2 Preliminaries

Let M be an n-dimensional $C^{\infty }$ connected manifold, $TM=\bigcup _{x \in M}T_{x}M$ the tangent bundle, and $TM_0:=TM-\{0\}$ the slit tangent bundle. Let $(M, F)$ be a Finsler manifold and $\mathbf {G}=y^i\delta /\delta x^i$ be its induced spray on $TM$ which in a standard coordinate $(x^i,y^i)$ for $TM_0$ is given by

$$\begin{align*}\mathbf{G}=y^i {{\partial} \over {\partial x^i}}-2G^i(x,y){{\partial} \over {\partial y^i}}, \ \ \ \ G^i:=\frac{1}{4}g^{il}\Big[\frac{\partial^2F^2}{\partial x^k \partial y^l}y^k-\frac{\partial F^2}{\partial x^l}\Big]. \end{align*}$$

Then, for a vector $y \in T_xM_{0}$ , the Riemann curvature is a family of linear transformation $\textbf {R}_y: T_xM \rightarrow T_xM$ which is defined by $\textbf {R}_y(u):=R^i_{k}(y)u^k {\partial / {\partial x^i}}$ , where

(2.1) $$ \begin{align} R^i_{k}=2{\partial G^i \over {\partial x^k}}-{\partial^2 G^i \over {{\partial x^j}{\partial y^k}}}y^j+2G^j{\partial^2 G^i \over {{\partial y^j}{\partial y^k}}}-{\partial G^i \over {\partial y^j}}{\partial G^j \over {\partial y^k}}. \end{align} $$

The family $\textbf {R}:=\{\textbf {R}_y\}_{y\in TM_0}$ is called the Riemann curvature. Let us put

(2.2) $$ \begin{align} R^i_{\ kl}:=\frac{1}{3}\Big\{\frac{\partial R^i_k}{\partial y^l}-\frac{\partial R^i_l}{\partial y^k}\Big\}, \ \ \ R^i_{j\ kl}:=\frac{1}{3}\Big\{\frac{\partial^2 R^i_k}{\partial y^j \partial y^l}-\frac{\partial^2 R^i_l}{\partial y^j \partial y^k}\Big\}. \end{align} $$

Then

(2.3) $$ \begin{align} R^i_{k}=R^i_{j\ kl}y^jy^l, \ \ \ \ R^i_{\ kl}=R^i_{j\ kl}y^j, \ \ \ \ R^i_{j\ kl}+R^i_{j\ lk}=0. \end{align} $$

Let $(M, F)$ be an n-dimensional Finsler manifold. Put

$$\begin{align*}\mathbf{Ric}:=\sum^n_{i,j=1}g^{ij}\Big(\mathbf{R}_y(b_i), b_j\Big), \end{align*}$$

where $\{b_i\}$ is a basis for $T_xM$ , $g_{ij}:=g(b_i, b_j)$ and $(g^{ij}):=(g_{ij})^{-1}$ . $\mathbf {Ric}$ is a well-defined scalar function on $TM_0$ . We call $\mathbf {Ric}$ the Ricci curvature. In a local coordinate system,

$$\begin{align*}\mathbf{Ric} =g^{ij}R_{ij}=R^m_m. \end{align*}$$

For a flag $\Pi :=\mathrm {span}\{y, u\} \subset T_xM$ with flagpole y, the flag curvature $\mathbf { K}=\mathbf {K}(\Pi , y)$ is defined by

(2.4) $$ \begin{align} \mathbf{K}(x, y, \Pi):= {\mathbf{g}_y \big(u, \mathbf{R}_y(u)\big) \over \mathbf{g}_y(y, y) \mathbf{g}_y(u,u) -\mathbf{g}_y(y, u)^2 }. \end{align} $$

The flag curvature $\mathbf {K}(x, y, \Pi )$ is a function of tangent planes $\Pi =\mathrm {span}\{ y, v\}\subset T_xM$ . This quantity tells us how curved the space is at a point. If F is a Riemannian metric, $\mathbf {K}(x, y, \Pi )=\mathbf {K}(x, \Pi )$ is independent of $y\in \Pi \setminus \{0\}$ . A Finsler manifold $(M, F)$ is said to have scalar flag curvature if $\mathbf {K}(\Pi _x, y) = \mathbf {K}(x, y)$ is only a function of $(x, y)\in TM_0$ . As a special case, $(M, F)$ is called of weakly isotropic flag curvature if

$$\begin{align*}\textbf{K} = \frac{3\theta}{F}+ \sigma, \end{align*}$$

where $\theta =\theta _i(x)y^i$ is a 1-form and $\sigma = \sigma (x)$ is a scalar function on M. Also, it is called of isotropic flag curvature if $\mathbf { K}(\Pi _x, y) = \mathbf {K}(x)$ – namely, the flag curvature is only a function of $x\in M$ . $(M, F)$ is said to have constant flag curvature if $\mathbf {K}(\Pi _x, y) = constant$ everywhere.

Let $(M, F)$ be a Finsler manifold. The following quadratic form $\textbf {g}_y:T_xM \times T_xM\rightarrow \mathbb {R}$ is called fundamental tensor:

$$\begin{align*}\textbf{g}_{y}(u,v)={1 \over 2}\frac{\partial^2}{\partial s\partial t} \Big[ F^2 (y+su+tv)\Big]_{s=t=0}, \ \ u,v\in T_xM. \end{align*}$$

Let $x\in M$ and $F_x:=F|_{T_xM}$ . To measure the non-Euclidean feature of $F_x$ , one can define $\mathbf {C}_y:T_xM\times T_xM\times T_xM\rightarrow \mathbb {R}$ by

$$\begin{align*}\mathbf{C}_{y}(u,v,w):={1 \over 2} \frac{d}{dt}\left[\textbf{g}_{y+tw}(u,v) \right]_{t=0}, \ \ u,v,w\in T_xM. \end{align*}$$

The family $\mathbf {C}:=\{\mathbf {C}_y\}_{y\in TM_0}$ is called the Cartan torsion.

Let $c=c(t)$ be a $C^{\infty }$ curve and $U(t)= U^i(t) {\partial }/{\partial } x^i|_{c(t)}$ be a vector field along c. Define the covariant derivative of $U(t)$ along c by

$$\begin{align*}D_{\dot{c}} U(t) := \Bigg \{ {dU^i\over dt}(t) + U^j (t) {{\partial} G^i\over {\partial} y^j} \big ( c(t), \dot{c}(t) \big ) \Bigg \} {{\partial} \over {\partial} x^i}\Big|_{c(t)}. \end{align*}$$

$U(t)$ is said to be linearly parallel if $D_{\dot {c}} U(t)=0$ .

For a vector $y\in T_xM$ , define

$$ \begin{align*} \mathbf{L}_y (u,v,w): = {d\over dt} \Big [ \mathbf{C}_{\dot{\sigma}(t) } \big ( U(t), V(t), W(t) \big )\Big ]|_{t=0}, \end{align*} $$

where $\sigma =\sigma (t)$ is the geodesic with $\sigma (0)=x$ , $\dot {\sigma }(0)=y$ , and $U(t), V(t), W(t)$ are linearly parallel vector fields along $\sigma $ with $U(0)=u, V(0)=v, W(0)=w$ . We call $\mathbf {L}_y$ the Landsberg curvature. The Landsberg curvature measures the rate of change of the Cartan torsion along geodesics.

Define $\mathbf {B}_y:T_xM\times T_xM \times T_xM\rightarrow T_xM$ by $\mathbf {B}_y(u, v, w):=B^i_{\ jkl}(y)u^jv^kw^l{{\partial }/ {\partial x^i}}|_x$ , where

$$\begin{align*}B^i_{\ jkl}:={{\partial^3 G^i} \over {\partial y^j \partial y^k \partial y^l}}. \end{align*}$$

The quantity $\mathbf {B}$ is called the Berwald curvature. F is called a Berwald metric if $\mathbf {B}=0$ . In this case, $G^i$ are quadratic in $y\in T_xM$ for all $x\in M$ ; that is, there exists $\Gamma ^i_{\ jk}=\Gamma ^i_{\ jk}(x)$ such that

$$\begin{align*}G^i=\Gamma^i_{\ jk} y^j y^k. \end{align*}$$

Taking a trace of Berwald curvature $\mathbf {B}$ give us the mean of Berwald curvature $\mathbf {E}$ which is defined by $\mathbf {E}_y:T_xM\times T_xM \rightarrow \mathbb {R}$ , where

(2.5) $$ \begin{align} \mathbf{E}_y (u, v) := {1\over 2} \sum_{i=1}^n g^{ij}(y) g_y \Big ( \mathbf{B}_y (u, v, e_i ) , e_j \Big ). \end{align} $$

The family $\mathbf {E}=\{ \mathbf {E}_y \}_{y\in TM\setminus \{0\}}$ is called the mean Berwald curvature. In local coordinates, $\mathbf {E}_y(u, v):=E_{ij}(y)u^iv^j$ , where

$$\begin{align*}E_{ij}:=\frac{1}{2}B^m_{\ mij}. \end{align*}$$

Taking a horizontal derivation of the mean of Berwald curvature $\mathbf {E}$ gives us the H-curvature $\mathbf {H}$ which is defined by $\mathbf { H}_y=H_{ij}dx^i\otimes dx^j$ , where

$$\begin{align*}H_{ij}:= E_{ij|m}y^m. \end{align*}$$

Here, $"|\text{"}$ denotes the horizontal covariant differentiation with respect to the Berwald connection.

3 Proof of Theorem 1.1

In this section, we are going to prove Theorem 1.1. For this aim, we will need some useful geometrical and topological facts about the homogeneous Finsler manifolds. In [Reference Tayebi and Najafi17], by using the property of exponential map, the following is proved.

Every two points of a homogeneous Finsler manifold $(M, F)$ map to each other under an isometry. This causes the norm of an invariant tensor under the isometries of a homogeneous Finsler manifold to be a constant function on the manifold M, and consequently, it has a bounded norm. Then, we have the following.

Now, we consider the flag curvature of homogeneous R-quadratic Finsler metrics and prove the following.

Here, we give a relation between homogeneous R-quadratic metrics and Landsberg metrics.

Proof The following Bianchi identity for the Berwald connection of F holds:

(3.1) $$ \begin{align} B^h_{\ mjk|i} - B^h_{\ mik|j}=R^h_{\ mij,k}. \end{align} $$

For more details, see page 136 in [Reference Shen14]. The R-quadratic Finsler metric is characterized by $R^h_{\ mij,k}=0$ . Then (3.1) reduces to

(3.2) $$ \begin{align} B^h_{\ mjk|i}=B^h_{\ mik|j}. \end{align} $$

Contacting (3.2) with $y_hy^i$ yields

(3.3) $$ \begin{align} L_{mjk|i}y^i=0. \end{align} $$

For any geodesic $c=c(t)$ and any parallel vector field $U=U(t)$ along c, we define

(3.4) $$ \begin{align} \mathbf{C}(t):=\mathbf{C}_{\dot c}\big(U(t),U(t),U(t)\big), \ \ \ \ \mathbf{L}(t):=\mathbf{L}_{\dot c}\big(U(t),U(t),U(t)\big). \end{align} $$

By (3.4) and the definition of $\mathbf {L}_y$ , we get

(3.5) $$ \begin{align} \mathbf{L}(t)=\mathbf{C}^{'}(t). \end{align} $$

By (3.3), we obtain

(3.6) $$ \begin{align} \mathbf{L}^{'}(t)=0. \end{align} $$

The equation (3.6) implies that

(3.7) $$ \begin{align} \mathbf{L}(t)=\mathbf{L}(0). \end{align} $$

Considering (3.5) and taking an integral of (3.7) yields

(3.8) $$ \begin{align} \mathbf{C}(t)=\mathbf{L}(0)t+\mathbf{C}(0). \end{align} $$

By using Lemma 3.1, one can put $t\rightarrow \infty $ in (3.8) and then Lemma 3.2 implies $\mathbf {L}(0)=0$ . Therefore, F reduces to a Landsberg metric.

Here, we consider homogeneous R-quadratic Randers metrics of nonzero flag curvature and prove the following rigidity result.

By using the celebrated Akbar-Zadeh theorem for Finsler metrics of vanishing flag curvature, we prove the following result which plays a key lemma in our investigation.

Now, we are ready to prove Theorem 1.1.

The condition of scalar flag curvature in Theorem 1.1 cannot be ignored. For example, see the following.

Also, in Theorem 1.1, the condition R-quadratic is necessary. Here, we give an example that certifies our claim.

We must mention that Theorem 1.1 does not hold for non-homogeneous Finsler metrics. See the following example.

Example 3 Let us consider the following Randers metric defined nearby the origin in $\mathbb {R}^n$ :

$$\begin{align*}F :=\frac{\sqrt{|y|^2(1-|xA|^2)+\langle y, xA\rangle^2}}{1-|xA|^2}-\frac{\langle y, xA\rangle}{1-|xA|^2}, \end{align*}$$

where $|.|$ and $\langle , \rangle $ denote the Euclidean norm and inner product in $\mathbb {R}^n$ , respectively, and $A:= (a^i_j)$ is a nonzero and anti-symmetric matrix. F is a R-quadratic and is of scalar flag curvature [Reference Li and Shen11]. However, it is not Riemannian nor locally Minkowskian.

Here, we give another alternative proof for a special case of Theorem 1.1 which is independent of Numata’s Theorem.

Randers metrics are special $(\alpha , \beta )$ -metrics [Reference Cheng and Shen5]. As an interesting result, we study homogeneous R-quadratic $(\alpha , \beta )$ -metrics and prove the following.

Let $(M, F)$ be an n-dimensional Finsler manifold, $TM$ its tangent bundle, and $(x^{i},y^{i})$ the coordinates in a local chart on $TM$ . Let F be a scalar function on $TM$ defined by $F=\sqrt [m]{A}$ , where A is given by $A:=a_{i_{1}\dots i_{m}}(x)y^{i_{1}}y^{i_{2}}\dots y^{i_{m}}$ , with $a_{i_{1}\dots i_{m}}$ symmetric in all its indices. F is called an m-th root Finsler metric.

An m-th root Finsler metric $F=\sqrt [m]{a_{i_{1}\dots i_{m}}(x)y^{i_{1}}y^{i_{2}}\dots y^{i_{m}}}$ is regarded as a direct generalization of Riemannian metric in the sense that the second root metric is a Riemannian metric $F=\sqrt {a_{ij}(x) y^iy^j}$ . The fourth root metrics $F=\sqrt [4]{a_{ijkl}(x) y^iy^jy^ky^l}$ are called quartic metric. The special 4-th root metric – namely, Berwald-Moór metric – plays an important role in the theory of space-time structure, gravitation, and general relativity.

Proof For a fourth root Finsler metric $F=\sqrt [4]{A}$ on an open subset $\mathcal {U}\subset \mathbb {R}^n$ , let us put $A_{ij}=[A]_{y^iy^j}$ . Suppose that the matrix $(A_{ij})$ defines a positive definite tensor and $(A^{ij})$ denotes its inverse. Then the spray coefficients $G^i$ of F are given by following:

(3.9) $$ \begin{align} G^k=\frac{1}{2}(A_{0j}-A_{x^j})A^{kj}, \end{align} $$

where $A_{x^i}=[A]_{x^i}$ and $A_{0l}:=A_{x^my^l}y^m$ . By (3.9), it follows that the spray coefficients of fourth root Finsler metrics are rational functions in y. Therefore, the Riemannian curvature $\mathbf {R}$ and Ricci curvature $\mathbf {Ric}$ are rational functions in y. By assumption, F has weakly isotropic flag curvature. Thus, we have

(3.10) $$ \begin{align} R^i_j=\left(\frac{3\theta}{F}+\sigma\right)F^2 h^i_j, \end{align} $$

where $\theta =\theta _i(x)y^i$ is a 1-form and $\sigma =\sigma (x)$ and is a scalar function on M. Taking a trace of (3.10) yields

(3.11) $$ \begin{align} \sigma F^2+3\theta F-\frac{1}{n-1}\mathbf{Ric}=0. \end{align} $$

We have two main cases as follows:

Case (i): Let at a point $x_0\in M$ , we have $\sigma (x_0)=0$ . In this case, (3.11) reduces to

(3.12) $$ \begin{align} \mathbf{Ric}(x_0, y)=3(n-1)\theta(x_0, y) F^2(x_0, y), \ \ \ \ \forall y\in T_{x_0}M. \end{align} $$

The left side of (3.12) is a rational function in y, while the right side is an irrational function in y. Thus, $\theta (x_0, y)=0$ and, by considering (3.10), $F|_{x_0}$ reduces to a R-flat metric.

Case (ii): Let at a point $x_0\in M$ , $\sigma (x_0)\neq 0$ holds. In this case, (3.11) is written as follows:

(3.13) $$ \begin{align} \sigma(x_0) F^2(x_0, y)+3\theta(x_0, y) F(x_0, y)-\frac{1}{n-1}\mathbf{Ric}(x_0, y)=0. \end{align} $$

For a fourth root metric F, we have $F(x_0, -y) = F(x_0, y)$ . By assumption, F is R-quadratic and then $R^i_{\ j}(x, -y)=R^i_{\ j}(x, y)$ . It follows that $\mathbf {Ric}(x_0, -y)=\mathbf {Ric}(x_0, y)$ . Thus, $y\rightarrow -y$ in (3.13) gives us

(3.14) $$ \begin{align} \sigma(x_0) F^2(x_0, y)-3\theta(x_0, y) F(x_0, y)-\frac{1}{n-1}\mathbf{Ric}(x_0, y)=0. \end{align} $$

By (3.13) and (3.14), we get

(3.15) $$ \begin{align} F^2(x_0, y)=\frac{1}{(n-1)\sigma(x_0)}\mathbf{Ric}(x_0, y). \end{align} $$

According to (3.15), we find that $F^2|_{x_0}$ is rational in y which is a contradiction. Thus, only the case (i) holds, and by Lemma 3.5, F is locally Minkowskian. The converse is trivial.

4 Proof of Theorems 1.2, 1.3, and 1.4

In this section, we are going to prove Theorems 1.2, 1.3, and 1.4. To prove Theorem 1.2, we need to extend Mo’s result in [Reference Mo12] to strongly Ricci-quadratic metrics. More precisely, we show that every strongly Ricci-quadratic Finsler metric satisfies $\mathbf {H}=0$ . Our approach is completely different from Mo. For this aim, we need the structure equations of the Berwald connection.

Throughout this paper, we use the Berwald connection on Finsler manifolds. Let $\{e_j\}$ be a local frame for $\pi ^*TM$ , $\{\omega ^i, \omega ^{n+i}\}$ be the corresponding local coframe for $T^*(TM_0)$ , and $\{\omega ^i_j\}$ be the set of local Berwald connection forms with respect to $\{e_j\}$ . Then the connection forms of the Berwald connection are characterized by the following structure equations:

• • Torsion freeness:

  (4.1) $$ \begin{align} d\omega^i = \omega^j \wedge \omega^i_{\ j}. \end{align} $$

• • Almost metric compatibility:

  (4.2) $$ \begin{align} dg_{ij}-g_{jk}\Omega^k_{\ i}-g_{ik}\Omega^k_{\ j}=-2L_{ijk}\omega^k+2C_{ijk}\omega^{n+k}, \end{align} $$
  where $\omega ^i := dx^i$ , $\omega ^{n+k}:= dy^k + y^j\omega ^k_{\ j}$ , and $\Omega ^i_{\ j}$ are the curvature forms of the Berwald connection defined by following:
  (4.3) $$ \begin{align} \Omega^i_{\ j}=d\omega^i_{\ j}-\omega^k_{\ j}\wedge\omega^i_{\ k}:=\frac{1}{2}R^i_{\ jkl}\omega^k\wedge\omega^l-B^i_{\ jkl}\omega^k\wedge\omega^{n+l}. \end{align} $$

The horizontal and vertical covariant derivations with respect to the Berwald connection are denoted by “ $|$ ” and “,” respectively. Thus,

$$\begin{align*}g_{ij|k}= -2L_{ijk}, \ \ \ g_{ij,k}= 2C_{ijk}. \end{align*}$$

For more details, one can see [Reference Shen14].

Proof Differentiating (4.2) yields the following Ricci identity:

(4.4) $$ \begin{align} g_{pj}\Omega^p_{\ i}-g_{pi}\Omega^p_{\ j}=-2L_{ijk|l}\omega^k\wedge\omega^l-2L_{ijk,l}\omega^k\wedge\omega^{n+l}- 2C_{ijl|k}\omega^k\wedge\omega^{n+l}\nonumber \\ -2C_{ijl,k}\omega^{n+k}\wedge\omega^{n+l}- 2C_{ijp}\Omega^p_{\ l}y^l. \end{align} $$

It follows from (4.4) that

(4.5) $$ \begin{align} C_{ijl|k}+ L_{ijk,l}=\frac{1}{2}g_{pj}B^p_{\ ikl}+\frac{1}{2}g_{ip}B^p_{\ jkl}. \end{align} $$

Contracting (4.5) with $y^j$ yields

(4.6) $$ \begin{align} L_{jkl}=-\frac{1}{2}y^mg_{im}B^i_{\ jkl}. \end{align} $$

Differentiating of (4.3) implies that

(4.7) $$ \begin{align} d\Omega^{\ j}_ i -\omega ^{\ k}_i \wedge \Omega^{\ j}_ k +\omega ^{\ j}_k \wedge\Omega^{\ k}_ i=0. \end{align} $$

Define $B^i_{\ jkl|m}$ and $B^i_{\ jkl,m}$ by

(4.8) $$ \begin{align} dB^i_{\ jkl} -B^i_{\ mkl}\omega^m_{\ i}-B^i_{\ jml}\omega^m_{\ k}-B^i_{\ jkm}\omega^m_{\ l}+B^i_{\ jkl}\omega^i_{\ m}:=B^i_{\ jkl|m}\omega^m+B^i_{\ jkl,m}\omega^{n+m}. \end{align} $$

Similarly, one can define $R^i_{\ jkl|m}$ and $R^i_{\ jkl,m}$ by following

(4.9) $$ \begin{align} dR^i_{\ jkl}- R^i_{\ mkl}\omega^m_{\ i}-B^i_{\ jml}\omega^m_{\ k}-R^i_{\ jkm}\omega^m_{\ l}+R^i_{\ jkl}\omega^i_{\ m}:= R^i_{\ jkl|m}\omega^m+R^i_{\ jkl,m}\omega^{n+m}. \end{align} $$

By (4.7), (4.8), and (4.9), one obtains the following Bianchi identities:

(4.10) $$ \begin{align} &R^h_{\ mij|k}+ R^h_{\ mjk|i} + R^h_{\ mki|j} = -B^h_{\ mir}R^r_{\ jk} - B^h_{\ mjr}R^r_{\ ki}- B^h_{\ mkr}R^r_{\ ij}, \end{align} $$

(4.11) $$ \begin{align} &B^i_{\ jkl|m}-B^i_{\ jkm|l}=R^i_{\ jkl,m}, \end{align} $$

(4.12) $$ \begin{align} &B^i_{\ jkl,m}=B^i_{\ jkm,l}. \end{align} $$

Letting $i=k$ in (4.11) yields

(4.13) $$ \begin{align} B^k_{\ jkl|m}-B^k_{\ jkm|l}=R^k_{\ jkl,m}. \end{align} $$

F is a strongly Ricci-quadratic metric; then from (4.13), we get

(4.14) $$ \begin{align} B^k_{\ jkl|m}=B^k_{\ jkm|l}. \end{align} $$

Multiplying (4.14) with $y^m$ implies that

$$\begin{align*}H_{jk}=E_{jk|m}y^m=0. \end{align*}$$

This completes the proof.

Generally, two-dimensional Finsler metrics have some different special Riemannian and non-Riemannian curvature properties from the higher dimensions. In [Reference Yan and Deng20], Yan-Deng studied homogeneous Einstein $(\alpha , \beta )$ -metrics and proved that any homogeneous Ricci-flat $(\alpha , \beta )$ -metric with vanishing S-curvature must be a Minkowski space. Here, we prove the following.

Proof Every Finsler surface has scalar flag curvature $\mathbf {K}=\mathbf {K}(x, y)$ – namely, F satisfies

(4.15) $$ \begin{align} R^i_j=\mathbf{K}F^2h^i_j. \end{align} $$

Taking a trace of (4.15) implies that $\mathbf {Ric}=\mathbf {K}F^2$ . From the assumption, we find that $\mathbf {K}=0$ and then F is a R-flat Finsler metric. By Lemma 3.5, we get the proof.

Proof Let F be a homogeneous Finsler metric of isotropic flag curvature $\mathbf {K}=\mathbf {K}(x)$ . Thus, we have $R^i_{\ j}=\mathbf {K}F^2h^i_j$ and taking a trace of it yields

(4.16) $$ \begin{align} \mathbf{Ric}=(n-1)\mathbf{K}F^2. \end{align} $$

By assumption, F is strongly Ricci-quadratic

(4.17) $$ \begin{align} \mathbf{Ric}=R^m_m=R^m_{j\ ml}(x)y^jy^l. \end{align} $$

Comparing (4.16) and (4.17) implies that

(4.18) $$ \begin{align} \mathbf{K}F^2=\frac{1}{n-1}R^m_{j\ ml}(x)y^jy^l. \end{align} $$

Let at a point $x_0\in M$ , $\mathbf {K}(x_0)= 0$ holds. In this case, the homogeneousness of $(M, F)$ implies that $\mathbf {K}(x)= 0$ , $\forall x\in M$ . By (4.16), F is Ricci-flat. By assumption, we get $R^i_{\ j}=0$ . Then, Lemma 3.5 implies that F is locally Minkowskian.

Now, let $\mathbf {K}(x)\neq 0$ , $\forall x\in M$ . Then (4.18) yields

$$\begin{align*}F=\sqrt{\frac{1}{(n-1)\mathbf{K}}R^m_{i\ mj}y^iy^j}. \end{align*}$$

In this case, F is Riemannian.

A Finsler metric F on an n-dimensional manifold M is called an Einstein metric if its Ricci curvature satisfies $\textbf {Ric}=(n - 1)\mu F^2$ , where $\mu = \mu (x)$ is a scalar function on M. In [Reference Deng and Hou7], Deng-Hou proved that a homogeneous Einstein-Randers space with negative Ricci curvature is Riemannian. Here, an extension of their result to two-dimensional homogeneous metrics is presented.

Proof Every Finsler surface of scalar flag curvature $\mathbf {K}=\mathbf {K}(x, y)$ satisfies $R^i_j=\mathbf {K}F^2h^i_j$ . Taking a trace of it yields

(4.19) $$ \begin{align} \mathbf{Ric}=\mathbf{K} F^2. \end{align} $$

By assumption, F is an Einstein Finsler metric

(4.20) $$ \begin{align} \mathbf{Ric}=\mu F^2, \end{align} $$

where $\mu =\mu (x)$ is a scalar function on M. By considering (4.19) and (4.20), we get $\mathbf {K}=\mu (x)$ (i.e., F is of isotropic flag curvature). Every scalar function on M which is invariant under isometries of $(M, F)$ is a constant function. Thus, the homogeneity of $(M, F)$ and invariancy of the flag curvature under isometries of F imply that $\mathbf {K}=constant$ . If $\mathbf {K}=0$ , then by Lemma 3.5, F is locally Minkowskian. If $\mathbf {K}<0$ , then by Lemmas 3.1 and 3.2, and Akbar-Zadeh’s theorem in [Reference Akbar-Zadeh1], F reduces to a Riemannian metric. This completes the proof.

In [Reference Yan and Deng20], Deng-Yan proved that a homogeneous $(\alpha , \beta )$ -space with vanishing S-curvature and negative Ricci curvature must be Riemannian. As a rigidity result, they showed that a homogeneous Ricci-flat $(\alpha , \beta )$ -space with vanishing S-curvature must be locally Minkowskian. Here, we extend their theorem for the Finsler metric satisfying $\mathbf {H}=0$ and prove the following.

The 4-th Hilbert problem is to characterize Finsler metrics on an open subset in $\mathbb {R}^n$ whose geodesics are straight lines. Such Finsler metrics are called projectively flat Finsler metrics. For homogeneous locally projectively flat Randers surfaces, we prove the following.

Proof Let $F = \alpha +\beta $ be a locally projectively flat Randers metric on an n-dimensional manifold M. Suppose that it has constant Ricci curvature

(4.21) $$ \begin{align} \mathbf{Ric} = (n -1)c F^2, \end{align} $$

where $c=constant$ . In this case, it is proved that $c\leq 0$ (see Theorem 8.1.2 in [Reference Cheng and Shen5]). However, every projectively flat Finsler metric is of scalar flag curvature – namely, it satisfies $R^i_j=\mathbf {K}F^2h^i_j$ , where $\mathbf {K}=\mathbf {K}(x,y)$ is a scalar function on $TM$ . Taking a trace of it yields

(4.22) $$ \begin{align} \mathbf{Ric}=(n-1)\mathbf{K}F^2. \end{align} $$

By (4.21) and (4.22), it follows that $\mathbf {K}=c$ . If $c=0$ , then by Lemma 4.2, F is locally Minkowskian. If $c<0$ , then by Akbar-Zadeh’s theorem, F reduces to a Riemannian metric.

As the final conclusion of Section 4, we consider three-dimensional homogeneous Randers metrics and prove the following.

It is interesting to characterize Einstein metrics with quadratic Ricci curvature. Then, we prove the following.

Proof Let F be an Einstein metric

(4.23) $$ \begin{align} \mathbf{Ric}=(n-1)\lambda F^2, \end{align} $$

where $\lambda =\lambda (x)$ is a scalar function on M. By (4.23), we have

$$\begin{align*}\lambda=\frac{1}{n-1}\frac{\mathbf{Ric}}{F^{2}}, \end{align*}$$

which shows that $\lambda $ is invariant under isometries. According to the assumption, F is strongly Ricci-quadratic. Thus,

(4.24) $$ \begin{align} \mathbf{Ric}=R^m_{j\ ml}(x)y^jy^l. \end{align} $$

Let at a point $x_0\in M$ , $\lambda (x_0)= 0$ holds. In this case, the homogeneousness of $(M, F)$ and the invariant property of $\lambda $ imply that $\lambda (x)= 0$ , $\forall x\in M$ . In this case, F reduces to a Ricci-flat metric. Now, suppose that $\lambda (x)\neq 0$ , $\forall x\in M$ . Then by (4.23) and (4.24), we get $F=\sqrt {a_{ij}(x)y^iy^j}$ , where

$$\begin{align*}a_{ij}:=\frac{1}{(n-1)\lambda}R^m_{i\ mj}(x). \end{align*}$$

It shows that F is Riemannian. In the case of $dim(M)=2$ , by Lemma 4.2, we get the result.

For 4-th root Finsler surfaces of Einstein-type, we get the following.

An $(\alpha , \beta )$ -metric is called of polynomial-type if $\phi (s) = c_ks^k +c_{k-1}s^{k-1}+\cdots + c_1s + 1$ , where $2\leq i\leq k$ are real constants, $c_k\neq 0$ and $k\geq 2$ . The well-known square metric $F=(\alpha +\beta )^2/\alpha ^2$ is a special polynomial-type $(\alpha , \beta )$ -metric. Here, we prove the following.

Let $(M, F)$ be an n-dimensional Finsler manifold. For a nonzero vector $y\in T_xM$ , define the Weyl tensor $\mathbf {W}_y:T_xM \rightarrow T_xM$ by $\mathbf { W}_y(u):=W^i_j(y)u^j{\partial }/{\partial x^i}\big |_x$ , where

(4.25) $$ \begin{align} W^i_j:=A^i_j-\frac{1}{n+1}\frac{\partial A^k_j}{\partial y^k}y^i, \end{align} $$

and

$$\begin{align*}A^i_j:=R^i_j-R \delta^i_j, \ \ \ R:=\frac{1}{n-1}R^m_{\ m}. \end{align*}$$

Then, F is called a Weyl metric if $\mathbf {W}=0$ . It is well-known that a Finsler metric is of scalar flag curvature $\mathbf {K}=\mathbf {K}(x, y)$ if and only if it is a Weyl metric. A Finsler metric F is said to be W-quadratic if $W ^i _k$ are quadratic in y. Every Weyl metric is a trivial W-quadratic metric. Also, R-quadratic Finsler metrics are W-quadratic. But the converse may not hold. In [Reference Li and Shen11], Li and Shen find the necessary and sufficient condition under which a Randers metric $F = \alpha + \beta $ is W-quadratic.

A Finsler metric is called a generalized Douglas-Weyl metric if its Douglas tensor satisfies $D^i_{\ jkl|m}y^m= T_{jkl}y^i$ for some tensor $T_{jkl}$ . All Douglas and Weyl metrics are generalized Douglas-Weyl metrics [Reference Cheng and Shen5]. Here, we prove Theorem 1.3.

In the class of non-homogeneous Finsler metrics, we have many W-quadratic Finsler metrics on a closed manifold. Indeed, by the uniformization theorem, every closed surface with genus $>1$ admits a Riemannian metric of constant negative sectional curvature. So, by perturbing it, we can get non-Riemannian Randers metrics of negative curvature. Using the same argument in Theorem 1.3, we find that these metrics are W-quadratic metrics.

Considering Cheng-Shen’s conjecture, another question arises:

Here, we show that there is not any pure homogeneous two-dimensional W-quadratic metric on a closed surface. More precisely, we prove Theorem 1.4 as follows.

5 Counterexamples to the Cheng-Shen conjecture

In [Reference Atashafrouz and Najafi2], Atashafrouz-Najafi constructed a three-dimensional homogeneous W-quadratic Randers metric which is not Berwald-type. Here, by using the navigation problem, we construct a four-dimensional W-quadratic Randers metric on a closed manifold which is nontrivial in the sense that it is not a Weyl metric.

Example 4 (four-dimensional W-quadratic Randers Metric)

Any Randers metric $F=\alpha +\beta $ on the manifold M is a solution of the following Zermelo navigation problem:

$$\begin{align*}\textbf{h}\Big(x,\frac{y}{F}-\mathcal{W}_x\Big)=1, \end{align*}$$

where $\textbf {h}=\sqrt {h_{ij}(x)y^iy^j}$ is a Riemannian metric and $\mathcal {W}=\mathcal {W}^i(x){\partial }/{\partial x^i}$ is a vector field such that

$$\begin{align*}\textbf{h}(x,-\mathcal{W}_x)=\sqrt{h_{ij}(x)\mathcal{W}^i(x)\mathcal{W}^j(x)}<1. \end{align*}$$

In fact, $\alpha $ and $\beta $ are given by

$$\begin{align*}\alpha=\frac{\sqrt{\lambda h^2+\mathcal{W}_0}}{\lambda},\qquad \beta=-\frac{\mathcal{W}_0}{\lambda}, \end{align*}$$

respectively, and moreover, $\lambda =1-\|\mathcal {W}\|^2_h$ and $W_0=h_{ij}\mathcal {W}^iy^j$ [Reference Cheng and Shen5]. Now, F can be written as follows:

(5.1) $$ \begin{align} F=\frac{\sqrt{\lambda \textbf{h}^2+\mathcal{W}_0^2}}{\lambda}-\frac{\mathcal{W}_0}{\lambda}. \end{align} $$

In this case, the pair $(\mathbf {h}, \mathcal {W})$ is called the navigation data of F. On the Lie group $G=SU(2)\times \mathbb {S}^1$ , we can find a non-Riemannian bi-invariant Randers metric F which is induced by navigation from a bi-invariant Riemannian metric $\mathbf {h}$ on G and the vector field $\mathcal {W}=c\partial /\partial t$ from $\mathbb {S}^1$ . Then F is a Douglas metric because it is of Berwald-type. Thus, it is W-quadratic. However, F is not of scalar flag curvature – namely,

$$\begin{align*}\mathbf{W}\neq 0. \end{align*}$$

At any point $x\in G$ , linearly independent vectors $y,z\in T_xG$ are tangent to the $SU(2)$ factor, and the nonzero vector $w\in T_xG$ is tangent to the $\mathbb {R}$ -factor. Then, we get

$$\begin{align*}\mathbf{K}(x,y,y\wedge z)>0, \ \ \ \mathbf{K}(x,y,y\wedge w)=0. \end{align*}$$

It follows that F is not projectively flat.

In the following example, we quote the well-known classification of two-dimensional Lie groups which admits a left invariant Randers metric.

Maybe, other than Example 5, there exist nontrivial homogeneous W-quadratic metrics of dimension $n\geq 3$ . During the preparation of this paper, Libing Huang informed me that Lun Zhang classified all three-dimensional Lie groups which admit a left invariant Randers metric of scalar flag curvature. His result shows that there are only three possibilities:

1. (i) The group $E(2)$ of all Euclidean motions on the plane with a locally Minkowski metric;

2. (ii) The group $SU(2)$ or $SO(3)$ with Bao-Shen metrics;

3. (iii) The hyperbolic model of a Riemannian space of constant sectional curvature -1, with a Randers metric, is given by this Riemannian metric plus a left invariant closed one-form.

Case (iii) of the above classification certifies our claim. It is remarkable that Zhang has excluded the trivial case of a commutative group $\mathbb {R}^3$ , with Minkowski metric. Case (i) can also be found by Huang (for more details, see Section 5.1 in [Reference Huang10]).

By considering Theorem 1.3, an interesting question arises:

Here, by using Theorems 1.1 and 1.2, we construct a family of two-dimensional homogeneous W-quadratic metrics that are not trivial.