General $(\alpha, \beta)$ metrics with relatively isotroic mean Landsberg curvature

TL;DR

This paper investigates a new class of Finsler metrics called general (,) metrics, establishing conditions under which metrics with relatively isotropic mean Landsberg curvature are Berwald, thus advancing understanding of Finsler geometry.

Contribution

It provides a necessary and sufficient condition for general (,) metrics with isotropic mean Landsberg curvature to be Berwald, under specific assumptions.

Findings

Derived a condition linking isotropic mean Landsberg curvature to Berwald metrics.

Extended the understanding of (,) Finsler metrics with conformal 1-forms.

Identified criteria for when these metrics exhibit Berwald properties.

Abstract

In this paper, we study a new class of Finsler metrics, F=\alpha\phi(b^2,s), s:=\beta/\alpha, defined by a Riemannian metric \alpha and 1-form \beta. It is called general (\alpha, \beta) metric. In this paper, we assume \phi be coefficient by s and \beta be closed and conformal. We find a nessecary and sufficient condition for the metric of relatively isotropic mean Landsberg curvature to be Berwald.

Equations

$$F:=\alpha \varphi (s),s:=\frac{\beta }{\alpha }$$

$$\varphi (s)>0,\varphi (s)-s{\varphi }^{\prime }(s)+(b^2-s^2){\varphi }^{\prime \prime }(s)>0,|s|\le b<b_0,$$

$$F:=\alpha \varphi (b^2,s),s:=\frac{\beta }{\alpha }$$

$$\varphi$$

$$F=\frac{{\left(\sqrt{(1-|x{|}^2)|y{|}^2+⟨x,y{⟩}^2}+⟨x,y⟩\right)}^2}{(1-|x{|}^2{)}^2\sqrt{(1-|x{|}^2)|y{|}^2+⟨x,y{⟩}^2}}.$$

$$F=\alpha \varphi (b^2,s)=\alpha {\left(\sqrt{1+b^2}+s^2\right)}^2$$

$$\alpha =\frac{\sqrt{(1-|x{|}^2)|y{|}^2+⟨x,y{⟩}^2}}{1-|x{|}^2},\beta =\frac{⟨x,y⟩}{(1-|x{|}^2{)}^{3/2}}$$

$$s:=\frac{\beta }{\alpha },b^2=\frac{|x{|}^2}{1-|x{|}^2}$$

$$F=|y|\varphi \left(|x{|}^2,\frac{⟨x,y⟩}{|y|}\right).$$

$$b_{i|j}=c(x)a_{ij},$$

$$\varphi (b^2,s)=c_0(b^2)+c_1(b^2)s+c_2(b^2)s^2+\cdots +c_m(b^2)s^m,$$

$$c_0(b^2)=\frac{a_0}{\sqrt{b^2}},c_1(b^2)=\frac{a_1}{b^2},c_2(b^2)=\frac{a_2}{(b^2{)}^{\frac{3}{2}}},...,c_m(b^2)=\frac{a_m}{(b^2{)}^{\frac{m+1}{2}}}$$

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

$$g_y(u,v)=g_{ij}{u}^i{v}^j,$$

$$C_{ijk}:=\frac{1}{4}[F^2{]}_{y^i{y}^j{y}^k}=\frac{1}{2}\frac{\partial g_{ij}}{\partial y^k}.$$

$$I_i:=g^{jk}{C}_{ijk}.$$

$$I_i=g^{jk}{C}_{ijk}=\frac{\partial }{\partial y^i}\left[\ln \sqrt{\det (g_{jk})}\right]$$

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

$$G^i=\frac{1}{4}{g}^{il}\left\{[F^2{]}_{x^m{y}^l}{y}^m-[F^2{]}_{x^l}\right\}.$$

$$B_{jkl}^i:=\frac{{\partial }^3{G}^i}{\partial y^j\partial y^k\partial y^l}.$$

$$L_{ijk}:=-\frac{1}{2}F{F}_{y^m}[G^m{]}_{y^i{y}^j{y}^k}.$$

$$J_i:=g^{jk}{L}_{ijk}.$$

$$\varphi >0,\varphi -s{\varphi }_2+(b^2-s^2){\varphi }_{22}>0,$$

$$(g_{ij}):=(\frac{1}{2}[F^2{]}_{y^i{y}^j}).$$

$$[F^2{]}_{y^i}$$

$$[F^2{]}_{y^i{y}^j}$$

$$+2\alpha {\varphi }_{22}{\varphi }_2{s}_{y^i}{s}_{y^j}+2\alpha [{\varphi }_2{]}^2{s}_{y^i}{s}_{y^j}$$

$$g_{ij}=\rho a_{ij}+{\rho }_0{b}_i{b}_j+{\rho }_1(b_i{\alpha }_{y^i}+b_j{\alpha }_{y^j})-s{\rho }_1{\alpha }_{y^i}{\alpha }_{y^j},$$

$$\rho =\varphi (\varphi -s{\varphi }_2),{\rho }_0=\varphi {\varphi }_{22}+({\varphi }_2{)}^2,{\rho }_1=(\varphi -s{\varphi }_2){\varphi }_2-s\varphi {\varphi }_{22}.$$

$$\det (g_{ij})={\varphi }^{n+1}(\varphi -s{\varphi }_2{)}^{n-2}(\varphi -s{\varphi }_2+(b^2-s^2){\varphi }_{22})\det (a_{ij}).$$

$$\varphi -s{\varphi }_2>0,|s|<b_0.$$

Full text

Keywords: Finsler geometry, Relatively isotropic mean Landsberg curvature , General ($\alpha$, $\beta$)-metrics.

1 Introduction

The $(\alpha,\beta)$ metrics were first introduced by Matsumoto [2]. They are Finsler metrics built from a Riemannian metric $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ and 1-form $\beta=b_{i}(x)y^{i}$ and a $C^{\infty}$ function $\phi(s)$ on a manifold $M$. A Finsler metric of $(\alpha,\beta)$ metrics is given by the form

[TABLE]

It is known that $F$ is positive and strongly convex on $TM\setminus\{0\}$ if and only if

[TABLE]

where $b=||\beta||_{\alpha}$.

The aim of this paper is to study a new class of Finsler metrics given by

[TABLE]

where $\phi=\phi(b^{2},s)$ is a $C^{\infty}$ positive function and $b=||\beta||_{\alpha}$ is its norm[8]. It is called general $(\alpha,\beta)$ metrics. This kind of metrics is first discussed by Yu and Zhu [3]. Many well-known Finsler metrics are general $(\alpha,\beta)$ metrics.

Example 1**.**

[8] The Randers metrics and the square metrics are defined by functions $\phi=\phi(b^{2},s)$ in the following form:

[TABLE]

Example 2**.**

[9] One Important example of $(\alpha,\beta)$ metric was given by L. Berwald

[TABLE]

It is a projectively flat Finsler metrics on $\mathbb{B}\subset\mathbb{R}^{n}$ with flag curvature $K=0$. Berwald’s metric can be expressed in form

[TABLE]

where

[TABLE]

Example 3**.**

[8] There is a special class of general $(\alpha,\beta)$ metrics called spherically symmetric metrics, which are defined on an open subset of $\mathbb{R}^{n}$ with $\alpha=|y|$ and $\beta=\langle x,y\rangle$,

[TABLE]

In Finsler geometry, there are several very important non-Riemannian quantities. The Cartan torsion $\mathbf{C}$ is a primary quantity. There is another quantity which is determined by the Busemann-Hausdorff volume form, that is the so-called distortion $\tau$. The vertical differential of $\tau$ on each tangent space gives rise to mean Cartan torsion $\mathbf{I}:=\tau_{y^{k}}dx^{k}$. $\mathbf{C}$, $\tau$ and $\mathbf{I}$ are the basic geometric quantities which characterize Riemannian metrics among Finsler metrics. Differentiating $\mathbf{C}$ along geodesics gives rise to the Landsberg curvature $\mathbf{L}$. The horizontal derivative of $\tau$ along geodesics is the so-called $S$-curvature $\mathbf{S}:=\tau_{|k}y^{k}$. The horizontal derivative of $\mathbf{I}$ along geodesics is called the mean Landsberg curvature $\mathbf{J}:=\mathbf{I}_{|k}y^{k}$.

By the definition, $\mathbf{J}/\mathbf{I}$ can be regarded as the relative growth rate of the mean Cartan torsion along geodesics. We call a Finsler metric $F$ is of relatively isotropic mean Landsberg curvature if $F$ satisfies $\mathbf{J}+\tilde{c}F\mathbf{I}=0$, where $\tilde{c}=\tilde{c}(x)$ is a scalar function on the Finsler manifold. In particular, when $\tilde{c}=0$, Finsler metrics with $\mathbf{J}=0$ are called weak Landsberg metrics [4].

We study general $(\alpha,\beta)$ metrics with relatively isotropic mean Landsberg curvature, where $\beta$ is a closed and conformal 1-form, i.e.

[TABLE]

where $b_{i|j}$ is the covariant derivation of $\beta$ with respect to $\alpha$ and $c=c(x)\neq 0$ is a scalar function on $M$. In [7], Zohrehvand and Maleki proved that, every Landsberg general $(\alpha,\beta)$ metric is a Berwald metric with condition (8). In [1], the authors showed that this result for the metric of mean Landsberg curvature.

In this paper, we prove the following

Theorem 1**.**

Let $F=\alpha\phi(b^{2},s),s:=\beta/\alpha$, be a non-Riemannian general $(\alpha,\beta)$ metric on an $n$-dimensional manifold $M$. Suppose that $\beta$ satisfies (8). If $\phi=\phi(b^{2},s)$ is a polynomial in $b^{2}$ and $s$, then $F$ is of relatively isotropic mean Landsberg curvature, $\mathbf{J}+\tilde{c}(x)F\mathbf{I}=0$, if and only if it is a Berwald metric. In this case,

[TABLE]

where

[TABLE]

and $a_{i}$, $1\leq i\leq n$ are constants.

Because every analytic function can be approximated by a series polynomials, we can assume that $\phi$ is a polynomial in $b^{2}$ and $s$.

2 Preliminary

Let $F=F(x,y)$ be a Finsler metric on an $n$-dimensional manifold $M$. Let

[TABLE]

and $(g^{ij}):=(g_{ij})^{-1}$. For a non-zero vector $y=y^{i}\frac{\partial}{\partial x^{i}}\mid_{x}\in T_{x}M$, $F$ induces an inner product on $T_{x}M$

[TABLE]

where $u=u^{i}\frac{\partial}{\partial x^{i}},v=v^{j}\frac{\partial}{\partial x^{j}}\in T_{x}M$. $g=\{g_{y}\}$ is called the fundamental tensor of $F$.

Let

[TABLE]

Define symmetric trilinear form $\mathbf{C}:=C_{ijk}(x,y)dx^{i}\otimes dx^{j}\otimes dx^{k}$ on $TM\setminus\{0\}$. We call $\mathbf{C}$ the Cartan torsion. The mean Cartan torsion $\mathbf{I}=I_{i}dx^{i}$ is defined by

[TABLE]

Further, we have ([4], [5])

[TABLE]

For a Finsler metric $F$, the geodesics are characterized locally by a system of 2nd ODEs:

[TABLE]

where

[TABLE]

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

For a tangent vector $y=y^{i}\dfrac{\partial}{\partial x^{i}}\in T_{x}M$, the Berwald curvature $\mathbf{B}:=B_{jkl}^{i}dx^{j}\otimes\dfrac{\partial}{\partial x^{i}}\otimes dx^{k}\otimes dx^{l}$ can be expressed by

[TABLE]

$F$ is a Berwald metric if $\mathbf{B}=0$. The Landsberg curvature $\mathbf{L}=L_{ijk}(x,y)dx^{i}\otimes dx^{j}\otimes dx^{k}$ is a horizontal tensor on $TM\setminus\{0\}$ defined by [6]

[TABLE]

$F$ is called a Landsberg metric if $\mathbf{L}=0$. The mean Landsberg curvature $\mathbf{J}=J_{i}dx^{i}$ is defined by

[TABLE]

We call $F$ a weak Landsberg metric if $\mathbf{J}=0$. We say that $F$ is of relatively isotropic mean Landsberg curvature if $J_{i}+\tilde{c}(x)FI_{i}=0$ for a scalar function $\tilde{c}=\tilde{c}(x)$ on $M$.

Now we consider a general $(\alpha,\beta)$ metric:

Definition 1**.**

Let $F$ be a Finsler metric on an $n$-dimensional manifold $M$. $F$ is called a general $(\alpha,\beta)$ metric if it can be expressed as the form (1) where $||\beta||_{\alpha}\leq b_{0}$ and $\phi=\phi(b^{2},s)$ is a positive $C^{\infty}$ function.

Proposition 1**.**

Let $M$ be an $n$-dimensional manifold. A function $F=\alpha\phi(b^{2},s)$ on $TM$ is a Finsler metric on $M$ for any Riemannian metric $\alpha$ and 1-form $\beta$ with $||\beta||_{\alpha}<b_{0}$ if and only if $\phi=\phi(b^{2},s)$ is a positve $C^{\infty}$ function satisfying

[TABLE]

where $s$ and $b$ are arbitrary numbers with $|s|\leq b<b_{0}$.

Proof.

It is easy to verify $F$ is a function with regularity and positive homogeneity. In the following we will verify strong convexity: The $n\times n$ Hessian matrix

[TABLE]

For the general $(\alpha,\beta)$ metric $F=\alpha\phi(b^{2},\frac{\beta}{\alpha})$, direct computations yield

[TABLE]

Direct computations yield

[TABLE]

where

[TABLE]

By Lemma 1.1.1 in [4], we find a formula for $\det(g_{ij})$

[TABLE]

Assume that (15) is satisfied. Then by taking $b=s$ in (15), we see that the following inequality holds for any $s$ with

[TABLE]

Using (15), (20) and (21), we get $\det(g_{ij})>0$, namely $(g_{ij}$ is positive-definite. The converse is obvious, so the proof is omitted here. ∎

By Lemma 1.1.1 in [4], we find a formula for $(g^{ij})$

[TABLE]

where $(g^{ij})=(g_{ij})^{-1}$, $(g_{ij})=\frac{1}{2}[F^{2}]_{y^{i}y^{j}}$, $(a^{ij})=(a_{ij})^{-1}$, $b^{i}=a^{ij}b_{j}$,

[TABLE]

Remark**.**

Note that $\phi_{1}$ means the derivation of $\phi$ with respect to the first variable $b^{2}$. In this paper, $\beta$ is closed and conformal 1-form, i.e. $b_{i|j}=c(x)a_{ij}$. Let [3]

[TABLE]

For a general $(\alpha,\beta)$ metric, its spray coefficients $G^{i}$ are related to the spray coefficients $G_{\alpha}^{i}$ of $\alpha$ by [3]

[TABLE]

where $l^{i}:=\dfrac{y^{i}}{\alpha}$ and

[TABLE]

When $\beta$ is closed and conformal one-form, i.e. satisfies (8), then

[TABLE]

Substituting this into (Remark) yields [3]

[TABLE]

If we have

[TABLE]

then from (25)

[TABLE]

Proposition 2**.**

[1]** Let $F=\alpha\phi(b^{2},s),~{}s=\beta/\alpha$, be a general $(\alpha,\beta)$-metric on an $n$-dimensional manifold $M$. Suppose that $\beta$ satisfies (8), then the weak Landsberg curvature of $F$ is given by

[TABLE]

where

[TABLE]

where $\rho$ and $\eta$ is defined in (19) and (23) and $l_{j}:=a_{ij}l^{i}$.

Proposition 3**.**

[1]** Let $F=\alpha\phi(b^{2},s),~{}s=\beta/\alpha$, be a general $(\alpha,\beta)$-metric on an $n$-dimensional manifold $M$. Suppose that $\beta$ satisfies (8), then $F$ is weak Landsberg metric if and only if the following equations hold:

[TABLE]

Theorem 2**.**

[1]** Let $F=\alpha\phi(b^{2},\frac{\beta}{\alpha})$ be a non-Riemannian general $(\alpha,\beta)$-metric on an $n$-dimentional manifold $M$ and $\beta$ satisfies (8). Then $F$ is a weak Landsberg metric if and only if it is Landsberg metric.

3 Proof of Theorem 1

In this section, we prove Theorem 1. From (10) and (18), we have

[TABLE]

We must mention the following lemmas firstly.

Lemma 1**.**

Let $F=\alpha\phi(b^{2},s),~{}s=\beta/\alpha$, be a general $(\alpha,\beta)$-metric on an $n$-dimensional manifold $M$ and $\beta$ satisfies (8). Then $F$ is of relatively isotropic mean Landsberg curvature if and only if $\phi$ satisfies the following ODE:

[TABLE]

where $W_{j}$ is defined in (2),and

[TABLE]

Proof.

By Proposition 2,

[TABLE]

∎

By use of Maple program, we can immediately get the following lemma.

Lemma 2**.**

Let $NJFI$ denote the numerator of the left of (34), then (34) holds if and only if

[TABLE]

Also, Let $NE_{22}$, $NH_{222}$ and $NP$ denote the numerators of $E_{22}$, $H_{222}$ and Eq. (32), respectively. Then from (31) and (32), we have

[TABLE]

By assumption, $F$ is of relatively isotropic mean Landsberg curvature. Express $\phi(b^{2},s)$ as below.

[TABLE]

Plugging (39) to $NJFI$ yields a polynomial in $s$. Denote the order of $NJFI$ by $r$. Then (36) can be rewritten as follows.

[TABLE]

where $v_{i}~{}(0\leq i\leq r)$ are independent of $s$.

By using Maple program, we can get following results:

Case 1. $m=1$: $\phi(b^{2},s)=c_{0}(b^{2})+c_{1}(b^{2})s$, where $c_{1}(b^{2})\neq 0$. We can get $r=2$ and

[TABLE]

In this case, because $v_{2}=0$, so $\tilde{c}$ must be zero.

In here, we can obtain the form of $c_{0}(b^{2})$ and $c_{1}(b^{2})$. Plugging the $\phi$ into (37) and (38) yields $NH_{222}=0$, and

[TABLE]

From (42) and (43), we obtain the following ODE:

[TABLE]

By solving the above ODE, we have

[TABLE]

Case 2. $m=2$. $\phi(b^{2},s)=c_{0}(b^{2})+c_{1}(b^{2})s+c_{2}(b^{2})s^{2}$, where $c_{2}(b^{2})\neq 0$. By using Maple, We can get $r=17$ and

[TABLE]

where $f_{17c}$ is independent of $s$ and

[TABLE]

In this case, because $v_{17}=0$, so $\tilde{c}$ must be zero.

We can obtain the form of $c_{0}(b^{2})$, $c_{1}(b^{2})$ and $c_{2}(b^{2})$ too. Plugging the $\phi$ into (37) and (38) and similar argument yields the following ODE:

[TABLE]

Then

[TABLE]

It is not hard to prove by induction that given any $m\geq 3$ in (39), the function $\tilde{c}$ must vanish.

In sum, we have proved that, if $F=\alpha\phi(b^{2},s)$ is of relatively isotropic mean Landsberg curvature and $\phi$ be polynomial in $b^{2}$ and $s$, then $F$ must be a weak Landsberg metric. Then $F$ is a Berwald metric by Theorem 2.

In this case, if

[TABLE]

then

[TABLE]

and $a_{i}$, $1\leq i\leq n$ are constants.