropic scalar curvature if and only if

$$\left\{\begin{array}{c}{\displaystyle {}^{\alpha }Ric=\frac{n-2}{b^4}(-g{\alpha }^2+c^2{\beta }^2+c\beta s_0-b^2{c}_0\beta ),}\hfill \\ {\displaystyle r_{00}=c\left(x\right){\alpha }^2,}\hfill \\ {\displaystyle (2s^m{s}_m-b^2{s}_{|m}^m){\alpha }^2=(n-1)(c\beta s_0+s_0^2-b^2{s}_{0|0}),}\hfill \\ {\displaystyle h{\alpha }^2=\left[\frac{3(n+1)}{n-2}{}^{\alpha }R-\frac{6(n+1)}{(n-2)b^2}{b}^k{b}^l{}^{\alpha }Ri{c}_{kl}+\frac{3(n-1)(n+1)}{b^2}{c}^2+\frac{n(3n+5)}{b^4}{s}^m{s}_m\right]{\beta }^2}\hfill \\ {\displaystyle +\left[\frac{(3n^2+4n-4)}{b^2}{s}_0^m{s}_m-3(n+1)s_{0|m}^m+\frac{3(n-1)(n+1)}{b^2}c{s}_0\right]\beta +2(n-1)s_0^m{s}_{m0},}\hfill \end{array}\right.$$

(1)

where $b={\left|\right|\beta \left|\right|}_{\alpha }$, g, h, $c^2$, $c_0$ are expressed by (15), (23), (26), (27), respectively. In this case, scalar curvature is

$$R=-\frac{1}{12}\left[\frac{n(1-b^2)}{b^2\beta }{s}_0^m{s}_m+\frac{2(3n+2)}{b^2}{s}^m{s}_m+(3n+4)s_t^m{s}_m^t\right].$$

2. Preliminaries

Let M be an n-dimensional $C^{\infty }$ manifold. A Finsler structure of M is a function

$$F:TM\to [0,\infty )$$

with the following properties:

(1)

Regularity: F is $C^{\infty }$ on the slit tangent bundle $TM\backslash \left\{0\right\}$;

(2)

Positive homogeneity: $F(x,\lambda y)=\lambda F(x,y)$, $\forall \lambda >0$;

(3)

Strong convexity:

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

is positive-definite at every point of $TM\backslash \left\{0\right\}$.

Let $(M,F)$ be an n-dimensional Finsler manifold. Suppose that $x\in M$. The geodesics of a Finsler metric $F=F(x,y)$ on M are classified by the following ODEs:

$$\frac{d^2{x}^i}{d{t}^2}+2G^i\left(x,\frac{dx}{dt}\right)=0,$$

where

$$G^i:=\frac{1}{4}{g}^{ik}\left[{\left(F^2\right)}_{x^j{y}^k}{y}^j-{\left(F^2\right)}_{x^k}\right],$$

$\left(g^{ij}\right):={\left(g_{ij}\right)}^{-1}$. The local functions $G^i=G^i(x,y)$ are called geodesic coefficients (or spray coefficients). Then, the S curvature with respect to a volume form $dV=\sigma \left(x\right)dx$ is defined by

$$S=\frac{\partial G^m}{\partial y^m}-y^m\frac{\partial ln\sigma }{\partial x^m}.$$

For $x\in M$ and $y\in T_{x}M\backslash \left\{0\right\}$, Riemann curvature $R_y:=R_k^i(x,y)\frac{\partial }{\partial x^i}\otimes d{x}^k$ is defined by

$$R_k^i:=2\frac{\partial G^i}{\partial x^k}-y^j\frac{\partial G^i}{\partial x^j\partial y^k}+2G^j\frac{\partial G^i}{\partial y^j\partial y^k}-\frac{\partial G^i}{\partial y^j}\frac{\partial G^j}{\partial y^k}.$$

The trace of Riemann curvature is called Ricci curvature of F, i.e., $Ric:=R_k^k$.

Riemann curvature tensor is defined by

$$R_{jkl}^i=\frac{1}{3}\left\{\frac{{\partial }^2{R}_k^i}{\partial y^l\partial y^j}-\frac{{\partial }^2{R}_l^i}{\partial y^k\partial y^j}\right\}.$$

Let ${\overline{Ric}}_{ij}:=R_{jkl}^k$, and

$$Ri{c}_{ij}:=\frac{1}{2}\left\{{\overline{Ric}}_{ij}+{\overline{Ric}}_{ji}\right\}$$

is called a Ricci curvature tensor. The scalar curvature R of F is defined by

$$R:=g^{ij}Ri{c}_{ij}.$$

Let $\kappa \left(x\right)$ be a scalar function on M, $\theta :={\theta }_i\left(x\right)y^i$ be a 1-form on M. If

$$R=n(n-1)\left[\frac{\theta }{F}+\kappa \left(x\right)\right],$$

then it is said that F is of weak isotropic scalar curvature. Especially when $\theta =0$, i.e., $R=n(n-1)\kappa \left(x\right)$, it is said that F is of isotropic scalar curvature.

Let F be a Finsler metric on M. If $F=\frac{{\alpha }^2}{\beta }$, where $\alpha =\sqrt{a_{ij}\left(x\right)y^i{y}^j}$ is a Riemannian metric, $\beta =b_i\left(x\right)y^i$ is a 1-form; then, F is a Kropina metric. Its fundamental tensor $g=g_{ij}d{x}^i\otimes d{x}^j$ is given by [4]

$$\begin{array}{c}\hfill g_{ij}=\frac{F}{\beta }\left\{2a_{ij}+\frac{3F}{\beta }{b}_i{b}_j-\frac{4}{\beta }(b_i{y}_j+b_j{y}_i)+\frac{4y_i{y}_j}{{\alpha }^2}\right\},\end{array}$$

where $y_i:=a_{ij}{y}^j$. Moreover,

$$\begin{array}{c}\hfill g^{ij}=\frac{\beta }{2F}\left\{a^{ij}-\frac{b^i{b}^j}{b^2}+\frac{2}{b^{2}F}(b^i{y}^j+b^j{y}^i)+2\left(1-\frac{2\beta }{b^{2}F}\right)\frac{y^i{y}^j}{{\alpha }^2}\right\},\end{array}$$

where $\left(a^{ij}\right):={\left(a_{ij}\right)}^{-1}$, $b^i:=a^{ij}{b}_j$.

Let $\nabla \beta =b_{i|j}{y}^{i}d{x}^j$ denote the covariant derivative of $\beta$ with respect to $\alpha$. Set

$$\begin{array}{cc}\hfill & r_{ij}=\frac{1}{2}\left(b_{i|j}+b_{j|i}\right),s_{ij}=\frac{1}{2}\left(b_{i|j}-b_{j|i}\right),r_{00}=r_{ij}{y}^i{y}^j,r_i=b^j{r}_{ij},r_0=r_i{y}^i,\hfill \\ \hfill & r^i=a^{ij}{r}_j,r=b^i{r}_i,{s^i}_0=a^{ij}{s}_{jk}{y}^k,s_i=b^j{s}_{ji},s_0=s_i{y}^i,s^i=a^{ij}{s}_j.\hfill \end{array}$$

The Ricci curvature of Kropina metrics is given by the following.

Lemma 1 

([3]). Let F be a Kropina metric on M. Then, the Ricci curvature of F is given by

$$Ric={}^{\alpha }Ric+T,$$

where ${}^{\alpha }Ric$ is the Ricci curvature of α, and

$$\begin{array}{cc}\hfill T=& \frac{3(n-1)}{b^4{\alpha }^4}{r}_{00}^2{\beta }^2+\frac{n-1}{b^2{\alpha }^2}{r}_{00\mid 0}\beta -\frac{4(n-1)}{b^4{\alpha }^2}{r}_{00}{r}_0\beta +\frac{2(n-1)}{b^4{\alpha }^2}{r}_{00}{s}_0\beta -\frac{r}{b^4}{r}_{00}\hfill \\ & +\frac{r^k{}_k}{b^2}{r}_{00}+\frac{2n}{b^2}{r}_{k0}{s}^k{}_0+\frac{1}{b^2}{b}^k{r}_{00\mid k}+\frac{1}{b^4}{r}_0^2-\frac{1}{b^2}{r}_{0\mid 0}-\frac{2(2n-3)}{b^4}{r}_0{s}_0+\frac{n-2}{b^2}{s}_{0\mid 0}\hfill \\ & -\frac{n-2}{b^4}{s}_0^2-\frac{1}{b^2\beta }{r}_{k0}{s}^k{\alpha }^2-\frac{1}{b^2\beta }{r}_k{s}^k{}_0{\alpha }^2-\frac{1}{b^4\beta }r{s}_0{\alpha }^2+\frac{1}{b^2\beta }{r}^k{}_k{s}_0{\alpha }^2+\frac{n-1}{b^2\beta }{s}^k{}_0{s}_k{\alpha }^2\hfill \\ & -\frac{1}{\beta }{s}^k{}_{0\mid k}{\alpha }^2+\frac{1}{b^2\beta }{b}^k{s}_{0\mid k}{\alpha }^2-\frac{1}{4{\beta }^2}{s}^j{}_k{s}^k{}_j{\alpha }^4-\frac{1}{2b^2{\beta }^2}{s}^k{s}_k{\alpha }^4.\hfill \end{array}$$

Lemma 2 

([4]). Let F be a Kropina metric on n-dimensional M. Then, the followings are equivalent:

(i) 

F has an isotropic S curvature, i.e., $S=(n+1)cF$;

(ii) 

$r_{00}=\sigma {\alpha }^2$;

(iii) 

$S=0$;

(iv) 

β is a conformal form with respect to α,

where $c=c\left(x\right)$ and $\sigma =\sigma \left(x\right)$ are functions on M.

3. Ricci Curvature Tensor and Scalar Curvature Tensor of Kropina Metrics

By the definition of Ricci curvature tensor and Lemma 1, we obtain the Ricci curvature tensor of Kropina metrics.

Proposition 1. 

Let F be a Kropina metric on an n-dimensional manifold M. Then, the Ricci curvature tensor of F is given by

$$\begin{array}{cc}\hfill Ri{c}_{kl}& ={}^{\alpha }Ri{c}_{kl}+F_{.k.l}\left\{-\frac{(n-5)}{b^4{F}^3}{r}_{00}^2+\frac{2}{b^4{F}^2}{r}_{00}(s_0-2r_0)+\frac{1}{b^2{F}^2}{r}_{00|0}+\frac{(3n+2)}{3b^{2}F}{s}_0^m{r}_{m0}\right.\hfill \\ \hfill & -\frac{(3n+4)}{3b^{4}F}{r}_0{s}_0-\frac{s_0^2}{3b^{4}F}+\frac{s_{0|0}}{3b^{2}F}+\frac{s_0^m{s}_{m0}}{3\beta }+\frac{1}{2b^2}(r_m^m{s}_0-s^m{r}_{m0}-s_0^m{r}_m-\frac{r{s}_0}{b^2}\hfill \\ \hfill & \left.-b^2{s}_{0|m}^m+b^m{s}_{0|m}-\frac{b^2}{2}F{s}_t^m{s}_m^t-F{s}^m{s}_m)+\frac{1}{6b^2}\left[3n-2-(1-b^2)\frac{F}{2\beta }\right]s_0^m{s}_m\right\}\hfill \\ \hfill & +F_{.k}{F}_{.l}\left\{\frac{3(n-5)}{b^4{F}^4}{r}_{00}^2-\frac{4}{b^4{F}^3}{r}_{00}(s_0-2r_0)-\frac{2}{b^2{F}^3}{r}_{00|0}-\frac{(3n+2)}{3b^2{F}^2}{s}_0^m{r}_{m0}\right.\hfill \\ \hfill & \left.+\frac{(3n+4)}{3b^4{F}^2}{r}_0{s}_0+\frac{s_0^2}{3b^4{F}^2}-\frac{s_{0|0}}{3b^2{F}^2}-\frac{1}{4}{s}_t^m{s}_m^t-\frac{(1-b^2)}{12b^2\beta }{s}_0^m{s}_m-\frac{1}{2b^2}{s}^m{s}_m\right\}\hfill \\ \hfill & -\frac{1}{b^4}\left[\frac{6(n-3)}{F^3}{r}_{00}-\frac{(n-7)}{F^2}{r}_0+\frac{(n-3)}{F^2}{s}_0\right]\left(F_{.k}{r}_{l0}+F_{.l}{r}_{k0}\right)+\frac{2}{b^2{F}^2}(F_{.k}{r}_{l0|0}\hfill \\ \hfill & +F_{.l}{r}_{k0|0})-\frac{(n-1)}{2b^2{F}^2}\left(F_{.k}{r}_{00|l}+F_{.l}{r}_{00|k}\right)+\frac{8(n-2)}{b^4{F}^2}{r}_{k0}{r}_{l0}-\frac{1}{2b^2}(F_{.k}{r}_{ml}+F_{.l}{r}_{mk})s^m\hfill \\ \hfill & +\frac{(3n+2)}{6b^{2}F}(F_{.k}{r}_{lm}+F_{.l}{r}_{km})s_0^m+\frac{(3n+2)}{6b^{2}F}(F_{.k}{s}_l^m+F_{.l}{s}_k^m)r_{m0}-\frac{(3n-5)}{b^{4}F}(r_{k0}{r}_l\hfill \\ \hfill & +r_{l0}{r}_k)+\frac{(n-3)}{b^{4}F}(r_{k0}{s}_l+r_{l0}{s}_k)-\frac{2}{b^{2}F}{r}_{kl|0}+\frac{(n-1)}{b^{2}F}\left(r_{k0|l}+r_{l0|k}\right)+\frac{1}{6b^{2}F}(F_{·k}{s}_{l|0}\hfill \\ \hfill & +F_{.l}{s}_{k|0})+\frac{1}{6b^{2}F}\left(F_{·k}{s}_{0|l}+F_{.l}{s}_{0|k}\right)+\frac{1}{b^2}{b}^m{r}_{kl|m}+\frac{1}{b^4}{r}_k{r}_l-\frac{1}{2b^2}(F_{.k}{s}_l^m+F_{.l}{s}_k^m)r_m\hfill \\ \hfill & +\frac{r_{kl}}{b^4}\left[\frac{4(n-2)r_{00}}{F^2}-\frac{2(n-3)}{F}{r}_0+\frac{2(n-1)}{F}{s}_0+(b^2{r}_m^m-r)\right]+\frac{(n-1)}{2b^2}(s_l^m{r}_{mk}\hfill \\ \hfill & +s_k^m{r}_{ml})-\frac{(3n-7)}{2b^4}\left(s_k{r}_l+s_l{r}_k\right)-\frac{1}{2b^2}\left(r_{k|l}+r_{l|k}\right)+\frac{1}{3\beta }(F_{.k}{s}_l^m+F_{.l}{s}_k^m)s_{m0}\hfill \\ \hfill & +\frac{(1-b^2)F}{24b^2{\beta }^2}(F_{.k}{b}_l+F_{.l}{b}_k)s_0^m{s}_m-\frac{1}{2}\left(F_{.k}{s}_{l|m}^m+F_{.l}{s}_{k|m}^m\right)+\frac{F}{3{\beta }^2}\left(b_k{s}_l^m+b_l{s}_k^m\right)s_{m0}\hfill \\ \hfill & -\frac{2F{b}_k{b}_l}{3{\beta }^3}{s}_0^m{s}_{m0}+\frac{1}{12b^2}\left[6n-5-(1-b^2)\frac{F}{2\beta }\right](F_{.k}{s}_l^m+F_{.l}{s}_k^m)s_m-\frac{(n-2)}{b^4}{s}_k{s}_l\hfill \\ \hfill & +\frac{(n-2)}{2b^2}\left(s_{k|l}+s_{l|k}\right)+\frac{b^m}{2b^2}(F_{.k}{s}_{l|m}+F_{.l}{s}_{k|m}),\hfill \end{array}$$

where ${}^{\alpha }Ri{c}_{kl}$ denotes the Ricci curvature tensor of α.

Contracting the Ricci curvature tensor with $g^{kl}$, we can obtain the expression of the scalar curvature R of Kropina metrics as following.

Proposition 2. 

Let F be a Kropina metric on an n-dimensional manifold M. Then, the scalar curvature of F is given by

$$\begin{array}{cc}\hfill R& =-\frac{24(n-2)}{b^6{F}^5}{r}_{00}^2\beta +\frac{1}{F^4}\left\{-\frac{(n-1)(n-8)}{b^4}{r}_{00}^2+\frac{8(n-2)}{b^6}{r}_{00}(5r_0-s_0)\beta \right.\hfill \\ \hfill & \left.-\frac{4(n-2)}{b^4}{r}_{00|0}\beta \right\}+\frac{2}{F^3}\left\{-\frac{1}{b^2}{}^{\alpha }Ric\beta -\frac{2(n-1)}{b^4}{r}_{00}(2r_0-s_0)+\frac{(n-1)}{b^2}{r}_{00|0}\right.\hfill \\ \hfill & -\frac{4(n-2)}{b^6}{r}_{00}r\beta +\frac{(n-3)}{b^4}{r}_m^m{r}_{00}\beta +\frac{(n-1)}{b^4}{r}_0^m{r}_{m0}\beta +\frac{4}{b^4}{s}_0^m{r}_{m0}\beta +\frac{(n-2)}{b^4}{b}^m{r}_{00|m}\beta \hfill \\ \hfill & \left.-\frac{7(n-2)}{b^6}{r}_0^2\beta +\frac{6(n-2)}{b^6}{r}_0{s}_0\beta +\frac{(n-2)}{b^4}{r}_{0|0}\beta +\frac{(n-2)}{b^6}{s}_0^2\beta -\frac{(n-2)}{b^4}{s}_{0|0}\beta \right\}\hfill \\ \hfill & +\frac{1}{F^2}\left\{{}^{\alpha }Ric+\frac{2}{b^2}{b}^k{y}^l{}^{\alpha }Ri{c}_{kl}\beta +\frac{1}{b^4}(r_m^m{b}^2-r)r_{00}+\frac{(3n^2+5n-3)}{3b^2}{s}_0^m{r}_{m0}+\frac{1}{b^2}{b}^m{r}_{00|m}\right.\hfill \\ \hfill & +\frac{r_0^2}{b^4}-\frac{(3n^2+13n-21)}{3b^4}{r}_0{s}_0-\frac{1}{b^2}{r}_{0|0}-\frac{2(2n-3)}{3b^4}(s_0^2-b^2{s}_{0|0})-\frac{(n-5)}{b^4}{r}_m^m{r}_0\beta \hfill \\ \hfill & +\frac{(n-1)}{b^4}{r}_m^m{s}_0\beta -\frac{4(n-2)}{b^4}{r}_0^m{r}_m\beta -\frac{3}{b^4}{r}_0^m{s}_m\beta +\frac{(n-1)}{b^2}{r}_{0|m}^m\beta -\frac{1}{b^2}{r}_{m|0}^m\beta \hfill \\ \hfill & \left.+\frac{(n-1)}{b^4}{s}_0^m{r}_m\beta +\frac{4(n-2)}{b^6}r(r_0-s_0)\beta +\frac{(n-2)}{b^4}{s}_0^m{s}_m\beta -\frac{(n-2)}{b^4}{b}^m(r_{0|m}-s_{0|m})\beta \right\}\hfill \\ \hfill & +\frac{1}{F}\left\{\frac{1}{2}\left({}^{\alpha }R-\frac{b^k{b}^l}{b^2}{}^{\alpha }Ri{c}_{kl}\right)\beta +\frac{(n-1)}{3\beta }{s}_0^m{s}_{m0}+\frac{(n+1)}{2b^2}(s_0{r}_m^m-s^m{r}_{m0}-s_0^m{r}_m\right.\hfill \\ \hfill & -\frac{r{s}_0}{b^2}+b^m{s}_{0|m}-b^2{s}_{0|m}^m)+\frac{(3n^2+n-7)}{6b^2}{s}_0^m{s}_m+\frac{1}{2b^6}{\left(r_m^m{b}^2-r\right)}^2\beta +\frac{(n-1)}{2b^2}{r}_m^k{s}_k^m\beta \hfill \\ \hfill & +\frac{b^m}{2b^2}{r}_{k|m}^k\beta +\frac{1}{2b^4}{r}^m{r}_m\beta -\frac{(3n-16)}{2b^4}{s}^m{r}_m\beta -\frac{1}{2b^2}{r}_{|m}^m\beta -\frac{1}{2b^6}{r}^2\beta +\frac{1}{2b^4}{b}^m{r}_{|m}\beta \hfill \\ \hfill & \left.-\frac{\beta }{2b^4}{r}^m{r}_{|m}-\frac{(n-2)}{b^4}{s}^m{s}_m\beta +\frac{(n-2)}{2b^2}{s}_{|m}^m\beta \right\}-\frac{n}{2b^2}{s}^m{s}_m-\frac{n}{4}{s}_t^m{s}_m^t-\frac{n(1-b^2)}{12b^2\beta }{s}_0^m{s}_m,\hfill \end{array}$$

(2)

where ${}^{\alpha }R$ denotes the scalar curvature of α.

4. The Proof of Main Theorem

In this section, we will prove Theorem 1.

Proof. 

“Necessity”. Assume Kropina metric F is of isotropic scalar curvature, i.e., $R=n(n-1)\kappa \left(x\right)$. Substituting (2) into $R=n(n-1)\kappa \left(x\right)$ yields

$$\begin{array}{c}\hfill {\alpha }^{10}{\mathrm{\Gamma }}_0+{\alpha }^8{\mathrm{\Gamma }}_1+{\alpha }^6{\mathrm{\Gamma }}_2+{\alpha }^4{\mathrm{\Gamma }}_3+{\alpha }^2{\mathrm{\Gamma }}_4+{\mathrm{\Gamma }}_5=0,\end{array}$$

(3)

where

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_0& =-n\left[(n-1)\kappa \left(x\right)+\frac{1}{2b^2}{s}^m{s}_m+\frac{1}{4}{s}_t^m{s}_m^t\right]\beta -\frac{n(1-b^2)}{12b^2}{s}_0^m{s}_m,\hfill \\ \hfill {\mathrm{\Gamma }}_1& =\left[\frac{1}{2}\left({}^{\alpha }R-\frac{b^k{b}^l}{b^2}{}^{\alpha }Ri{c}_{kl}\right)+\frac{1}{2b^6}{\left(r_m^m{b}^2-r\right)}^2+\frac{(n-1)}{2b^2}{r}_m^k{s}_k^m+\frac{5}{2b^4}{r}^m{r}_m-\frac{(n-1)}{b^4}{s}^m{r}_m\right.\hfill \\ \hfill & \left.-\frac{1}{2b^6}{r}^2+\frac{b^m}{2b^2}{r}_{k|m}^k-\frac{1}{2b^2}{r}_{|m}^m-\frac{(n-2)}{2b^4}{s}^m{s}_m+\frac{(n-2)}{2b^2}{s}_{|m}^m-\frac{(n-2)b^k{b}^l{s}_{k|l}}{2b^4}\right]{\beta }^3\hfill \\ \hfill & +\left[\frac{(3n^2+n-7)}{6b^2}{s}_0^m{s}_m+\frac{(n+1)}{2b^2}(r_m^m{s}_0-s^m{r}_{m0}-s_0^m{r}_m-\frac{r{s}_0}{b^2}+b^m{s}_{0|m}-b^2{s}_{0|m}^m)\right]{\beta }^2\hfill \\ \hfill & +\frac{(n-1)}{3}{s}_0^m{s}_{m0}\beta ,\hfill \\ \hfill {\mathrm{\Gamma }}_2& =\left\{{}^{\alpha }Ric+\frac{2}{b^2}{b}^k{y}^l{}^{\alpha }Ri{c}_{kl}\beta +\frac{1}{b^4}(r_m^m{b}^2-r)r_{00}+\frac{1}{b^2}{b}^m{r}_{00|m}+\frac{(3n^2+5n-3)}{3b^2}{s}_0^m{r}_{m0}\right.\hfill \\ \hfill & +\frac{r_0^2}{b^4}-\frac{(3n^2+13n-21)}{3b^4}{r}_0{s}_0-\frac{2(2n-3)}{3b^4}{s}_0^2-\frac{1}{b^2}{r}_{0|0}+\frac{2(2n-3)}{3b^2}{s}_{0|0}\hfill \\ \hfill & -\frac{(n-5)}{b^4}{r}_m^m{r}_0\beta +\frac{(n-1)}{b^4}{r}_m^m{s}_0\beta -\frac{(2n-1)}{b^4}{r}^m{r}_{m0}\beta -\frac{(2n-3)}{b^4}{s}^m{r}_{m0}\beta \hfill \\ \hfill & +\frac{(n-2)}{b^4}{s}_0^m{r}_m\beta +\frac{4(n-2)}{b^6}r(r_0-s_0)\beta -\frac{1}{b^2}{r}_{m|0}^m\beta +\frac{(n-1)}{b^2}{r}_{0|m}^m\beta -\frac{(n-2)}{b^4}{b}^m{r}_{0|m}\beta \hfill \\ \hfill & \left.-\frac{(n-2)}{b^4}{s}_0^m{s}_m\beta +\frac{(n-2)}{b^4}{b}^m{s}_{0|m}\beta \right\}{\beta }^3,\hfill \\ \hfill {\mathrm{\Gamma }}_3& =\left\{-\frac{1}{b^2}{}^{\alpha }Ric\beta -\frac{4(n-1)}{b^4}{r}_{00}{r}_0+\frac{2(n-1)}{b^4}{r}_{00}{s}_0+\frac{(n-1)}{b^2}{r}_{00|0}+\frac{(n-3)}{b^4}{r}_m^m{r}_{00}\beta \right.\hfill \\ \hfill & +\frac{(n-1)}{b^4}{r}_0^m{r}_{m0}\beta -\frac{2(n-2)}{b^4}{s}_0^m{r}_{m0}\beta -\frac{4(n-2)}{b^6}{r}_{00}r\beta +\frac{(n-2)}{b^4}{b}^m{r}_{00|m}\beta \hfill \\ \hfill & \left.-\frac{7(n-2)}{b^6}{r}_0^2\beta +\frac{6(n-2)}{b^6}{r}_0{s}_0\beta +\frac{1}{b^4}{r}_{0|0}\beta +\frac{(n-2)}{b^6}{s}_0^2\beta -\frac{(n-2)}{b^4}{s}_{0|0}\beta \right\}{\beta }^4,\hfill \\ \hfill {\mathrm{\Gamma }}_4& =\left[-\frac{(n-1)(n-8)}{b^4}{r}_{00}^2+\frac{40(n-2)}{b^6}{r}_{00}{r}_0\beta -\frac{8(n-2)}{b^6}{r}_{00}{s}_0\beta -\frac{4(n-2)}{b^4}{r}_{00|0}\beta \right]{\beta }^5,\hfill \\ \hfill {\mathrm{\Gamma }}_5& =-\frac{24(n-2)}{b^6}{r}_{00}^2{\beta }^7.\hfill \end{array}$$

By (3), we have that ${\alpha }^2$ divides ${\mathrm{\Gamma }}_5$. Thus, there exists a scalar function $c=c\left(x\right)$ such that $r_{00}=c{\alpha }^2$, which is the second formula of (1). Thus, we deduce that

$$\begin{array}{cc}\hfill & r_{ij}=c{a}_{ij};r_{i0}=c{y}_i;r_{ij|m}=c_m{a}_{ij};r_{i0|m}=c_m{y}_i;r_{i0|0}=c_0{y}_i;\hfill \\ \hfill & r_{00|k}=c_k{\alpha }^2;r_{00|0}=c_0{\alpha }^2;r_k^k=nc;r_{k|0}^k=n{c}_0;r_i=c{b}_i;\hfill \\ \hfill & r_0=c\beta ;r_{i|j}=c_j{b}_i+c{s}_{ij}+c^2{a}_{ij};r=c{b}^2;r_{i|0}=c_0{b}_i+c{s}_{i0}+c^2{y}_i;\hfill \\ \hfill & r_{0|j}=c_j\beta +c{s}_{0j}+c^2{y}_j;r_{0|0}=c_0\beta +c^2{\alpha }^2;r_{|k}^k=c_b+n{c}^2,\hfill \end{array}$$

where $c_i=\frac{\partial c}{\partial x^i}$, $c_b=c_i{b}^i$, $c_0=c_i{y}^i$.

Substituting the above equations into (3) yields

$$\begin{array}{c}\hfill {\alpha }^6{\mathrm{\Delta }}_0+{\alpha }^4{\mathrm{\Delta }}_1+{\alpha }^2{\mathrm{\Delta }}_2+{\mathrm{\Delta }}_3=0,\end{array}$$

(4)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Delta }}_0& =-n\left[(n-1)\kappa \left(x\right)+\frac{1}{2b^2}{s}^m{s}_m+\frac{1}{4}{s}_t^m{s}_m^t\right]\beta -\frac{n(1-b^2)}{12b^2}{s}_0^m{s}_m,\hfill \\ \hfill {\mathrm{\Delta }}_1& =\frac{1}{2b^4}\{b^2(b^2{}^{\alpha }R-b^k{b}^l{}^{\alpha }Ri{c}_{kl})+(n+1)b^2[(n-2)c^2+c_b]\hfill \\ \hfill & -(n-2)(2s^m{s}_m-b^2{s}_{|m}^m)\}{\beta }^3,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Delta }}_2& =\frac{(n-2)}{b^4}(c_b-c^2){\beta }^5+\frac{1}{b^4}\left\{2b^2{b}^k{y}^l{}^{\alpha }Ri{c}_{kl}-\frac{2(8n-15)}{3}c{s}_0+(n-2)[2b^2{c}_0\right.\hfill \\ \hfill & +b^m{s}_{0|m}-s_0^m{s}_m\left]\right\}{\beta }^4+\frac{1}{b^4}\left[{}^{\alpha }Ric+\frac{2(2n-3)}{3}(b^2{s}_{0|0}-s_0^2)\right]{\beta }^3,\hfill \\ \hfill {\mathrm{\Delta }}_3& =\frac{2}{b^6}\{-b^4{}^{\alpha }Ric+(n-2)[c^2{\beta }^2+(2c{s}_0-b^2{c}_0)\beta +(s_0^2-b^2{s}_{0|0})]\}{\beta }^5.\hfill \end{array}\end{array}$$

By (4), we have that ${\alpha }^2$ divides ${\mathrm{\Delta }}_3$, i.e., there exists a scalar function $f=f\left(x\right)$ such that

$$\begin{array}{c}\hfill f{\alpha }^2=-b^4{}^{\alpha }Ric+(n-2)[c^2{\beta }^2+(2c{s}_0-b^2{c}_0)\beta +(s_0^2-b^2{s}_{0|0})].\end{array}$$

(5)

Differentiating the above equation with respect to $y^i{y}^j$ yields

$$\begin{array}{cc}\hfill 2f{a}_{ij}=& -2b^4{}^{\alpha }Ri{c}_{ij}+(n-2)[2c^2{b}_i{b}_j+2c(s_i{b}_j+s_j{b}_i)-b^2(c_i{b}_j+c_j{b}_i)\hfill \\ \hfill & +2s_i{s}_j-b^2(s_{i|j}+s_{i|j})].\hfill \end{array}$$

Contracting this formula with $b^i{b}^j$ or $a^{ij}$ yields, respectively,

$$\begin{array}{cc}\hfill f& =-b^2{b}^i{b}^j{}^{\alpha }Ri{c}_{ij}+(n-2)b^2{c}^2-(n-2)b^2{c}_b-(n-2)s^m{s}_m,\hfill \\ \hfill nf& =-b^4{}^{\alpha }R+(n-2)b^2{c}^2-(n-2)b^2{c}_b+(n-2)(s^m{s}_m-b^2{s}_{|m}^m).\hfill \end{array}$$

Combining the above two formulas, we obtain

$$\begin{array}{c}\hfill f=\frac{b^2}{n-1}\left(b^i{b}^j{}^{\alpha }Ri{c}_{ij}-b^2{}^{\alpha }R\right)+\frac{n-2}{n-1}(2s^m{s}_m-b^2{s}_{|m}^m)\end{array}$$

(6)

and

$$\begin{array}{c}\hfill s_{|m}^m=-\frac{1}{n-2}(b^2{}^{\alpha }R-n{b}^i{b}^j{}^{\alpha }Ri{c}_{ij})+(n-1)(c_b-c^2)+\frac{(n+1)}{b^2}{s}^m{s}_m.\end{array}$$

(7)

Substituting (5)–(7) into (4), we obtain

$$\begin{array}{c}\hfill {\alpha }^4{\mathrm{\Theta }}_0+{\alpha }^2{\mathrm{\Theta }}_1+{\mathrm{\Theta }}_2=0,\end{array}$$

(8)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Theta }}_0& =-n\left[(n-1)\kappa \left(x\right)+\frac{1}{2b^2}{s}^m{s}_m+\frac{1}{4}{s}_t^m{s}_m^t\right]\beta -\frac{n(1-b^2)}{12b^2}{s}_0^m{s}_m,\hfill \\ \hfill {\mathrm{\Theta }}_1& =\frac{1}{6b^4}\left[n(3n+5)\left(\frac{b^2{b}^k{b}^l}{n-2}{}^{\alpha }Ri{c}_{kl}+s^m{s}_m+b^2{c}_b\right)-2n{b}^2{c}^2\right]{\beta }^3\hfill \\ \hfill & +\frac{1}{6}\left[-\frac{6(n+1)}{n-2}{b}^k{y}^l{}^{\alpha }Ri{c}_{kl}+\frac{3n^2+4n-4}{b^2}{s}_0^m{s}_m-3(n+1)(s_{0|m}^m-\frac{n}{b^2}c{s}_0+c_0)\right]{\beta }^2\hfill \\ \hfill & +2(n-1)s_0^m{s}_{m0}\beta ,\hfill \\ \hfill {\mathrm{\Theta }}_2& =\frac{n}{3b^4}\left(-\frac{b^4}{n-2}{}^{\alpha }Ric+c^2{\beta }^2+c\beta s_0-b^2{c}_0\beta \right){\beta }^3.\hfill \end{array}\end{array}$$

By (8), we have that ${\alpha }^2$ divides ${\mathrm{\Theta }}_2$. Then, there exists a scalar function $g=g\left(x\right)$ such that

$$\begin{array}{c}\hfill g{\alpha }^2=-\frac{b^4}{n-2}{}^{\alpha }Ric+c^2{\beta }^2+c\beta s_0-b^2{c}_0\beta ,\end{array}$$

(9)

which is the first formula of (1). Differentiating the above equation with respect to $y^i$ or $y^i{y}^j$, respectively, we obtain

$$\begin{array}{cc}\hfill 2g{y}_i& =-\frac{2b^4}{n-2}{y}^l{}^{\alpha }Ri{c}_{il}+2c^2\beta b_i+c{b}_i{s}_0-b^2{c}_0{b}_i+c\beta s_i-b^2{c}_i\beta ,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill 2g{a}_{ij}& =-\frac{2b^4}{n-2}{}^{\alpha }Ri{c}_{ij}+2c^2{b}_i{b}_j+c(b_i{s}_j+b_j{s}_i)-b^2(c_i{b}_j+c_j{b}_i).\hfill \end{array}$$

(11)

Contracting (10) with $b^i$ yields

$$\begin{array}{c}\hfill 2g\beta =-\frac{2b^4}{n-2}{b}^i{y}^l{}^{\alpha }Ri{c}_{il}+b^2(2c^2-c_b)\beta +b^{2}c{s}_0-b^4{c}_0.\end{array}$$

(12)

Contracting (11) with $b^i{b}^j$ or $a^{ij}$, respectively, we obtain

$$\begin{array}{cc}\hfill g& =-\frac{b^2}{n-2}{b}^i{b}^j{}^{\alpha }Ri{c}_{ij}+b^2(c^2-c_b),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill ng& =-\frac{b^4}{n-2}{}^{\alpha }R+b^2(c^2-c_b).\hfill \end{array}$$

(14)

Comparing (13) and (14) yields

$$\begin{array}{c}\hfill g=\frac{b^2}{(n-1)(n-2)}\left(b^i{b}^j{}^{\alpha }Ri{c}_{ij}-b^2{}^{\alpha }R\right)\end{array}$$

(15)

and

$$\begin{array}{c}\hfill c_b=c^2+\frac{1}{(n-1)(n-2)}\left(b^2{}^{\alpha }R-n{b}^i{b}^j{}^{\alpha }Ri{c}_{ij}\right).\end{array}$$

(16)

Substituting (15) and (16) into (12) yields

$$\begin{array}{c}\hfill c_0=-\frac{2b^i{y}^j}{n-2}{}^{\alpha }Ri{c}_{ij}+\frac{\beta }{n-1}\left[\frac{1}{n-2}{}^{\alpha }R+\frac{b^i{b}^j}{b^2}{}^{\alpha }Ri{c}_{ij}+\frac{n-1}{b^2}{c}^2\right]+\frac{c}{b^2}{s}_0.\end{array}$$

(17)

Combining (5), (6), (9), and (15), we obtain the third formula of (1).

Substituting (9), (15)–(17) into (8) yields

$$\begin{array}{c}\hfill {\alpha }^2{\mathrm{\Omega }}_0+{\mathrm{\Omega }}_1=0,\end{array}$$

(18)

where

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_1& =\frac{1}{6}\left\{\frac{3(n+1)}{b^2}\left[\frac{1}{(n-2)}(b^2{}^{\alpha }R-2b^k{b}^l{}^{\alpha }Ri{c}_{kl})+(n-1)c^2+\frac{n(3n+5)}{3(n+1)b^2}{s}^m{s}_m\right]{\beta }^2\right.\hfill \\ \hfill & +\frac{1}{b^2}\left[(3n^2+4n-4)s_0^m{s}_m-3(n+1)b^2{s}_{0|m}^m+3(n-1)(n+1)c{s}_0\right]\beta \hfill \\ \hfill & +2(n-1)s_0^m{s}_{m0}\}\beta ,\hfill \\ \hfill {\mathrm{\Omega }}_0& =-n\left[(n-1)\kappa \left(x\right)+\frac{1}{2b^2}{s}^m{s}_m+\frac{1}{4}{s}_t^m{s}_m^t\right]\beta -\frac{n(1-b^2)}{12b^2}{s}_0^m{s}_m.\hfill \end{array}$$

By (18), we have that ${\alpha }^2$ divides ${\mathrm{\Omega }}_1$. Then, there exists a scalar function $h=h\left(x\right)$ such that

$$\begin{array}{cc}\hfill h{\alpha }^2=& \frac{3(n+1)}{b^2}\left[\frac{1}{n-2}(b^2{}^{\alpha }R-2b^k{b}^l{}^{\alpha }Ri{c}_{kl})+(n-1)c^2+\frac{n(3n+5)}{3(n+1)b^2}{s}^m{s}_m\right]{\beta }^2\hfill \\ \hfill & +\frac{1}{b^2}\left[(3n^2+4n-4)s_0^m{s}_m-3(n+1)b^2{s}_{0|m}^m+3(n-1)(n+1)c{s}_0\right]\beta \hfill \\ \hfill & +2(n-1)s_0^m{s}_{m0},\hfill \end{array}$$

(19)

which is the fourth formula of (1). Differentiating the above equation with respect to $y^i{y}^j$ yields

$$\begin{array}{cc}\hfill 2h{a}_{ij}=& \frac{6(n+1)}{b^2}\left[\frac{1}{n-2}(b^2{}^{\alpha }R-2b^k{b}^l{}^{\alpha }Ri{c}_{kl})+(n-1)c^2+\frac{n(3n+5)}{3(n+1)b^2}{s}^m{s}_m\right]b_i{b}_j\hfill \\ \hfill & +\frac{1}{b^2}\left[(3n-2)(n+2)s_m(s_j^m{b}_i+s_i^m{b}_j)+3(n-1)(n+1)c(s_j{b}_i+s_i{b}_j)\right]\hfill \\ \hfill & -3(n+1)(s_{j|m}^m{b}_i+s_{i|m}^m{b}_j)+4(n-1)s_i^m{s}_{mj}.\hfill \end{array}$$

(20)

Contracting (20) with $b^i{b}^j$ or $a^{ij}$, respectively, we have

$$\begin{array}{cc}\hfill h=& \frac{3(n+1)}{(n-2)}\left[b^2{}^{\alpha }R-2b^i{b}^j{}^{\alpha }Ri{c}_{ij}\right]+\frac{(3n+2)}{b^2}{s}^m{s}_m+3(n+1)[(n-1)c^2+s_{|m}^m+s_t^m{s}_m^t],\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill nh=& \frac{3(n+1)}{(n-2)}\left[b^2{}^{\alpha }R-2b^i{b}^j{}^{\alpha }Ri{c}_{ij}\right]+\frac{(n+4)}{b^2}{s}^m{s}_m+3(n+1)[(n-1)c^2+s_{|m}^m]+(n+5)s_t^m{s}_m^t.\hfill \end{array}$$

(22)

Comparing (21) and (22) yields

$$\begin{array}{c}\hfill h=-2\left(\frac{s^m{s}_m}{b^2}+s_t^m{s}_m^t\right),\end{array}$$

(23)

$$\begin{array}{c}\hfill s_{|m}^m=\frac{1}{n-2}[2b^i{b}^j{}^{\alpha }Ri{c}_{ij}-b^2{}^{\alpha }R]-(n-1)c^2-\frac{3n+4}{3(n+1)b^2}{s}^m{s}_m-\frac{3n+5}{3(n+1)}{s}_t^m{s}_m^t.\end{array}$$

(24)

By (7) and (24), we obtain

$$\begin{array}{c}\hfill c_b=-\frac{1}{n-1}\left[b^i{b}^j{}^{\alpha }Ri{c}_{ij}+\frac{3n^2+9n+7}{3(n+1)b^2}{s}^m{s}_m+\frac{3n+5}{3(n+1)}{s}_t^m{s}_m^t\right].\end{array}$$

(25)

Therefore, by (16), we have

$$\begin{array}{c}\hfill c^2=\frac{2b^i{b}^j{}^{\alpha }Ri{c}_{ij}-b^2{}^{\alpha }R}{(n-1)(n-2)}-\frac{1}{3(n+1)(n-1)}\left[\frac{3n^2+9n+7}{b^2}{s}^m{s}_m+(3n+5)s_t^m{s}_m^t\right].\end{array}$$

(26)

Substituting (26) into (17) yields

$$\begin{array}{cc}\hfill c_0& =\frac{\beta }{(n-1)b^2}\left\{\frac{n{b}^i{b}^j{}^{\alpha }Ri{c}_{ij}}{(n-2)}-\frac{1}{3(n+1)}\left[\frac{3n^2+9n+7}{b^2}{s}^m{s}_m+(3n+5)s_t^m{s}_m^t\right]\right\}\hfill \\ \hfill & -\frac{2b^i{y}^j}{n-2}{}^{\alpha }Ri{c}_{ij}+\frac{c{s}_0}{b^2}.\hfill \end{array}$$

(27)

Substituting (23) into (18) yields

$$\begin{array}{c}\hfill R=n(n-1)\kappa \left(x\right)=-\frac{n(1-b^2)}{12b^2\beta }{s}_0^m{s}_m-\frac{3n+2}{6b^2}{s}^m{s}_m-\frac{3n+4}{12}{s}_t^m{s}_m^t.\end{array}$$

“Sufficiency”. It is obviously true.

This completes the proof of Theorem 1. □

5. Other Related Results

In this section, we consider $s_0=0$ in Theorem 1.

Corollary 1. 

Let F be a Kropina metric on an n$(\ge 3)$-dimensional manifold M. Assume $s_0=0$. Then F is of isotropic scalar curvature if and only if

$$\left\{\begin{array}{c}{\displaystyle b^4{}^{\alpha }Ric=\frac{1}{n-1}(b^2{}^{\alpha }R-b^i{b}^j{}^{\alpha }Ri{c}_{ij})(b^2{\alpha }^2-{\beta }^2)-b^i{b}^j{}^{\alpha }Ri{c}_{ij}{\beta }^2-2b^2{b}^i{y}^j{}^{\alpha }Ri{c}_{ij}\beta ,}\hfill \\ {\displaystyle r_{00}=c\left(x\right){\alpha }^2,}\hfill \\ {\displaystyle 0=s_t^m{s}_m^t(2{\alpha }^2-\frac{3n+5}{b^2}{\beta }^2)-3(n+1)s_{0|m}^m\beta +2(n-1)s_0^m{s}_{m0}.}\hfill \end{array}\right.$$

(28)

In this case, $R=-\frac{3n+4}{12}{s}_m^t{s}_t^m$.

Proof. 

Sufficiency is obviously true. Next we prove necessity. Assume that F is of isotropic scalar curvature, i.e., $R=n(n-1)\kappa \left(x\right)$. By Theorem 1, obviously, $r_{00}=c\left(x\right){\alpha }^2$, (5), (19), (26) and (27) are true. When $s_0=0$, (26) and (27) can be simplified as

$$\begin{array}{cc}\hfill c^2=& \frac{(2b^i{b}^j{}^{\alpha }Ri{c}_{ij}-b^2{}^{\alpha }R)}{(n-1)(n-2)}-\frac{(3n+5)}{3(n+1)(n-1)}{s}_t^m{s}_m^t,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill c_0=& \frac{\beta }{(n-1)b^2}\left[\frac{n{b}^i{b}^j{}^{\alpha }Ri{c}_{ij}}{(n-2)}-\frac{(3n+5)}{3(n+1)}{s}_t^m{s}_m^t\right]-\frac{2}{n-2}{b}^i{y}^j{}^{\alpha }Ri{c}_{ij}.\hfill \end{array}$$

(30)

Substituting $s_0=0$, (29) and (30) into (5), we obtain

$$f{\alpha }^2=-b^4{}^{\alpha }Ric-2b^2{b}^i{y}^j{}^{\alpha }Ri{c}_{ij}\beta -\frac{b^2{}^{\alpha }R+(n-2)b^i{b}^j{}^{\alpha }Ri{c}_{ij}}{n-1}{\beta }^2,$$

where $f=\frac{b^2}{n-1}(b^i{b}^j{}^{\alpha }Ri{c}_{ij}-b^2{}^{\alpha }R)$. This is the first formula of (28).

Substituting $s_0=0$ and (29) into (19), we obtain

$$h{\alpha }^2=-\frac{3n+5}{b^2}{s}_t^m{s}_m^t{\beta }^2-3(n+1)s_{0|m}^m\beta +2(n-1)s_0^m{s}_{m0},$$

where $h=-2s_t^m{s}_m^t$. This is the third formula of (28).

By Theorem 1, in this case, $R=-\frac{3n+4}{12}{s}_m^t{s}_t^m$. □

Corollary 2. 

Let F be a Kropina metric on an n$(\ge 3)$-dimensional compact manifold M. Then F is of isotropic scalar curvature if and only if

$$\left\{\begin{array}{c}{\displaystyle b^4{}^{\alpha }Ric=\frac{1}{n-1}(b^2{}^{\alpha }R-b^i{b}^j{}^{\alpha }Ri{c}_{ij})(b^2{\alpha }^2-{\beta }^2)-b^i{b}^j{}^{\alpha }Ri{c}_{ij}{\beta }^2-2b^2{b}^i{y}^j{}^{\alpha }Ri{c}_{ij}\beta ,}\hfill \\ {\displaystyle r_{00}=c\left(x\right){\alpha }^2,}\hfill \\ {\displaystyle 0=(2{\alpha }^2-\frac{3n+5}{b^2}{\beta }^2)s_t^m{s}_m^t-3(n+1)s_{0|m}^m\beta +2(n-1)s_0^m{s}_{m0}.}\hfill \end{array}\right.$$

In this case, $R=-\frac{3n+4}{12}{s}_m^t{s}_t^m$.

Proof. 

Sufficiency is obviously true. Next we prove necessity. Assume that F is of isotropic scalar curvature, i.e., $R=n(n-1)\kappa \left(x\right)$. By Theorem 1, (24) and (26) are true. Substituting (26) into (24), we obtain

$$\begin{array}{c}\hfill s_{|m}^m=\frac{n+1}{b^2}{s}^m{s}_m.\end{array}$$

Using the divergence theorem, when M is a compact manifold, $s_0=0$. By Corollary 1, Corollary 2 is true. □

Based on Lemma 2 and Theorem 1, we obtain the following result.

Theorem 2. 

Let a Kropina metric F be of isotropic scalar curvature. Then, F is of isotropic S curvature if and only if S = 0.

Proof. 

Assume that F is of isotropic scalar curvature. By Theorem 1, we know that $r_{00}=c{\alpha }^2$. By Lemma 2, the result is obviously true. □

Lemma 3 

([6]). For a Finsler metric or a spray on a manifold M, $R_{imj}^m=R_{jmi}^m$ if and only if ${\chi }_i=0$.

Remark 1. 

Li–Shen defined $\chi ={\chi }_{i}d{x}^i$ with the S curvature in [11], where ${\chi }_i:=\frac{1}{2}\{S_{.i|m}{y}^m-S_{|i}\}.$ Based on Theorem 2, we know that ${\chi }_i$ for a Kropina metric with isotropic scalar curvature vanishes, i.e., $R_{imj}^m=R_{jmi}^m$. This means that $Ri{c}_{ij}={\overline{Ric}}_{ij}$.

6. Conclusions

In this paper, we study the Kropina metric with isotropic scalar curvature. Firstly, we obtain the expressions of Ricci curvature tensor and scalar curvature. Based on these, we characterize Kropina metrics with isotropic scalar curvature by tensor analysis in Theorem 1. In Corollary 2, we discuss the case of a compact manifold. Kropina metrics with isotropic scalar curvature deserve further study by the navigation method.