A class of Randers metrics of scalar flag curvature

TL;DR

This paper investigates the classification of Randers metrics with scalar flag curvature in Finsler geometry, providing necessary conditions under the assumption that a certain 1-form is Killing.

Contribution

It introduces new necessary conditions for Randers metrics of scalar flag curvature when a specific 1-form is Killing, advancing the classification problem.

Findings

Derived necessary conditions for Randers metrics with scalar flag curvature

Identified the role of Killing 1-forms in the classification

Contributed to the understanding of scalar flag curvature in Finsler geometry

Abstract

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that $\beta$ is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.

Equations

$$F=\alpha +\beta ,$$

$$V^i{\eta }_{j;i}+{\eta }^i{V}_{i;j}=2c{\eta }_j,$$

$$F\left(x,\frac{y}{\tilde{F}}-V\right)=1.$$

$${\mathbf{K}}_{\tilde{F}}(x,y)={\mathbf{K}}_F(x,\tilde{y})-c^2,$$

$$b_{i;j}+b_{j;i}=0,$$

$$s_{0;m}^m=-(n-1)b^{-2}c(x)\beta ,$$

$$t_{00}+s_{0;0}=c(x)({\alpha }^2-b^{-2}{\beta }^2),$$

$$t_0=-\frac{n-1}{n+1}(\lambda +c{b}^{-2})\beta ,$$

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

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

$$\mathbf{Ric}=R_m^m.$$

$${\mathbf{Ric}}_{ij}:=\frac{1}{2}{\mathbf{Ric}}_{y^i{y}^j}.$$

$${\rm\bf K}(x,y,P):=\frac{g_{y}({\rm\bf R}_{y}(u),u)}{g_{y}(y,y)g_{y}(u,u)-\big{[}g_{y}(u,y)\big{]}^{2}},$$

$$R_k^i=\mathbf{K}(x,y)\left(F^2{\delta }_k^i-F{F}_{y^k}{y}^i\right),$$

$$\mathbf{Ric}=(n-1)\mathbf{K}(x,y)F^2.$$

$$d{V}_{BH}:={\sigma }_{BH}(x){\omega }^1\wedge \cdots \wedge {\omega }^n,$$

$$\sigma_{BH}(x):=\frac{Vol\big{(}{\bf B}^{n}(1)\big{)}}{Vol\left\{(y^{i})\in R^{n}|F(x,y^{i}{\bf b}_{i})<1\right\}}.$$

$$\tau (x,y):=\ln \left[\frac{\sqrt{det(g_{ij}(x,y))}}{{\sigma }_{BH}(x)}\right].$$

$$\mathbf{S}(x,y):=\frac{d}{dt}\left[\tau \left(\sigma (t),\dot{\sigma }(t)\right)\right]{|}_{t=0}.$$

$$\mathbf{S}(x,y):={\tau }_{|m}(x,y)y^m,$$

$$I_i(x,y):=\frac{\partial \tau }{\partial y^i}(x,y)=\frac{1}{2}{g}^{jk}(x,y)\frac{\partial g_{jk}}{\partial y^i}(x,y).$$

$$L_{ijk}(x,y):=-\frac{1}{2}F{F}_{y^m}\frac{{\partial }^3{G}^m}{\partial y^i\partial y^j\partial y^k}.$$

$${\mathbf{S}}_{\cdot i|m}{y}^m-{\mathbf{S}}_{|i}=-\frac{n+1}{3}{\mathbf{K}}_{\cdot i}{F}^2,$$

$$J_{i|m}{y}^m+\mathbf{K}{F}^2{I}_i=-\frac{n+1}{3}{F}^2{\mathbf{K}}_{\cdot i}.$$

$$r_{ij}:=\frac{1}{2}(b_{i;j}+b_{j;i}),s_{ij}:=\frac{1}{2}(b_{i;j}-b_{j;i}),$$

$$r_j^i:=a^{ik}{r}_{kj},s_j^i:=a^{ik}{s}_{kj},r_j:=b^i{r}_{ij},s_j:=b^i{s}_{ij},$$

$$q_{ij}:=r_{im}{s}_j^m,t_{ij}:=s_{im}{s}_j^m,q_j:=b^i{q}_{ij},t_j:=b^i{t}_{ij},$$

$$I_i=$$

$$J_i=$$

$$\displaystyle+\frac{\alpha}{b^{2}-s^{2}}\Big{[}\Psi_{1}+s\frac{\Phi}{\Delta}\Big{]}(r_{00}-2\alpha Qs_{0})h_{i}+\alpha\Big{[}-\alpha Q^{\prime}s_{0}h_{i}+\alpha Q(\alpha^{2}s_{i}-y_{i}s_{0})$$

$$\displaystyle+\alpha^{2}\Delta s_{i0}+\alpha^{2}(r_{i0}-2\alpha Qs_{i})-(r_{00}-2\alpha Qs_{0})y_{i}\Big{]}\frac{\Phi}{\Delta}\bigg{\}}.$$

$$\bar{J}:=J_i{b}^i=-\frac{1}{2\Delta {\alpha }^2}\left\{{\Psi }_1(r_{00}-2\alpha Q{s}_0)+\alpha {\Psi }_2(r_0+s_0)\right\},$$

Full text

Keywords: Randers metric, flag curvature, Killing 1-form, S-curvature, mean Landsberg curvature.

MR(2000) Subject Classification: 53B40, 53C60

1 Introduction

In Finsler geometry, the flag curvature is the natural analogue of sectional curvature in Riemannian geometry and is an important Riemannian geometric quantity. The flag curvature characterizes the shape of Finsler spaces, which was firstly introduced by L. Berwald ([3, 4]). For a Finsler manifold $(M,F)$, the flag curvature ${\bf K=K}(x,y,P)$ of $F$ is a function of “flag” $P\in T_{x}M$ and “flagpole” $y\in T_{x}M$ at $x$ with $y\in P$. A Finsler metric $F$ is said to be of scalar flag curvature if for any non-zero vector $y\in T_{x}M,{\bf K=K}(x,y)$ is independent of $P$ containing $y\in T_{x}M$ (that is, the flag curvature is just a scalar function on the slit tangent bundle). In particular, a Finsler metric $F$ is said to be of weakly isotropic flag curvature if its flag curvature is a scalar function on $TM$ in the form ${\bf K}=\frac{3c_{x^{m}}y^{m}}{F(x,y)}+\sigma(x)$, where $c=c(x)$ and $\sigma=\sigma(x)$ are scalar functions on $M$. A Finsler metric $F$ is of constant flag curvature if ${\bf K=}constant$. In 1929, Berwald proved that a projectively flat Finsler metric must be of scalar flag curvature ([5]). However, we can find infinitely many Finsler metrics of scalar flag curvature, which are not projective flat. Therefore, one of the most important problems in Finsler geometry is to study and characterize Finsler metrics of scalar flag curvature. As we known, every two-dimensional Finsler metric is of scalar flag curvature. It is then a natural problem to understand Finsler metrics of scalar flag curvature in dimension $n\geq 3$.

Randers metrics are the simplest non-Riemannian Finsler metrics that can be expressed in the following special form

[TABLE]

where $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric and $\beta=b_{i}y^{i}$ is a non-zero 1-form on $M$ such that the norm of $\beta$ with respect to $\alpha$ satisfies that $||\beta||_{\alpha}<1$. Randers metrics were introduced by physicist G. Randers in 1941 from the standpoint of general relativity ([14]). Later on, these metrics were used in the theory of the electron microscope by R. S. Ingarden in 1957, who first named them Randers metrics. Randers metrics form an important class of Finsler metrics with a strong presence in both the theory and applications of Finsler geometry, and studying Randers metrics is an important step to understand general Finsler metrics. It is an important and fundamental problem in Finsler geometry to characterize and classify Randers metrics of scalar flag curvature. Matsumoto and Shimada characterized Randers metrics of constant flag curvature ([13]). In 2003, Z. Shen classified locally projectively flat Randers metrics of constant flag curvature ([16]). Further, D. Bao, Z. Shen and C. Robles showed that a Randers metric $F$ with navigation data $(h,W)$ is of constant flag curvature if and only if $h$ is a Riemannian metric of constant sectional curvature and $W$ is homothetic with respect to $h$. Based on this, they classified Randers metrics of constant flag curvature via navigation data ([1]). Furthermore, the first author, X. Mo and Z. Shen classified projectively flat Randers metrics with isotropic S-curvature ([7]). In this case, the metrics must be of weakly isotropic flag curvature ${\bf K}=\frac{3c_{x^{m}}y^{m}}{F(x,y)}+\sigma(x)$. Later, the first author and Z. Shen classified completely Randers metrics of weakly isotropic flag curvature on the manifolds of dimension $n\geq 3$ ([9]).

For the current research on the classification problem of Randers metrics of scalar flag curvature, it is very important to find and construct some new examples. B. Chen and L. Zhao constructed a new class of Randers metrics of scalar flag curvature ([6]). Further, Z. Shen and Q. Xia proved firstly the following theorem:

Theorem A([17]) Let $\bar{F}=\alpha+\bar{\beta}$ be a Randers metric of weakly isotropic flag curvature on a manifold $M$ with navigation data $(\bar{h},\bar{W})$ and $V$ be a homothetic vector field on $(M,\bar{F})$ with dilation $c$. Let $F=\alpha+\beta$ be a Randers metric on $M$ such that $\eta:=\beta-\bar{\beta}$ is closed. Assume that $\eta$ satisfies the following

[TABLE]

where “;” denotes the covariant derivative with respect to the Levi-Civita connection of $\bar{h}$ and $V_{i}:=\bar{h}_{ij}V^{j},\ \eta^{i}:=\bar{h}^{ij}\eta_{j}$. Then

• •

$V$ is also homothetic vector field with respect to $F$.

• •

With this $V$ such that $F(x,-V_{x})<1$, define $\tilde{F}=\tilde{\alpha}+\tilde{\beta}$ by

[TABLE]

Then $\tilde{F}$ is of scalar flag curvature and its flag curvature is related to that of $F$ by

[TABLE]

where $\tilde{y}:=y-\tilde{F}(x,y)V$.

Then, based on Theorem A, they found some new examples of Randers metrics of scalar flag curvature ([17]).

Soon after, Q. Xia studied Randers metrics of Douglas type and proved the following theorem:

Theorem B ([18]) Let $F=\alpha+\beta$ be a Randers metric of Douglas type on an $n$-dimensional manifold $M(n\geq 3)$ and $V$ be a conformal vector field on $(M,F)$ with a conformal factor $c(x)$. Let $\tilde{F}=\tilde{\alpha}+\tilde{\beta}$ be a Randers metric defined from $(F,V)$ given by (1.1). Then each two of the following imply the third one:

1. (1)

$F$ is of scalar flag curvature. 2. (2)

$\tilde{F}$ is of scalar flag curvature. 3. (3)

$V$ is homothetic or $\beta=0$.

Then, by using a special Randers metric $F=|y|+\langle x,y\rangle$ and a conformal vector field $V=xQ$ with $|xQ|<1$ on $R^{n}$, where $Q$ is an antisymmetric matrix, Q. Xia constructed a new Randers metric $\tilde{F}=\tilde{\alpha}+\tilde{\beta}$ by solving (1.1) such that $\tilde{F}$ is of scalar flag curvature but $\tilde{F}$ is not locally projectively flat. In this case, $V$ is actually a Killing vector field with respect to $F$ ([18]).

However, although a significant progress has been made in studying Randers metrics of scalar flag curvature as mentioned above, it is still an important open problem in Finsler geometry to characterize and classify Randers metrics of scalar flag curvature.

For a 1-form $\beta=b_{i}y^{i}$ on $M$, we say that $\beta$ is a Killing 1-form with respect to a Riemannian metric $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ if it satisfies

[TABLE]

where $``\ ;"$ is the covariant derivative with respect to Levi-Civita connection of $\alpha$. In this paper, we study classification problem of Randers metrics of scalar flag curvature. Under the condition that $\beta$ is a Killing 1-form, we obtain some necessary conditions for Randers metrics to be of scaler flag curvature. Firstly, we have the following theorem.

Theorem 1.1

Let $F=\alpha+\beta$ be a Randers metric on an $n$-dimensional manifold $M$. Suppose that $\beta$ is a Killing 1-form with respect to $\alpha$. If $F$ is of scalar flag curvature ${\bf K=K}(x,y)$ , then $\alpha$ and $\beta$ satisfy the following equations

[TABLE]

where $t_{ij}=s_{im}s^{m}_{\ j}$, $t_{00}=t_{ij}y^{i}y^{j}$ and $b:=||\beta_{x}||_{\alpha}$ denotes the norm of $\beta$ with respect to $\alpha$ and $c=c(x)$ is a scalar function on $M$.

Further, based on Theorem 1.1 and Proposition 4.1 below, we can obtain the following theorem.

Theorem 1.2

Let $F=\alpha+\beta$ be a Randers metric on an $n$-dimensional manifold $M$. Suppose that $\beta$ is a Killing 1-form with respect to $\alpha$. If $F$ is of scalar flag curvature ${\bf K=K}(x,y)$, then $\alpha$ and $\beta$ satisfy the following equation

[TABLE]

where $t_{i}=b^{j}t_{ji}$, $t_{0}=t_{i}y^{i}$, $c=c(x)$ and $\lambda=\lambda(x)$ are scalar functions on $M$.

As an application of Theorem 1.1 and Theorem 1.2, we study Randers metric on $S^{3}$ constructed by D. Bao and Z. Shen in [2]. Bao-Shen have proved that this Randers metric is of constant flag curvature $K>1$. We will show that this metric also satisfies (1.2), (1.3) and (1.4) (see Section 5). Besides, Theorem 1.1 and Theorem 1.2 tell us, if a Randers metric $F=\alpha+\beta$ with Killing 1-form $\beta$ does not satisfy one of (1.2), (1.3) and (1.4), then $F$ is not of scalar flag curvature.

2 Preliminaries

Let $F$ be a Finsler metric on an $n$-dimensional manifold $M$. The geodesics of $F=F(x,y)$ are characterized by

[TABLE]

where $G^{i}$ are the geodesic coefficients of $F$.

For any $x\in M$ and $y\in T_{x}M\backslash\{0\}$, the Riemann curvature ${\bf R}_{y}=R^{i}_{\ k}\frac{{\partial}}{{\partial}x^{i}}\otimes dx^{k}$ is defined by

[TABLE]

The trace of the Riemann curvature is Ricci curvature Ric, which is defined by

[TABLE]

Obviously, the Ricci curvature is a positive homogenous function of degree two in $y$. The Ricci tensor is defined by

[TABLE]

The flag curvature of Finsler manifold $(M,F)$ is the function ${\bf K}={\bf K}(x,y,P)$ of a two-dimensional plane called “flag” $P\subset T_{x}M$ and a “flagpole” $y\in P\backslash\{0\}$ defined by

[TABLE]

where $P=Span\{y,u\}$.

It is known that $F$ is of scalar flag curvature if and only if, in a standard local coordinate system we have ([11])

[TABLE]

which means

[TABLE]

In Finsler geometry, there are some important quantities which all vanish for Riemannian metrics. Hence they are said to be non-Riemannian. Let $\{{\bf b}_{i}\}$ be a basis for $T_{x}M$ and $\{\omega^{i}\}$ be the basis for $T^{*}_{x}M$ dual to $\{{\bf b}_{i}\}$. Define the Busemann-Hausdorff volume form by

[TABLE]

where

[TABLE]

Here $Vol\{\cdot\}$ denotes the Euclidean volume function on subsets in ${\bf R}^{n}$ and ${\bf B}^{n}(1)$ denotes the unit ball in ${\bf R}^{n}$.

Define

[TABLE]

$\tau=\tau(x,y)$ is well-defined, which is called the distortion of $F$. The distortion $\tau$ characterizes the geometry of tangent space $(T_{x}M,F_{x})$. It is natural to study the rate of change of the distortion along geodesics. For a vector $y\in T_{x}M\setminus\{0\}$, let $\sigma=\sigma(x)$ be the geodesic with $\sigma(0)=x$ and $\dot{\sigma}(0)=y$. Put

[TABLE]

Equivalently,

[TABLE]

where $``|"$ denotes the horizontal covariant derivative with respect to Chern connection of $F$. ${\bf S}={\bf S}(x,y)$ is called the S-curvature of Finsler metric $F$. The S-curvature ${\bf S}(x,y)$ measures the rate of change of $(T_{x}M,F_{x})$ in the direction $y\in T_{x}M$ ([11, 15]). In particular, ${\bf S}=0$ for Riemannian metrics.

There are other several important non-Riemannian quantities in Finsler geometry. Let

[TABLE]

The tensor ${\bf I}_{y}:=I_{i}(x,y)dx^{i}$ is called the mean Cartan torsion of $F$. According to Deicke’s theorem, a Finsler metric $F$ is Riemannian if and only if ${\bf I}_{y}=0$.

Another important non-Riemannian quantity is the Landsberg curvature ${\bf L}=L_{ijk}(x,y)dx^{i}\otimes dx^{y}\otimes dx^{k}$ defined by

[TABLE]

A Finsler metric $F$ is called the Landsberg metric if the Landsberg curvature vanishes, i.e. ${\bf L}=0$. Further, let $J_{k}:=g^{ij}L_{ijk}$. Then ${\bf J}=J_{k}dx^{k}$ is called the mean Landsberg curvature. A Finsler metric $F$ is said to be weakly Landsbergian if ${\bf J}=0$.

Now we assume that Finsler metric $F$ is of scalar flag curvature ${\bf K}={\bf K}(x,y)$. Firstly, we have the following identity ([7, 11])

[TABLE]

where ${{\bf K}_{\cdot i}:={\bf K}_{y^{i}}}$. Further, we have following important identity which connects the mean Cartan torsion $I_{i}$ and the mean Landsberg curvature $J_{i}$ with the flag curvature ([11])

[TABLE]

In the following, let us consider $(\alpha,\beta)$-metrics in the form $F=\alpha\phi(\beta/\alpha)$, where $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric and $\beta=b_{i}y^{i}$ is a 1-form on a manifold. Let

[TABLE]

[TABLE]

[TABLE]

where $b^{i}:=a^{ij}b_{j}$, $(a^{ij})$ is the inverse of $(a_{ij})$ and “ ; ” denotes the covariant derivative with respect to Levi-Civita connection of $\alpha$. We use $(a^{ij})$ and $(a_{ij})$ to raise or lower the indices of a tensor.

For an $(\alpha,\beta)$-metric $F=\alpha\phi(s),\ s=\beta/\alpha$, we have the formulas for the mean Landsberg curvature ([12]) and the mean Cartan torsion ([8]) of $F$ as follows

[TABLE]

Besides, we also have ([12])

[TABLE]

where

[TABLE]

3 Some properties of Randers metrics of scalar flag curvature

Let $F=\alpha+\beta$ be a Randers metric on an $n$-dimensional manifold $M$ and $\beta$ be a Killing 1-form (i.e., $r_{ij}=0$) with respect to $\alpha$. In this case, $\phi=1+s$ and we have

[TABLE]

From these and (2.6)-(2.8), we get

[TABLE]

and

[TABLE]

In the following, we always assume that Randers metric $F=\alpha+\beta$ is of scalar flag curvature. In this case, by the formulas (4.15) and (5.30) for Ricci curvature of $F$ in [10], we have

[TABLE]

where $\kappa=\kappa(x)$ is a scalar function on $M$. Obviously, by (2.3) and (3.3), we can get

Lemma 3.1

Let $F=\alpha+\beta$ be a Randers metric on an $n$-dimensional manifold $M$. If $F$ is of scalar flag curvature ${\bf K=K}(x,y)$ and $\beta$ is a Killing 1-form with respect to $\alpha$, then

[TABLE]

where

[TABLE]

Further, contracting the (2.4) by $b^{i}$ yields the following equation

[TABLE]

Let $G^{i}$ and $\bar{G^{i}}$ denote the geodesic coefficients of $F$ and $\alpha$ respectively. The horizontal covariant derivatives ${\bf S}_{\cdot i|m}$ and ${\bf S}_{\cdot i;m}$ of ${\bf S}_{\cdot i}$ with respect to $F$ and $\alpha$ respectively are given by

[TABLE]

where $\Gamma^{l}_{\ im}=\frac{{\partial}^{2}G^{l}}{{\partial}y^{i}y^{m}}-L^{l}_{\ im}$,  $N^{l}_{m}:=\frac{{\partial}^{2}G^{l}}{{\partial}y^{m}},\ L^{l}_{\ im}:=g^{jl}L_{jim}$ and ${\bar{\Gamma}}^{l}_{\ im}:=\frac{{\partial}^{2}{\bar{G}}^{l}}{{\partial}y^{i}y^{m}}$, ${\bar{N}}^{l}_{m}:=\frac{{\partial}^{2}{\bar{G}}^{l}}{{\partial}y^{m}}$. From (3.7) and (3.8), we have

[TABLE]

Similarly, we have

[TABLE]

Substituting (3.9) (3.10) into (3.6) yields

[TABLE]

As we known, the relationship between the geodesic coefficients $G^{i}$ and ${\bar{G}}^{i}$ is as follows ([10, 11])

[TABLE]

From this, one obtains

[TABLE]

Besides, the formula for S-curvature of $F$ is given by the following ([11])

[TABLE]

where $\rho:=ln\sqrt{1-||\beta||^{2}_{\alpha}}$, $\rho_{i}=\rho_{x^{i}}$ and $\rho_{0}=\rho_{i}y^{i}$.

By a series of direct computations, we have the following

[TABLE]

From (3.12) and (3.14)-(3.17), one obtains

[TABLE]

Plugging (3.18)-(3.20) into (3.11), we obtain the following lemma.

Lemma 3.2

Let $F=\alpha+\beta$ be a Randers metric on an $n$-dimensional manifold $M$. If $F$ is of scalar flag curvature ${\bf K=K}(x,y)$ and $\beta$ is a Killing 1-form with respect to $\alpha$, then

[TABLE]

On the other hand, contracting (2.5) by $b^{i}$ yields the following equation

[TABLE]

Similar to the methods to get Lemma 3.2, we have

[TABLE]

Substituting (3.23) into (3.22) yields

[TABLE]

where ${\bar{J}}=J_{i}b^{i}$.

It is easy to see that

[TABLE]

From (3.1), (3.2), (3.12) and (3.13), (3.25), by a series of direct computations, one obtains the following

[TABLE]

Substituting (3.26)-(3.30) into (3.24) , we have the following lemma.

Lemma 3.3

Let $F=\alpha+\beta$ be a Randers metric on an $n$-dimensional manifold $M$. If $F$ is of scalar flag curvature ${\bf K=K}(x,y)$ and $\beta$ is a Killing 1-form with respect to $\alpha$, then

[TABLE]

Lemmas 3.1, 3.2 and 3.3 are fundamental for the proofs of our main theorems.

4 Proofs of The Theorems

In this section, we will prove Theorem 1.1 and 1.2. Firstly, we will prove the following proposition which plays an important role in our proofs.

Proposition 4.1

Let $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ be a Riemann metric and $\beta=b_{i}y^{i}$ be a 1-form on a manifold. Then

[TABLE]

where $``;"$ denotes the covariant derivative with respect to Levi-Civita connection of $\alpha$, ${{}^{\alpha}\bf{Ric}}$ is the Ricci curvature of $\alpha$ and ${{}^{\alpha}\bf{Ric}}_{0b}:=(\frac{1}{2}{{}^{\alpha}\bf{Ric}})_{y^{i}y^{j}}y^{i}b^{j}$.

**Proof **: By the definition of $r_{ij}$, we have

[TABLE]

On the other hand, by Ricci identity, we get

[TABLE]

Here, ${{}^{\alpha}R}_{imjk}$ denote the Riemann curvature tensor of $\alpha$. Adding all the equations from (4.2) to (4.6), we can get

[TABLE]

Contracting (4.7) with $a^{pi}$ yields

[TABLE]

where ${{}^{\alpha}R}_{j\ km}^{\ p}:={{}^{\alpha}R}_{jikm}a^{ip}$ and ${{}^{\alpha}R}^{\ p}_{j\ kb}={{}^{\alpha}R}^{\ p}_{j\ km}b^{m}$. Contracting (4.8) with $y^{j}$ yields

[TABLE]

Letting $p=k$ in (4.9), we obtain (4.1). Q.E.D.

Now we are in the position to prove our main theorems.

Proof of Theorem 1.1: Let $F=\alpha+\beta$ be a Randers metric. Assume that $F$ is of scalar flag curvature ${\bf{K=K}}(x,y)$ and $\beta$ is a Killing 1-form. Then $(\ref{lem3.1})$, $(\ref{lem3.2})$ and $(\ref{lem3.3})$ all hold by Lemma 3.1, Lemma 3.2 and Lemma 3.3.

By $(\ref{lem3.3})-(\ref{lem3.2})+(\ref{lem3.1})\ \frac{b^{2}-s^{2}}{2(n-1)\alpha(1+s)}$ and plugging $s=\frac{\beta}{\alpha}$ into the resulting equation, we can obtain

[TABLE]

which is equivalent to the following

[TABLE]

where

[TABLE]

Further, by the definition of $\rho$ and because of $r_{ij}=0$ , we have $\rho_{i}=\rho_{x^{i}}=-\frac{s_{i}}{1-b^{2}}$. Then, we can get

[TABLE]

On the other hand, it is clear that $(b^{2})_{i;j}=(b^{2})_{j;i}$. By a direct computation, we get $s_{i;j}=s_{j;i}$, that is, $s_{i;0}=s_{0;i}$. Hence, $(\ref{Xi2}),(\ref{Xi1})$ can be rewritten as

[TABLE]

From (4.10) we obtain the following fundamental equations

[TABLE]

Rewrite (4.17) as

[TABLE]

Since $\alpha^{2}$ is an irreducible polynomial in $y$, from (4.19) and (4.14), (4.15), we have

[TABLE]

where $c=c(x)$ is a scalar function on $M$. Substituting (4.20) into (4.19) yields

[TABLE]

which means that (1.2) holds. Plugging (4.21) into (4.20), we get

[TABLE]

which is just (1.3). This completes the proof of Theorem 1.1. Q.E.D.

Finally, let us give the proof of Theorem 1.2. Proof of Theorem 1.2: Let $F=\alpha+\beta$ be a Randers metric. Assume that $F$ is of scalar flag curvature ${\bf K=K}(x,y)$. By (5.30) in [10], we have the following

[TABLE]

where $\lambda:=\lambda(x)$ is a scalar function on $M$. Further, by a direct computation, we have

[TABLE]

From Proposition 4.1 and by (4.22) and the assumption that $\beta$ is a Killing 1-form, we get

[TABLE]

On the other hand, by Theorem 1.1, we have

[TABLE]

Comparing (4.23) with (4.24), we obtain (1.4). Q.E.D.

5 Example: Bao-Shen metric on $S^{3}$

In this section, we will prove that, for Bao-Shen metric on $S^{3}$ constructed in [2], $\alpha$ and $\beta$ satisfy (1.2), (1.3) and (1.4).

We view $S^{3}$ as a Lie group and let $\eta^{1}$, $\eta^{2}$ and $\eta^{3}$ be the standard right invariant 1-form on $S^{3}$ such that

[TABLE]

For a constant $K>1$, write $\varepsilon:=\sqrt{K},\ \delta:=\pm\sqrt{K-1}.$ Let $\omega^{1}:=\varepsilon\eta^{1},\ \omega^{2}:=\eta^{2},\ \omega^{3}:=\eta^{3}$. Then

[TABLE]

Consider the following Riemannian metric on $S^{3}$:

[TABLE]

The Levi-Civita connection forms $(\omega_{i}^{\ j})$ of $\alpha$ are given by $d\omega^{i}=\omega^{j}\wedge\omega_{j}^{\ i}$ and $\omega_{ij}=-\omega_{ji}$, where $\omega_{ij}:=\omega_{i}^{\ k}\delta_{kj}$. Concretely, we have

[TABLE]

Further, let

[TABLE]

Then we have

[TABLE]

Obviously, this $\beta$ has constant length with respect to Riemannian metric $\alpha$:

[TABLE]

By the Riemannian metric $\alpha$ defined by (5.1) and the 1-form $\beta$ defined by (5.2), D. Bao and Z. Shen constructed the following family of Randers metrics on $S^{3}$:

[TABLE]

Further, they have proved that this family of Randers metrics has constant flag curvature $K$ but nonzero Douglas tensor (see [2]). This family of Randers metrics is the first example of Finsler metrics of constant flag curvature which are not projectively flat.

In the following, we carry out our calculations in the moving coframe $\{\omega^{1},\omega^{2},\omega^{3}\}$ and the dual moving frame $\{e_{1},e_{2},e_{3}\}$. The covariant differentiation formulas of $\beta$ with respect to $\alpha$ are given by

[TABLE]

Straightforward calculations give:

[TABLE]

Then, we have $r_{ij}=0$, that is, $\beta$ is a Killing 1-form with respect to $\alpha$. Further, we have

[TABLE]

It is obvious that $s_{1}=s_{2}=s_{3}=0$. Hence, we have

[TABLE]

By the definition, $t_{ij}=s_{im}s^{m}_{\ j}$, we can get

[TABLE]

Then we have

[TABLE]

and $t_{1}=t_{2}=t_{3}=0$, which means

[TABLE]

Further, by $s_{ij;k}=(ds_{ij}-s_{pj}\omega_{i}^{\ p}-s_{ip}\omega_{j}^{\ p})(e_{k})$, we have the following

[TABLE]

It is easy to see that

[TABLE]

Then, from (5.1), (5.2) and (5.4), (5.5) and (5.7), we find that (1.2) and (1.3) hold for

[TABLE]

At the same time, by (5.6), it is easy to check that (1.4) holds for $\lambda=K$ and $c(x)=-(K-1)$.