On general $(\alpha, \beta)$-metrics of weak Landsberg type

TL;DR

This paper proves that all weak Landsberg general $(eta,eta)$-metrics are Berwald metrics with closed, conformal one-forms, and establishes the equivalence between Landsberg and weak Landsberg metrics in this class.

Contribution

It demonstrates that weak Landsberg general $(eta,eta)$-metrics are Berwald and characterizes when such metrics are Landsberg, revealing no existence of generalized unicorn metrics.

Findings

Weak Landsberg general $(eta,eta)$-metrics are Berwald.

Landsberg and weak Landsberg conditions are equivalent for these metrics.

No generalized unicorn metrics exist in this class.

Abstract

In this paper, we study general $(\alpha,\beta)$-metrics which $\alpha$ is a Riemannian metric and $\beta$ is an one-form. We have proven that every weak Landsberg general $(\alpha,\beta)$-metric is a Berwald metric, where $\beta$ is a closed and conformal one-form. This show that there exist no generalized unicorn metric in this class of general $(\alpha,\beta)$-metric. Further, We show that $F$ is a Landsberg general $(\alpha,\beta)$-metric if and only if it is weak Landsberg general $(\alpha,\beta)$-metric, where $\beta$ is a closed and conformal one-form.

Equations

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

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

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

$$b_{i|j}=c{a}_{ij},$$

$$F=\alpha \phi \left(\frac{s^2}{e^{\int (\frac{1}{b^2}-b^2\theta )d{b}^2}+s^2\int \theta e^{\int (\frac{1}{b^2}-b^2\theta )d{b}^2}d{b}^2}\right)e^{\int (\frac{1}{2}{b}^2\theta -\frac{1}{b^2})d{b}^2}s$$

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

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

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

$$L_{jkl}:=-\frac{1}{2}{y}^m{g}_{im}{B}_{jkl}^i$$

$$y^m{g}_{im}{B}_{jkl}^i=0.$$

$$J_k:=g^{ij}{L}_{ijk},$$

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

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

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

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

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

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

$$g^{ij}=\frac{1}{\rho }\{a^{ij}+\eta b^i{b}^j+{\eta }_0{\alpha }^{-1}(b^i{y}^j+b^j{y}^i)+{\eta }_1{\alpha }^{-2}{y}^i{y}^j\},$$

$$\eta$$

$${\eta }_1$$

$$r_{ij}$$

$$s_{ij}$$

$$G^i=$$

$$+\{\Psi (-2\alpha Q{s}_0+r_{0}0+2{\alpha }^{2}Rr)+\alpha \Pi (r_0+s_0)\}{b}^i-{\alpha }^{2}R(r^i+s^i),$$

$$Q$$

$$\Theta$$

$$\Pi$$

$$r_{00}=c{\alpha }^2,r_0=c\beta ,r=c{b}^2,r^i=c{b}^i,s_0^i=s_0=s^i=0$$

$$G^i:=G_{\alpha }^i+c{\alpha }^2\{\Theta (1+2R{b}^2)+s\Omega \}{l}^i+c{\alpha }^2\{\Psi (1+2R{b}^2)+s\Pi -R\}{b}^i$$

$$E$$

$$H$$

$$G^i:=G_{\alpha }^i+c{\alpha }^{2}E{l}^i+c{\alpha }^{2}H{b}^i.$$

$$B_{jkl}^i=\frac{c}{\alpha }{U}_{jkl}^i$$

$$U_{jkl}^i:=$$

Full text

Keywords: Finsler geometry; general $(\alpha,\beta)$-metrics; Weak Landsberg metric; generalized Unicorn problem.

1 Introduction

A Finsler manifold $(M,F)$ is a $C^{\infty}$ manifold equipped with a Finsler metric which is a continuous function $F:TM\rightarrow[0,\infty)$ with the following properties:

(1) Smoothness: $F(x,y)$ is $C^{\infty}$ on $TM-\{0\}$.

(2)Positive homogeneity: $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$.

(3) Strong convexity: The fundamental tensor $\left(g_{ij}(x,y)\right)$ is positive definite at all $(x,y)\in TM-\{0\}$, where

[TABLE]

There are a lot of non-Riemannian metrics in Finsler geometry. Randers metric is the simplest non-Riemannian Finsler metric, which was introduced by G. Randers in [6]. As a generalization of Randers metric, $(\alpha,\beta)$-metric is defined by the following form

[TABLE]

which $\alpha$ is a Riemannian metric, $\beta$ is a one-form and $\phi(s)$ is a $C^{\infty}$ positive function. A more general metric class called general $(\alpha,\beta)$-metric was first introduced by C. Yu and H. Zhu in [9]. By definition, it is a Finsler metric expressed in the following form

[TABLE]

This class of Finsler metrics include some Finsler metrics constructed by Bryant (see [3], [4] and [5]).

In Finsler geometry, There are several classes of metrics, such as Berwald metric, Landsberg metric, and weak Landsberg metric. We know that Berwald metric is a bit more general than Riemannian and Minkowskian metric. However, every Berwald metric is not only a Landsberg metric but also a weak Landsberg metric. For $(\alpha,\beta)$-metrics, by definitions, we have the following relations

$\{$ Riemannian $\}\&\{$ locally Minkowskian $\}\subset\{$ Berwald $\}\subset\{$ Landsberg $\}$,

 

and

$\{$ Landsberg metrics $\}\subset\{$ weak Landsberg metrics $\}$.

The pivotal question is, is there a Landsberg metric that is not Berwaldian? This problem was called ”unicorn” by D. Bao [2]. Morever, is there weak Landsberg metric that is not Berwaldian? This problem was called ”generalized unicorn” by Zou and Cheng [12].

In 2009, Z. Shen has proved that a regular $(\alpha,\beta)$-metric $F=\alpha\phi(\beta/\alpha)$-metric on $M$ of dimension $n>2$ is a Landsberg metric if and only if $F$ is a Berwald metric [8]. On the other hand, Z. Shen and G.S. Asanov have constructed almost regular $(\alpha,\beta)$-metrics which are Landsberg metrics but not Berwald metrics respectively (see [1] and [8]). In 2014, Y. Zou and Cheng have showed that if $\phi=\phi(s)$ is a polynomial in $s$, then $F$ is a weak Landsberg metric if and only if $F$ is a Berwald metric. They generalized the main theorem on unicorn problem for regular $(\alpha,\beta)$-metrics in [12].

In general $(\alpha,\beta)$-metrics, Zohrehvand and Maleki in [11] showed that hunting for an unicorn cannot be successful in the class of metrics where $\beta$ is a closed and conformal one-form, i.e. $b_{i|j}=ca_{ij}$, where $b_{i|j}$ is the covariant derivation of $\beta$ with respect to $\alpha$ and $c=c(x)$ is a scalar function on $M$. In this paper, we show that Landsberg metric and weak Landsberg metric are equivalent in the class of general $(\alpha,\beta)$-metrics where $\beta$ is a closed and conformal one-form and thus hunting for an generalized unicorn cannot be successful.

Theorem 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

[TABLE]

where $b_{i|j}$ is the covariant derivation of $\beta$ with respect to $\alpha$ and $c=c(x)$ is a scalar function on $M$. Then $F$ is a weak Landsberg metric if and only if it is Landsberg metric.

Corollary 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 $\beta$ is closed and conformal. Then $F$ is a weak Landsberg metric if and only if it is Berwald metric.

Thus, under the certain condition, the generalized unicorn’s problem cannot be successful in the class of general $(\alpha,\beta)$-metrics. In this case $F$ can be expressed by

[TABLE]

where $\varphi(.)$ is any positive continuously differentiable function and $\theta$ is a smooth function of $b^{2}$ [10].

2 Preliminaries

For a given Finsler metric $F=F(x,y)$, the geodesic of $F$ satisfies the following differential equation:

[TABLE]

where $G^{i}=G^{i}(x,y)$ are called the geodesic coefficients defined by

[TABLE]

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

[TABLE]

Thus, a Finsler metric $F$ is a Berwald metric if and only if $\textbf{B}=0$.

The Landsberg curvature can be expressed by

[TABLE]

A Finsler metric $F$ is a Landsberg metric if and only if $L_{jkl}=0$, i.e.

[TABLE]

Thus, Berwald metrics are always Landsberg metrics.

There is a weaker non-Riemannian quantity than the Landsberg curvature L in Finsler geometry, $\textbf{J}=J_{k}dx^{k}$, where

[TABLE]

and $(g^{ij})=(g_{ij})^{-1}$. We call J the mean Landsberg curvature of Finsler metric $F$. A Finsler metric $F$ is called weak Landsberg metric if its mean Landsberg curvature J vanishes [7].

Let $F$ be a Finsler metric on a manifold $M$. $F$ is called a general $(\alpha,\beta)$-metrics, if $F$ can be expressed as the form

[TABLE]

where $\alpha$ is a Riemannian metric and $\beta:=b_{i}(x)y^{i}$ is an one-form with $||\beta||_{\alpha}<b_{0}$ for every $x\in M$. The function $\phi=\phi(b^{2},s)$ is a positive $C^{\infty}$ function satisfying

[TABLE]

when $n\geq 3$ or

[TABLE]

when $n=2$, where $s$ and $b$ are arbitrary numbers with $|s|\leq b<b_{0}$, for some $0<b_{0}\leq+\infty$. In this case, the fundamental tensor is given by [9]

[TABLE]

where

[TABLE]

Moreover,

[TABLE]

where $(g^{ij})=(g_{ij})^{-1},~{}(a^{ij})=(a_{ij})^{-1},~{}b^{i}=a^{ij}b_{j}$,

[TABLE]

Not that, we use the indices 1 and 2 as the derivation with respect to $b^{2}$ and $s$, respectively [9].

Let $b_{i|j}$ denote the coefficients of the covariant derivative of $\beta$ with respect to $\alpha$. Let [9]

[TABLE]

$\beta$ is a closed one-form if and only if $s_{ij}=0$, and it is a conformal one-form with respect to $\alpha$ if and only if $b_{i|j}+b_{j|i}=ca_{ij}$, where $c=c(x)$ is a nonzero scalar function on $M$. Thus, $\beta$ is closed and conformal with respect to $\alpha$ if and only if $b_{i|j}=ca_{ij}$, where $c=c(x)$ is a nonzero scalar function on $M$.

For a general $(\alpha,\beta)$-metric, its spray coefficients $G^{i}$ are related to the spray coefficients $G_{\alpha}^{i}$ of $\alpha$ by [9]

[TABLE]

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

[TABLE]

When $\beta$ is closed and conformal one-form, i.e. satisfies (1), then

[TABLE]

Substituting this into (2) yields [10]

[TABLE]

If we have

[TABLE]

then from (10)

[TABLE]

The Berwald curvature of a general $(\alpha,\beta)$-metric, when $\beta$ is a closed and conformal one-form, is computed in [10]:

Proposition 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 (1), then the Berwald curvature of $F$ is given by

[TABLE]

where

[TABLE]

where $E$ and $H$ is defined in (11) and (12), $l_{j}:=a_{ij}l^{i}$ and $(k\rightarrow l\rightarrow j\rightarrow k)$ denotes cyclic permutation.

3 The mean Landsberg curvature of General $(\alpha,\beta)$-metrics

By use of Proposition 1, M. Zohrehvand and H. Maleki calculated the Landsberg curvature of a general $(\alpha,\beta)$-metric in [11], when $\beta$ is a closed and conformal one-form:

Proposition 2**.**

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 (1), then the Landsberg curvature of $F$ is given by

[TABLE]

where

[TABLE]

By use of Proposition 2 and Maple, we can calculate the mean Landsberg curvature of a general $(\alpha,\beta)$-metric, when $\beta$ is a closed and conformal one-form:

Proposition 3**.**

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 (1), then the Landsberg curvature of $F$ is given by

[TABLE]

where

[TABLE]

where $\rho$ and $\eta$ is defined in (5) and (8) and $l_{j}:=a_{ij}l^{i}$.

Proof.

To compute the mean Landsberg curvature $J_{j}:=g^{kl}L_{jkl}$ using (15), we need to (7). Using $L_{jkl}y^{k}=0$, we get

[TABLE]

We need

[TABLE]

and

[TABLE]

Substituting these in (19), we obtain (17).∎∎

Now, we can obtain the necessary and sufficient conditions for a general $(\alpha,\beta)$-metric to be weak Landsbergian.

Proposition 4**.**

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 (1), then $F$ is weak Landsberg metric if and only if the following equations hold:

[TABLE]

Proof.

Let $F=\alpha\phi(b^{2},s),~{}s=\beta/\alpha$, be a weak Landsberg metric, where $\beta$ is a closed and conformal one-form. From Proposition 2, it concluded

[TABLE]

where $W_{j}$ is defined in (3). We can rewrite (22) as following:

[TABLE]

Thus

[TABLE]

Since $1+n+3(b^{2}-s^{2})\eta\neq 0$, we have from (24)

[TABLE]

From (25), we know $(b^{2}-s^{2})[1+(b^{2}-s^{2})\eta]\neq 0$, then $s\phi+(b^{2}-s^{2})\phi_{2}=0$ or $H_{222}=0$. If $s\phi+(b^{2}-s^{2})\phi_{2}=0$, we have

[TABLE]

where $\sigma(b^{2})$ is any positive smooth function. This is a Riemannian case and we have

[TABLE]

In (26), we have $3(b^{2}-s^{2})[1+(b^{2}-s^{2})\eta]\phi_{2}-[1+n+3(b^{2}-s^{2})\eta]s\phi\neq 0$ and

[TABLE]

∎∎

**Proof of Theorem 1.**Since in [11] is obtained exactly (20) and (21) for Landsberg metrics, according to 4, it is obvious. ∎

Proof of Corollary 1. Here, by [11], we have $E-sE_{2}=0$ and $H_{2}-sH_{22}=0$ also. In [10], it is proved that a general $(\alpha,\beta)$-metric where $\beta$ is closed and conformal one-form, is a Berwald metric if and only if

[TABLE]

and in this case $F$ can be expressed by

[TABLE]

where $\varphi(.)$ is any positive continuously differentiable function and $\theta$ is a smooth function of $b^{2}$ [10]. This complete the proof.∎

4 Examples

In this section, we will explicitly construct some new examples.

Example 1**.**

Take $\varphi(t):=1+\xi t$ and $\theta(b^{2}):=1$, then by (31)

[TABLE]

We can see that $\phi(b^{2},s)$ satisfies in (20) and (21). Moreover, the corresponding general $(\alpha,\beta)$-metrics

[TABLE]

are Landsberg and weak Landsberg metrics, i.e. $L_{jkl}=0$ and $J_{j}=0$.

Example 2**.**

Take $\varphi(t):=\frac{\mu}{1+\xi t}$ and $\theta(b^{2}):=\frac{\varepsilon}{1+\xi b^{2}}$ in (31), we have

[TABLE]

The corresponding general $(\alpha,\beta)$-metrics are Landsberg and weak Landsberg metrics.

Acknowledgment

The authors would like to express their special thanks to Dr. M. Zohrehvand for his valuable opinions on this paper.