Some curvature properties of Cartan spaces with mth root metrics

TL;DR

This paper investigates curvature properties of m-th root metrics in Cartan geometry, establishing conditions under which these metrics exhibit specific curvature behaviors and transformations.

Contribution

It provides new conditions for m-th root metrics to be weakly Berwald, weakly Landsberg, and locally Minkowskian under conformal $eta$-changes.

Findings

m-th root metrics with isotropic mean Berwald curvature are weakly Berwald

m-th root metrics with isotropic mean Landsberg curvature are weakly Landsberg

conformal $eta$-change conditions lead to locally Minkowskian metrics

Abstract

In this paper, we prove that every m-th root metric with isotropic mean Berwald curvature reduces to a weakly Berwald metric. Then we show that an m-th root metric with isotropic mean Landsberg curvature is a weakly Landsberg metric. We find necessary and sufficient condition under which conformal $\beta$-change of an m-th root metric be locally dually flat. Finally, we prove that the conformal $\beta$-change of locally projectively flat m-th root metrics are locally Minkowskian.

Equations

$$\bar{F}=e^{\alpha (x)}F+\beta ,$$

$$A_i=\frac{\partial A}{\partial y^i},A_{ij}=\frac{{\partial }^{2}A}{\partial y^j\partial y^j},A_{x^i}=\frac{\partial A}{\partial x^i},A_0=A_{x^i}{y}^i.$$

$$g_{ij}=\frac{A^{\frac{2}{m}}-2}{m^2}[mA{A}_{ij}+(2-m)A_i{A}_j],$$

$$g^{ij}=A^{-\frac{2}{m}}[mA{A}^{ij}+\frac{m-2}{m-1}{y}^i{y}^j],$$

$$y^i{A}_i=mA,y^i{A}_{ij}=(m-1)A_j,A^{ij}{A}_i=\frac{1}{m-1}{y}^j,$$

$$y_i=\frac{1}{m}{A}^{\frac{2}{m}-1}{A}_i,A_i{A}_j{A}^{ij}=\frac{m}{m-1}A.$$

$${\beta }_{0l}\beta +{\beta }_l{\beta }_0=2\beta {\beta }_{x^l},$$

$$A_{x^l}=\frac{1}{3m}[mA{\theta }_l+2\theta A_l+2({\alpha }_0{A}_l-{\alpha }_{x^l}A)],$$

$$\beta [(\frac{1}{m}-2)A_l{A}^{-1}{A}_0-4A_{x^l}+{\alpha }_0{A}_l]+2[A_l{\beta }_0+(A_0\beta {)}_l]=-2m{e}^{\alpha }A\Psi ,$$

$${\mathbf{g}}_y(u,v):=\frac{1}{2}\frac{{\partial }^2}{\partial s\partial t}\left[F^2(y+su+tv)\right]{|}_{s,t=0},u,v\in T_{x}M.$$

$${\mathbf{C}}_y(u,v,w):=\frac{1}{2}\frac{d}{dt}\left[{\mathbf{g}}_{y+tw}(u,v)\right]{|}_{t=0},u,v,w\in T_{x}M.$$

$$B_{jkl}^i(y):=\frac{{\partial }^3{G}^i}{\partial y^j\partial y^k\partial y^l}(y),E_{jk}(y):=\frac{1}{2}{B}_{jkm}^m(y),$$

$$\tau(x,y):=\ln\Big{[}{\sqrt{\det\Big{(}g_{ij}(x,y)\Big{)}}\over{\rm Vol}({\rm B}^{n}(1))}\cdot{\rm Vol}\Big{\{}(y^{i})\in\mathbb{R}^{n}\Big{|}\ F\Big{(}y^{i}{\partial\over\partial x^{i}}|_{x}\Big{)}<1\Big{\}}\Big{]}.$$

$${\bf S}(x,y):={d\over dt}\Big{[}\tau\Big{(}\sigma(t),\dot{\sigma}(t)\Big{)}\Big{]}_{t=0},$$

$$\mathbf{S}(x,y)=(n+1)c(x)F(x,y).$$

$$G^{i}:=\frac{1}{4}g^{il}\Big{[}\frac{\partial^{2}F^{2}}{\partial x^{k}\partial y^{l}}y^{k}-\frac{\partial F^{2}}{\partial x^{l}}\Big{]},\ \ \ \ y\in T_{x}M.$$

$$G^i=\frac{1}{2}(A_{0j}-A_{x^j})A^{ij}.$$

$$\mathbf{E}=\frac{n+1}{2}cF\mathbf{h},$$

$$\Theta (x,y):=\frac{\sqrt{|y{|}^2-(|x{|}^2|y{|}^2-<x,y{>}^2)}+<x,y>}{1-|x{|}^2},y\in T_x{B}^n(1)\simeq {\mathbb{R}}^n,$$

$$B_{jkl}^i=c\{F_{y^j{y}^k}{\delta }_l^i+F_{y^k{y}^l}{\delta }_j^i+F_{y^l{y}^j}{\delta }_k^i+F_{y^j{y}^k{y}^l}{y}^i\},$$

$$E_{ij}=\frac{n+1}{2}c{F}^{-1}{h}_{ij}.$$

$$B_{jkl}^i=\frac{2}{n+1}\{E_{jk}{\delta }_l^i+E_{kl}{\delta }_j^i+E_{lj}{\delta }_k^i+E_{jk,l}{y}^i\}.$$

$$I_i$$

$$J_i$$

$$A_{s}G^{s}_{ijk}=-2cA^{\frac{1}{m}-2}\Big{[}A^{2}A_{ijk}+(\frac{2}{m}-1)\{(\frac{2}{m}-2)A_{i}A_{j}A_{k}+A[A_{i}A_{jk}+A_{j}A_{ki}+A_{k}A_{ij}]\}\Big{]}.$$

$$A^2{A}_{ijk}+(\frac{2}{m}-1)(\frac{2}{m}-2)A_i{A}_j{A}_k+(\frac{2}{m}-1)A\{A_i{A}_{jk}+A_j{A}_{ki}+A_k{A}_{ij}\}=0.$$

$$G^i=-\frac{1}{2}{g}^{ij}{H}_{y^j},$$

$$(F^2{)}_{x^k{y}^l}{y}^k=2(F^2{)}_{x^l}.$$

$$A:=a_{i_1}..{.}_{i_m}(x)y^{i_1}...y^{i_m},B:=b_{ij}(x)y^i{y}^j.$$

$$e^{2\alpha }[2B_{x^l}+4{\alpha }_{x^l}B-B_{0l}-2{\alpha }_0{B}_l]=2({\beta }_l{\beta }_0+\beta {\beta }_{0l}-2\beta {\beta }_{x^l}),$$

$$A_{x^l}=\frac{1}{3m}[mA{\theta }_l+2\theta A_l+2({\alpha }_0{A}_l-{\alpha }_{x^l}A)],$$

$${{\rm Y}}_l{{\rm Y}}_0\beta =2{\rm Y}[(({{\rm Y}}_0\beta {)}_l+{{\rm Y}}_l{\beta }_0+{\alpha }_0\beta {{\rm Y}}_l-2{{\rm Y}}_{x^l}\beta )+2e^{\alpha }{\rm Y}\Psi ],$$

$${{\rm Y}}_p=\frac{2}{m}{A}^{\frac{2}{m}-1}{A}_p+B_p,$$

$$\displaystyle\Upsilon_{0p}=\frac{2}{m}A^{\frac{2}{m}-2}\big{[}(\frac{2}{m}-1)A_{p}A_{0}+AA_{0p}\big{]}+B_{0p},$$

Full text

Keywords: Conformal change, $m$-th root metric, $\beta$-change, Locally dually flat metric, projectively flat metric.111 2000 Mathematics subject Classification: 53C60, 53C25.

1 Introduction

Let $(M,F)$ be a Finsler manifold of dimension $n$, $TM$ its tangent bundle and $(x^{i},y^{i})$ the coordinates in a local chart on $TM$. Let $F$ be the following function on $M,$ 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 [4][9][14][15][16]. Then $F$ is called an $m$-th root Finsler metric. The theory of $m$-th root metric has been developed by Shimada [14], and applied to Biology as an ecological metric [2]. It is regarded as a direct generalization of Riemannian metric in a sense, i.e., the second root metric is a Riemannian metric.

Let $(M,F)$ be a Finsler manifold of dimension $n$. Denote by $\tau(x,y)$ the distortion of the Minkowski norm $F_{x}$ on $T_{x}M_{0}$, let $\sigma(t)$ be the geodesic with $\sigma(0)=x$ and $\dot{\sigma}(0)=y$. The rate of change of $\tau(x,y)$ along Finslerian geodesics $\sigma(t)$ called $S$-curvature. $F$ is said to have isotropic $S$-curvature and almost isotropic $S$-curvature if ${\bf S}=(n+1)cF$ and ${\bf S}=(n+1)cF+dh$, respectively, where $c=c(x)$ and $h=h(x)$ are scalar functions on $M$ and $dh=h_{x^{i}}(x)y^{i}$ is the differential of $h$ [19]. Taking twice vertical covariant derivatives of the $S$-curvature gives rise the $E$-curvature. The Finsler metric $F$ is called weakly Berwald metric if ${\bf E}=0$ and is said to have isotropic mean Berwald curvature if ${\bf E}=\frac{n+1}{2}cF{\bf h}$, where $c=c(x)$ is a scalar function on $M$ and ${\bf h}=h_{ij}dx^{i}dx^{j}$ is the angular metric.

Theorem 1.1**.**

*Let $F=\sqrt[m]{A}$ be an m-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$.

(i) For a scalar function $c=c(x)$ on $M$, the following are equivalent:*

(ia)   ${\bf S}=(n+1)cF+\eta$;

(ib)   ${\bf S}=\eta$.

(ii) For a scalar function $c=c(x)$ on $M$, the following are equivalent:

(iia)  ${\bf E}=\frac{n+1}{2}cF{\bf h}$;

(iib)  ${\bf E}=0$.

Let $(M,F)$ be a Finsler manifold. There are two basic tensors on Finsler manifolds: fundamental metric tensor ${\bf g}_{y}$ and the Cartan torsion $\textbf{C}_{y}$, which are second and third order derivatives of ${1\over 2}F_{x}^{2}$ at $y\in T_{x}M_{0}$, respectively. Taking a trace of Cartan torsion ${\bf C}_{y}$ give us the mean Cartan torsion ${\bf I}_{y}$. The rate of change of the Cartan torsion along Finslerian geodesics, $\textbf{L}_{y}$ is said to be Landsberg curvature [17][18]. Taking a trace of Landsberg curvature ${\bf L}_{y}$, yields the mean Landsberg curvature ${\bf J}_{y}$. $F$ is called isotropic mean Landsberg curvature if ${\bf J}=cF{\bf I}$, where $c=c(x)$ is a scalar function on $M$.

Theorem 1.2**.**

Let $(M,F)$ be an non-Riemannian $m$-th root Finsler manifold. For a scalar function $c=c(x)$ on $M$, the following are equivalent:

(ia)   ${\bf J}+cF{\bf I}=0$;

(ib)   ${\bf J}=0$.

There are two important transformation in Finsler geometry: conformal change and $\beta$-change. Two metric functions $F$ and $\bar{F}$ on a manifold $M$ are called conformal if the length of an arbitrary vector in the one is proportional to the length in the other, that is if $\bar{g_{ij}}=\varphi g_{ij}$. The length of vector $\varepsilon$ means here the fact that $\varphi g_{ij}$, as well as $g_{ij}$, must be Finsler metric tensor. He showed that $\varphi$ falls into a point function. A change of Finsler metric $F\rightarrow\bar{F}$ is called a $\beta$-change of $F$, if $\bar{F}(x,y)=F(x,y)+\beta(x,y)$, where $\beta(x,y)=b_{i}(x)y^{i}$ is a $1$-form on a smooth manifold $M$. It is easy to see that, if $\sup_{F(x,y)=1}|b_{i}(x)y^{i}|<1$, then $\bar{F}$ is again a Finsler metric. The notion of a $\beta$-change has been proposed by Matsumoto, named by Hashiguchi-Ichijyō and studied in detail by Shibata [6][8][13]. If the Finsler metric $F$ reduces to a Riemannian metric then ${\bar{F}}$ reduces to a Randers metric. Due to this reason, the $\beta$-change has been called the Randers change of Finsler metric, also.

Let $(M,F)$ be a Finsler manifold. In this paper, we are consider the conformal $\beta$-chenges of Finsler metrics

[TABLE]

where $\beta(x,y)=b_{i}(x)y^{i}$ is a $1$-form on a smooth manifold $M$ and $\alpha=\alpha(x)$ is the conformal factor. It is easy to see that, if $\sup_{F(x,y)=1}||\beta||<1$, then $\bar{F}$ is again a Finsler metric.

Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$. Put

[TABLE]

Suppose that $A_{ij}$ define a positive definite tensor and $A^{ij}$ denotes its inverse. The the following hold

[TABLE]

In [1], Amari-Nagaoka introduced the notion of dually flat Riemannian metrics when they study the information geometry on Riemannian manifolds. A Finsler metric $F$ on an open subset $U\subset\mathbb{R}^{n}$ is called dually flat if it satisfies $(F^{2})_{x^{k}y^{l}}y^{k}=2(F^{2})_{x^{l}}$ [12][19].

In this paper, we consider conformal $\beta$-change of locally dually flat $m$-th root Finsler metrics and prove the following.

Theorem 1.3**.**

Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$, where $A$ is irreducible. Suppose that $\bar{F}=e^{\alpha}F+\beta$ be conformal $\beta$-change of $F$ where $\beta=b_{i}(x)y^{i}$ and $\alpha=\alpha(x)$. Then $\bar{F}$ is locally dually flat if and only if there exists a 1-form $\theta=\theta_{l}(x)y^{l}$ on U such that the following hold

[TABLE]

where $\beta_{0l}=\beta_{x^{k}y^{l}}y^{k}$, $\alpha_{0}=\alpha_{x^{l}}y^{l}$, $\beta_{x^{l}}=(b_{i})_{x^{l}}y^{i}$, $\beta_{0}=\beta_{x^{l}}y^{i}$, $\beta_{0l}=(b_{l})_{0}$ and $\Psi=\alpha_{0}\beta_{l}+\beta_{0l}-2\beta_{x^{l}}-2\alpha_{x^{l}}\beta$.

A Finsler metric is said to be locally projectively flat if at any point there is a local coordinate system in which the geodesics are straight lines as point sets. It is known that a Finsler metric $F(x,y)$ on an open domain $U\subset\mathbb{R}^{n}$ is locally projectively flat if and only if $G^{i}=Py^{i}$, where $P(x,\lambda y)=\lambda P(x,y)$, $\lambda>0$ [7]. Finally, we study conformal $\beta$-change of locally projectively flat $m$-th root metrics and prove the following.

Theorem 1.4**.**

Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$, where $A$ is irreducible. Suppose that $\bar{F}=e^{\alpha}F+\beta$ be conformal $\beta$-change of $F$ where $\beta=b_{i}(x)y^{i}$ and $\alpha=\alpha(x)$. Then $\bar{F}$ is locally projectively flat if and only if it is locally Minkowskian.

2 Preliminaries

Let $M$ be a n-dimensional $C^{\infty}$ manifold. Denote by $T_{x}M$ the tangent space at $x\in M$, by $TM=\cup_{x\in M}T_{x}M$ the tangent bundle of $M$ and by $TM_{0}=TM\setminus\{0\}$ the slit tangent bundle. A Finsler metric on $M$ is a function $F:TM\rightarrow[0,\infty)$ which has the following properties: (i) $F$ is $C^{\infty}$ on $TM_{0}$; (ii) $F$ is positively 1-homogeneous on the fibers of tangent bundle $TM$; (iii) for each $y\in T_{x}M$, the following quadratic form ${\bf g}_{y}$ on $T_{x}M$ is positive definite,

[TABLE]

Let $x\in M$ and $F_{x}:=F|_{T_{x}M}$. To measure the non-Euclidean feature of $F_{x}$, define ${\bf C}_{y}:T_{x}M\otimes T_{x}M\otimes T_{x}M\rightarrow\mathbb{R}$ by

[TABLE]

The family ${\bf C}:=\{{\bf C}_{y}\}_{y\in TM_{0}}$ is called the Cartan torsion. It is well known that C=0 if and only if $F$ is Riemannian.

Given a Finsler manifold $(M,F)$, then a global vector field $G$ is induced by $F$ on $TM_{0}$, which in a standard coordinate $(x^{i},y^{i})$ for $TM_{0}$ is given by ${\bf{G}}=y^{i}{{\partial}\over{\partial x^{i}}}-2G^{i}(x,y){{\partial}\over{\partial y^{i}}}$, where $G^{i}(y)$ are local functions on $TM$. $\bf{G}$ is called the associated spray to $(M,F)$. The projection of an integral curve of $\bf{G}$ is called a geodesic in $M$. In local coordinates, a curve $c(t)$ is a geodesic if and only if its coordinates $(c^{i}(t))$ satisfy $\ddot{c}^{i}+2G^{i}(\dot{c})=0$.

Define ${\bf B}_{y}:T_{x}M\otimes T_{x}M\otimes T_{x}M\rightarrow T_{x}M$ and ${\bf E}_{y}:T_{x}M\otimes T_{x}M\rightarrow\mathbb{R}$ by ${\bf B}_{y}(u,v,w):=B^{i}_{\ jkl}(y)u^{j}v^{k}w^{l}{{\partial}\over{\partial x^{i}}}|_{x}$, ${\bf E}_{y}(u,v):=E_{jk}(y)u^{j}v^{k}$, where

[TABLE]

$u=u^{i}{{\partial}\over{\partial x^{i}}}|_{x}$, $v=v^{i}{{\partial}\over{\partial x^{i}}}|_{x}$ and $w=w^{i}{{\partial}\over{\partial x^{i}}}|_{x}$. $\bf B$ and $\bf E$ are called the Berwald curvature and mean Berwald curvature, respectively. A Finsler metric is called a Berwald metric and mean Berwald metric if $\textbf{B}=0$ or ${\bf E}=0$, respectively.

Let

[TABLE]

$\tau=\tau(x,y)$ is a scalar function on $TM\setminus\{0\}$, which is called the distortion.

Let

[TABLE]

where $\sigma(t)$ is the geodesic with $\sigma(0)=x$ and $\dot{\sigma}(0)=y$. ${\bf S}$ is called the S-curvature. ${\bf S}$ said to be isotropic if there is a scalar functions $c(x)$ on $M$ such that

[TABLE]

3 Proof of the Theorem 1.1

In local coordinates $(x^{i},y^{i})$, the vector filed ${\bf G}=y^{i}\frac{\partial}{\partial x^{i}}-2G^{i}\frac{\partial}{\partial y^{i}}$ is a global vector field on $TM_{0}$, where $G^{i}=G^{i}(x,y)$ are local functions on $TM_{0}$ given by following

[TABLE]

By a simple calculation, we have the following.

Lemma 3.1**.**

([22])* Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$. Then the spray coefficients of $F$ are given by:*

[TABLE]

Thus the spray coefficients of an $m$-th root Finsler metric are rational functions with respect to $y$.

Lemma 3.2**.**

Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$. Then the following are equivalent

a)

${\bf S}=(n+1)cF+\eta$;

b)

${\bf S}=\eta$;

where $c=c(x)$ is a scalar function and $\eta=\eta_{i}(x)y^{i}$ is a 1-form on $M$.

Proof.

By Lemma 3.1, the $E$-curvature of an $m$-th root metric is a rational function in $y$. On the other hand, by taking twice vertical covariant derivatives of the S-curvature, we get the $E$-curvature. Thus $S$-curvature is a rational function in $y$. Suppose that $F$ has almost isotropic $S$-curvature, $\textbf{S}=(n+1)c(x)F+\eta$, where $c=c(x)$ is a scalar function and $\eta=\eta_{i}(x)y^{i}$ is a 1-form on $M$. Then the left hand side of $\textbf{S}-\eta=(n+1)c(x)F$ is a rational function in $y$ while the right hand is irrational function. Thus $c=0$ and ${\bf S}=\eta$. ∎

Lemma 3.3**.**

Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$. Then the following are equivalent

a)

${\bf E}=\frac{n+1}{2}cF{\bf h}$;

b)

${\bf E}=0$;

where $c=c(x)$ is a scalar function on $M$.

Proof.

Suppose that $F=\sqrt[m]{A}$ has isotopic mean Berwald curvature

[TABLE]

where $c=c(x)$ is a scalar function on $M$. The left hand side of ${\bf E}=\frac{n+1}{2}cF{\bf h}$ is a rational function in $y$ while the right hand is irrational function. Thus $c=0$ and ${\bf E}=0$. ∎

Proof of Theorem 1.1: By Lemmas 3.2 and 3.3, we get the proof. ∎

By the Theorem 1.1, we have the following:

Corollary 3.4**.**

Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$. Suppose that $F$ has isotropic $S$-curvature ${\bf S}=(n+1)cF$, for some scalar function $c=c(x)$ on $M$. Then ${\bf S}=0$.

A Finsler metric $F$ satisfying $F_{x^{k}}=FF_{y^{k}}$ is called a Funk metric. The standard Funk metric on the Euclidean unit ball $B^{n}(1)$ is denoted by $\Theta$ and defined by

[TABLE]

where $<,>$ and $|.|$ denote the Euclidean inner product and norm on $\mathbb{R}^{n}$, respectively. In [5], Chen-Shen introduce the notion of isotropic Berwald metrics. A Finsler metric $F$ is said to be isotropic Berwald metric if its Berwald curvature is in the following form

[TABLE]

for some scalar function $c=c(x)$ on $M$. Berwald metrics are trivially isotropic Berwald metrics with $c=0$. Funk metrics are also non-trivial isotropic Berwald metrics. In (4), putting $i=l$ yields

[TABLE]

Plugging it in (4) implies that

[TABLE]

This means that every isotropic Berwald metric is a Douglas metric. For the definition of Douglas metrics see [3].

Now, let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$. Suppose that $F$ has isotropic Berwald curvature (4). By Lemma 3.1, the left hand side of (4) is a rational function in $y$ while the right hand is irrational function. Thus $c=0$ and we have the following.

Theorem 3.5**.**

Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$. Suppose that $F$ has isotropic Berwald curvature. Then $F$ is a Berwald metric.

In [21], Tayebi-Rafie Rad proved that every isotropic Berwald metric (4) on a manifold $M$ has isotopic $S$-curvature ${\bf S}=(n+1)cF$, for some scalar function $c=c(x)$ on $M$. Thus by the Theorem 3.5, we have the following.

Corollary 3.6**.**

Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$. Suppose that $F$ has isotropic Berwald curvature. Then ${\bf S}=0$.

4 Proof of the Theorem 1.2

The quotient ${\bf J}/{\bf I}$ is regarded as the relative rate of change of mean Cartan torsion ${\bf I}$ along Finslerian geodesics. Then $F$ is said to be isotropic mean Landsberg metric if ${\bf J}=cF\bf I$, where $c=c(x)$ is a scalar function on $M$. In this section, we are going to prove the Theorem 1.2. More precisely, we show that every $m$-th root isotropic mean Landsberg metric reduces to a weakly Landsberg metric.

Proof of Theorem 1.2: The mean Cartan tensor of $F$ is given by following

[TABLE]

The mean Landsberg curvature of $F$ is given by

[TABLE]

Since ${\bf J}=cF\bf I$, then we have

[TABLE]

By the Lemma 3.1, the left hand side is a rational function in $y$, while its right-hand side is an irrational function in $y$. Thus, either $c=0$ or $A$ satisfies the following PDE:

[TABLE]

That implies that $C_{ijk}=0$. Hence, by Deike’s theorem, $F$ is Riemannian metric, which contradicts our assumption. Therefore, $c=0$. This completes the proof. ∎

By the similarly method, we have the following.

Theorem 4.1**.**

Let $F=\sqrt[m]{A}$ be an non-Riemannian m-th root Finsler metric on an open subset $U\subseteq\mathbb{R}^{n}$. Suppose that $F$ has isotropic Landsberg curvature, i.e., ${\bf L}=cF{\bf C}$; where $c=c(x)$ is a scalar function on $M$. Then $F$ reduces to a Landsberg metric.

5 Proof of the Theorem 1.3

A Finsler metric $F=F(x,y)$ on a manifold $M$ is said to be locally dually flat if at any point there is a coordinate system $(x^{i})$ in which the spray coefficients are in the following form

[TABLE]

where $H=H(x,y)$ is a $C^{\infty}$ scalar function on $TM_{0}=TM\setminus\{0\}$ satisfying $H(x,\lambda y)=\lambda^{3}H(x,y)$ for all $\lambda>0$. Such a coordinate system is called an adapted coordinate system [15]. Recently, Shen proved that the Finsler metric $F$ on an open subset $U\subset\mathbb{R}^{n}$ is dually flat if and only if it satisfies

[TABLE]

In this case, $H=-\frac{1}{6}[F^{2}]_{x^{m}}y^{m}$.

In this section, we will prove a generalized version of Theorem 1.3. Indeed we find necessary and sufficient condition under which a conformal $\beta$-change of an generalized $m$-th root metric be locally dually flat. Let $F$ be a scalar function on $TM$ defined by $F=\sqrt{A^{2/m}+B}$, where $A$ and $B$ are given by

[TABLE]

Then $F$ is called generalized $m$-th root Finsler metric. Suppose that the matrix $(A_{ij})$ defines a positive definite tensor and $(A^{ij})$ denotes its inverse.

Now, we are going to prove the following:

Theorem 5.1**.**

Let $F=\sqrt{A^{2/m}+B}$ be an generalized $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$, where $A$ is irreducible. Suppose that $\bar{F}=e^{\alpha}F+\beta$ be conformal $\beta$-change of $F$ where $\beta=b_{i}(x)y^{i},\alpha=\alpha(x)$. Then $\bar{F}$ is locally dually flat if and only if there exists a 1-form $\theta=\theta_{l}(x)y^{l}$ on U such that the following holds

[TABLE]

where $\Upsilon:=A^{\frac{2}{m}}+B$, $\beta_{0l}=\beta_{x^{k}y^{l}}y^{k}$ , $\alpha_{0}=\alpha_{x^{l}}y^{l}$, $\beta_{x^{l}}=(b_{i})_{x^{l}}y^{i}$, $\beta_{0}=(b_{i})_{0}y^{i}$, $\beta_{0l}=(b_{l})_{0}$, and

[TABLE]

To prove Theorem 5.1, we need the following.

Lemma 5.2**.**

Suppose that the equation $\Phi A^{\frac{2}{m}-2}+\Psi A^{\frac{1}{m}-1}+\Theta=0$ holds, where $\Phi,\Psi,\Theta$ are polynomials in $y$ and $m>2$. Then $\Phi=\Psi=\Theta=0$.

Proof of Theorem 5.1: The following hold

[TABLE]

Then

[TABLE]

Since $\bar{F}$ be a locally dually flat metric, then

[TABLE]

By Lemma 5.2, we have

[TABLE]

One can rewrite (9) as follows

[TABLE]

Irreducibility of $A$ and $deg(A_{l})=m-1$ imply that there exists a 1-form $\theta=\theta_{l}y^{l}$ on $U$ such that

[TABLE]

By (13), we get

[TABLE]

Substituting (13) and (14) into (12) yields (7). The converse yields by a direct computation. This completes the proof. ∎

6 Proof of the Theorem 1.4

It is known that a Finsler metric $F(x,y)$ on ${\cal U}$ is projective if and only if its geodesic coefficients $G^{i}$ are in the form

[TABLE]

where $P:T{\cal U}={\cal U}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ is positively homogeneous with degree one, $P(x,\lambda y)=\lambda P(x,y)$, $\lambda>0$. We call $P(x,y)$ the projective factor of $F(x,y)$. The following lemma plays an important role.

Lemma 6.1**.**

(Rapcsák)* Let $F(x,y)$ be a Finsler metric on an open subset ${\cal U}\subset\mathbb{R}^{n}$. $F(x,y)$ is projective on ${\cal U}$ if and only if it satisfies*

[TABLE]

In this case, the projective factor $P(x,y)$ is given by

[TABLE]

Much earlier, G. Hamel proved that a Finsler metric $F(x,y)$ on ${\cal U}\subset\mathbb{R}^{n}$ is projective if and only if

[TABLE]

Thus (16) and (15) are equivalent.

In this section, we will prove a generalized version of Theorem 1.4. Indeed we study the conformal $\beta$-change of an generalized $m$-th root metric $F=\sqrt{A^{\frac{2}{m}}+B}$, where $A$ is irreducible. More precisely, we prove the following:

Theorem 6.2**.**

Let $F=\sqrt{A^{2/m}+B}$ be an generalized $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$, where $A$ is irreducible. Suppose that $\bar{F}=e^{\alpha}F+\beta$ be conformal $\beta$-change of $F$ where $\beta=b_{i}(x)y^{i},\alpha=\alpha(x)$. Then $\bar{F}$ is locally projectively flat if and only if it is locally Minkowskian.

To prove Theorem 1.4, we need the following.

Lemma 6.3**.**

Let $(M,F)$ be a Finsler manifold. Suppose that $\bar{F}=e^{\alpha}F+\beta$ be a conformal $\beta$-change of $F$. Then $\bar{F}$ is a projectively flat Finsler metric if and only if the following holds

[TABLE]

Proof.

The following hold

[TABLE]

This completes the proof. ∎

By using the Lemma 6.3, we are going to prove the following.

Proposition 6.4**.**

Let $F=\sqrt{A^{2/m}+B}$ be an generalized $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$, where $A$ is irreducible, $m>4$ and $B\neq 0$. Suppose that $\bar{F}=e^{\alpha}F+\beta$ be conformal $\beta$-change of $F$ where $\beta=b_{i}(x)y^{i},\alpha=\alpha(x)$. In that case, if $\bar{F}$ is projectively flat metric then $F$ reduces to a Berwald metric.

Proof.

By Lemma 6.3, we get

[TABLE]

Then we have

[TABLE]

Thus

[TABLE]

By (18), we obtain the following:

[TABLE]

where

[TABLE]

By Lemma 5.2, we have

[TABLE]

By (20), it follows that

[TABLE]

Then irreducibility of $A$ and $deg(A_{l})=m-1<deg(A)$ implies that $A_{0}$ is divisible by $A$. This means that, there is a 1-form $\theta=\theta_{l}y^{l}$ on $U$ such that,

[TABLE]

Substituting (23) into (22), yields

[TABLE]

Plugging (23) and (24) into (19), we get

[TABLE]

Clearly, the right side of (25) is divisible by $A$. Since $A$ is irreducible, deg($A_{l}$) and deg($2\theta B-\frac{1}{2}B$) are both less than deg$(A)$, then we have

[TABLE]

By (23) and (26), we get the spray coefficients $G^{i}=Py^{i}$ with $P=\theta$. Thus $F$ is a Berwald metric. ∎

The Riemann curvature ${\bf K}_{y}=R^{i}_{\ k}dx^{k}\otimes{\partial\over\partial x^{i}}|_{x}:T_{x}M\to T_{x}M$ is a family of linear maps on tangent spaces, defined by

[TABLE]

For a flag $P={\rm span}\{y,u\}\subset T_{x}M$ with flagpole $y$, the flag curvature ${\bf K}={\bf K}(P,y)$ is defined by

[TABLE]

When $F$ is Riemannian, ${\bf K}={\bf K}(P)$ is independent of $y\in P$, which is just the sectional curvature of $P$ in Riemannian geometry. We say that a Finsler metric $F$ is of scalar curvature if for any $y\in T_{x}M$, the flag curvature ${\bf K}={\bf K}(x,y)$ is a scalar function on the slit tangent bundle $TM_{0}$. One of the important problems in Finsler geometry is to characterize Finsler manifolds of scalar flag curvature [10][11]. If ${\bf K}=constant$, then the Finsler metric $F$ is said to be of constant flag curvature.

Proof of Theorem 6.2: By Proposition 6.4, $F$ is a Berwald metric. On the other hand, according to Numata’s Theorem every Berwald metric of non-zero scalar flag curvature ${\bf K}$ must be Riemaniann. This is contradicts with our assumption. Then ${\bf K}=0$, and in this case $F$ reduces to a locally Minkowskian metric. ∎