On m-th root Finsler metrics

TL;DR

This paper characterizes specific properties of m-th root Finsler metrics, showing that under certain curvature conditions, these metrics simplify to weakly Berwald metrics, advancing understanding in Finsler geometry.

Contribution

It provides a characterization of locally dually flat and Antonelli m-th root Finsler metrics and shows their reduction to weakly Berwald metrics under isotropic mean Berwald curvature.

Findings

Characterization of locally dually flat m-th root Finsler metrics

Characterization of Antonelli m-th root Finsler metrics

Reduction of isotropic mean Berwald curvature m-th root metrics to weakly Berwald metrics

Abstract

In this paper, we characterize locally dually flat and Antonelli $m$-th root Finsler metrics. Then, we show that every $m$-th root Finsler metric of isotropic mean Berwald curvature reduces to a weakly Berwald metric.

Equations

$$A:=a_{i_1\dots i_m}(x)y^{i_1}{y}^{i_2}\dots y^{i_m},$$

$$A_i=\frac{\partial A}{\partial y^i},\text{ and}{A}_{ij}=\frac{{\partial }^{2}A}{\partial y^i\partial y^j}.$$

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

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

$$y^i{A}_i=mA,y^i{A}_{ij}=(m-1)A_j,y_i=\frac{1}{m}{A}^{\frac{2}{m}-1}{A}_i,$$

$$A^{ij}{A}_{jk}={\delta }_k^i,A^{ij}{A}_i=\frac{1}{m-1}{y}^j,A_i{A}_j{A}^{ij}=\frac{m}{m-1}A,$$

$$A_0:=A_{x^m}{y}^m,A_{0l}:=A_{x^m{y}^l}{y}^m.$$

$$A_{x^{l}}=\frac{1}{2A}\Big{\{}(\frac{2}{m}-1)A_{l}A_{0}+AA_{0l}\Big{\}}.$$

$$A_{x^{l}}=\frac{1}{3m}\Big{\{}2\theta A_{l}+mA\theta_{l}\Big{\}}.$$

$$A_{x^l}=[{\Gamma }_{lk}^i{y}^k+\frac{1}{2}{\Gamma }_{jk,l}^i{y}^j{y}^k]A_i,$$

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

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

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

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

$$L_{x^k{y}^l}{y}^k=2L_{x^l}.$$

$$(A^{\frac{2}{m}}{)}_{x^k{y}^l}{y}^k=2(A^{\frac{2}{m}}{)}_{x^l}.$$

$$(A^{\frac{2}{m}}{)}_{x^l}$$

$$(A^{\frac{2}{m}}{)}_{x^k{y}^l}$$

$$A(2A_{x^l}-A_{0l})=(\frac{2}{m}-1)A_l{A}_0.$$

$$A_0=\theta A.$$

$$A_{0l}=A{\theta }_l+\theta A_l-A_{x^l}.$$

$$A_{x^l}=\frac{1}{3}(\theta A{)}_{y^l}.$$

$$G^i=\frac{1}{12}{\theta }^i{F}^2+\frac{1}{6}\theta y^i,$$

$$3\frac{\partial a_{ij}}{\partial x^l}={\theta }_l{a}_{ij}+{\theta }_i{a}_{lj}+{\theta }_j{a}_{il}.$$

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

$$A_{x^l}=[{\Gamma }_{lk}^i{y}^k+\frac{1}{2}{\Gamma }_{jk,l}^i{y}^j{y}^k]A_i,$$

$$G^i=\frac{1}{2}{\Gamma }_{jk}^i(y)y^j{y}^k.$$

$${\Gamma }_{jk}^i{y}^j{y}^k=(A_{0j}-A_{x^j})A^{ij}.$$

$${\Gamma }_{jk}^i{y}^j{y}^k{A}_{il}$$

$${\Gamma }_{jk}^i{y}^j{y}^k{A}_i$$

$$[{\Gamma }_{jk,l}^i{y}^j{y}^k+2{\Gamma }_{lk}^i{y}^k]A_i+{\Gamma }_{jk}^i{y}^j{y}^k{A}_{il}=A_{0l}+A_{x^l}.$$

$$A_{x^l}=[{\Gamma }_{lk}^i{y}^k+\frac{1}{2}{\Gamma }_{jk,l}^i{y}^j{y}^k]A_i.$$

$$A_{x^l}=[{\Gamma }_{lk}^i{y}^k+\frac{1}{2}{\Gamma }_{jk,l}^i{y}^j{y}^k]A_i,$$

Full text

Keywords: Antonelli metric, locally dually flat metric, isotropic mean Berwald metric.111 2000 Mathematics subject Classification: 53C60, 53C25.

1 Introduction

The theory of $m$-th root metrics has been developed by H. Shimada [14], and applied to Biology as an ecological metric [3]. It is regarded as a direct generalization of Riemannian metric in the sense that the second root metric is a Riemannian metric. The third and fourth root metrics are called the cubic metric and quartic metric, respectively.

Recently studies show that the theory of $m$-th root Finsler metrics plays a very important role in physics, theory of space-time structure, gravitation, general relativity and seismic ray theory [4][7][10][11][12]. For quartic metrics, a study of the geodesics and of the related geometrical objects is made by S. Lebedev [8]. Also, Einstein equations for some relativistic models relying on such metrics are studied by V. Balan and N. Brinzei in two papers [5], [6]. Tensorial connections for such spaces have been recently studied by L. Tamassy [15]. B. Li and Z. Shen study locally projectively flat fourth root metrics under some irreducibility condition [9]. Y. Yu and Y. You show that an $m$-th root Einstein Finsler metric is Ricci-flat [17].

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 a scalar function on $TM$ defined by $F=\sqrt[m]{A}$, where $A$ is given by

[TABLE]

with $a_{i_{1}\dots i_{m}}$ symmetric in all its indices [14]. Then $F$ is called an $m$-th root Finsler metric.

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

[TABLE]

Suppose that the matrix $(A_{ij})$ defines a positive definite tensor and $(A^{ij})$ denotes its inverse. Then the following hold

[TABLE]

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 standard coordinate system $(x^{i},y^{i})$ in $TM$ such that $[F^{2}]_{x^{k}y^{l}}y^{k}=2[F^{2}]_{x^{l}}$. In this case, the coordinate $(x^{i})$ is called an adapted local coordinate system [13]. Here, we characterize locally dually flat $m$-th root Finsler metrics.

Theorem 1.1**.**

Let $F$ be an $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$. Then, $F$ is a locally dually flat metric if and only if the following holds

[TABLE]

Moreover, suppose that $A$ is irreducible. Then $F$ 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]

A Finsler metric $F$ is called an Antonelli metric, if there is a local coordinate system in $TM$, in which the spray coefficients are dependent on direction alone. In this case, the spray coefficients of $F$ are given by $G^{i}=\frac{1}{2}\Gamma^{i}_{jk}(y)y^{j}y^{k}$.

Antonelli metrics were first introduced by P. L. Antonelli for some studies in Biology and Ecology [3]. Antonelli calls them $y$-Berwald metrics. This class of metrics arises in time sequencing change models in the evolution of colonial systems. Here, we characterize $m$-th root Antonelli metrics. More precisely, we prove the following.

Theorem 1.2**.**

Let $F$ be an $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$. Then, $F$ is an Antonelli metric if and only if there exist functions $\Gamma^{i}_{lk}$ dependent only on direction such that the following holds

[TABLE]

where $\Gamma^{i}_{jk,l}=\frac{\partial\Gamma^{i}_{jk}}{\partial y^{l}}$.

A Finsler metric $F$ is called a Berwald metric if $G^{i}$ are quadratic in $y\in T_{x}M$ for any $x\in M$ or equivalently Berwald curvature vanishes. In this case, we have $G^{i}={1\over 2}\Gamma^{i}_{jk}(x)y^{j}y^{k}$. Hence, Berwald metrics can be considered as $x$-Berwald metrics. The $E$-curvature is defined by the trace of the Berwald curvature. A Finsler metric $F$ is called of isotropic mean Berwald curvature if ${\bf E}=\frac{n+1}{2}cF^{-1}{\bf h}$, where $c=c(x)$ is a scalar function on $M$ and ${\bf h}$ is the angular metric. If $c=0$, then $F$ is called weakly Berwald metric.

Theorem 1.3**.**

Let $F$ be an $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$ with $n\geq 2$. Suppose that $F$ is of isotropic mean Berwald curvature. Then $F$ is a weakly Berwald metric.

2 Preliminaries

Let $M$ be an $n$-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 of $M$.

A Finsler metric on a manifold $M$ is a function $F:TM\rightarrow[0,\infty)$ with the following properties:

(i) $F$ is $C^{\infty}$ on $TM_{0}$;

(ii) $F(x,\lambda y)=\lambda F(x,y)\,\,\,\forall\lambda>0$,  $y\in TM$;

(iii) for each $y\in T_{x}M$, the following quadratic form $g_{y}$ on $T_{x}M$ is positive definite,

[TABLE]

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

[TABLE]

The vector field ${\bf G}$ is called the associated spray to $(M,F)$ [16].

For $y\in T_{x}M_{0}$, 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 ${\bf B}_{y}(u,v,w):=B^{i}_{\ jkl}(y)u^{j}v^{k}w^{l}{{\partial}\over{\partial x^{i}}}|_{x}$ and ${\bf E}_{y}(u,v):=E_{jk}(y)u^{j}v^{k}$ where

[TABLE]

$\bf B$ and $\bf E$ are called Berwald curvature and mean Berwald curvature, respectively. Then $F$ is called a Berwald metric and weakly Berwald metric if ${\bf{B}}=0$ and ${\bf{E}}=0$, respectively [16].

A Finsler metric $F$ on an $n$-dimensional manifold $M$ is said to be isotropic mean Berwald metric or of isotropic $E$-curvature if

[TABLE]

where $h_{ij}=g_{ij}-g_{ip}y^{p}g_{jq}y^{q}$ is the angular metric.

3 Proof of Theorem 1.1

Information geometry has emerged from investigating the geometrical structure of a family of probability distributions and has been applied successfully to various areas including statistical inference, control system theory and multi-terminal information theory [1][2]. Dually flat Finsler metrics form a special and valuable class of Finsler metrics in Finsler information geometry, which plays a very important role in studying flat Finsler information structure [13].

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 standard coordinate system $(x^{i},y^{i})$ in $TM$ such that $L=F^{2}$ satisfies

[TABLE]

In this case, the coordinate $(x^{i})$ is called an adapted local coordinate system. Every locally Minkowskian metric $F$ satisfies trivially the above equation, hence $F$ is locally dually flat.

In order to find explicit examples of locally dually flat metrics, we consider $m$-th root Finsler metrics.

Proof of Theorem 1.1: Let $F$ be a locally dually flat metric

[TABLE]

We have

[TABLE]

By (11), (12) and (13), we have (7). The converse is trivial.

Now, suppose that $A$ is irreducible. One can rewrite (7) 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]

Plugging (15) into (7), we get

[TABLE]

Substituting (15) and (16) into (7) yields (8). The converse is a direct computation. This completes the proof. ∎

Corollary 3.1**.**

Let $F=\sqrt{A}$ be a Riemannian metric on open subset $U\subset\mathbb{R}^{n}$. Then, $F$ is a locally dually flat metric if and only if there exists a 1-form $\theta$ on $U$ such that the following holds

[TABLE]

In this case, the spray coefficients of $F$ are given by

[TABLE]

where $\theta^{i}=2A^{ik}\theta_{k}$.

Proof.

Putting $m=2$ in (8), we get (17). Plugging $A=a_{ij}y^{i}y^{j}$ into (17), one can obtain

[TABLE]

Substituting (19) into (10), we get (18). ∎

4 Proof of Theorem 1.2

In this section, we deal with Antonelli metrics. First, we remark the following.

Lemma 4.1**.**

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

[TABLE]

Now, we are going to prove Theorem 1.2. First, we prove the following.

Lemma 4.2**.**

Let $F$ be an $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$. If $F$ is an Antonelli metric, then the following holds

[TABLE]

where $\Gamma^{i}_{jk,l}=\frac{\partial\Gamma^{i}_{jk}}{\partial y^{l}}$.

Proof.

Let $F$ be an Antonelli metric metric. Then

[TABLE]

Plugging (22) into (20), we get

[TABLE]

Multiplying (22) with $A_{il}$ and $A_{i}$ implies that

[TABLE]

Differentiating of (25) with respect to $y^{l}$, we have

[TABLE]

Subtracting (26) from (24) yields

[TABLE]

Then we get the proof. ∎

Lemma 4.3**.**

Let $F$ be an $m$-th root Finsler metric on an open subset $U\subset\mathbb{R}^{n}$. Suppose that $F$ satisfies the following equation

[TABLE]

where $\Gamma^{i}_{lk}$ are functions only of direction. Then, $F$ is an Antonelli metric.

Proof.

Now suppose that (28) holds. By differentiating of (28) with respect to $y^{h}$, we have

[TABLE]

Contracting (29) with $y^{l}$ and 0-homogeneity of functions $\Gamma^{i}_{lk}$ yield

[TABLE]

Substituting (28) into (30) implies that

[TABLE]

Multiplying (31) with $A^{hj}$, we have

[TABLE]

By (20) and (32), one can obtain that

[TABLE]

This means that $F$ is an Antonelli metric. ∎

Proof of Theorem 1.2: By Lemma 4.2 and Lemma 4.3, we get the proof of Theorem 1.2. ∎

5 Proof of Theorem 1.3

Proof of Theorem 1.3: Let $F=\sqrt[m]{A}$ be an $m$-th root Finsler metric and be of isotropic mean Berwald curvature, i.e.,

[TABLE]

where $c=c(x)$ is a scalar function on $M$. A direct computation implies that the angular metric $h_{ij}=g_{ij}-F^{2}y_{i}y_{j}$ are given by the following

[TABLE]

Plugging (35) into (34), we get

[TABLE]

By (20), one can see that $E_{ij}$ is rational with respect to $y$. Thus, (36) implies that $c=0$ or

[TABLE]

Suppose that $c\neq 0$. Contracting (37) with $A^{jk}$ yields

[TABLE]

which implies that $mnA=mA$. This contradicts our assumption $n\geq 2$. Thus $c=0$ and consequently $E_{ij}=0$. ∎

Acknowledgments. The authors express their sincere thanks to referees and Professor Z. Shen for their valuable suggestions and comments.