On a class of weakly Landsberg metrics composed by a Riemannian metric and a conformal 1-form

1.   Introduction

Without the quadratic restriction, Finsler geometry has many non-Riemannian properties compared to Riemannian geometry. There are many non-Riemannian quantities in Finsler geometry, such as the Berwald curvature, the Landsberg curvature, the mean Landsberg curvature and more. A Finsler metric is called a Berwald metric if its Berwald curvature vanishes, it is called a Landsberg metric if its Landsberg curvature vanishes, or it is called a weakly Landsberg metric if its mean Landsberg curvature vanishes. It is a fact that all Berwald metrics are Landsberg metrics, and all Landsberg metrics are weakly Landsberg metrics. M. Matsumoto showed a list of reduction theorems of certain Landsberg metrics to Berwald metrics ^[10]. Afterward, a natural problem arose in Finsler geometry:

The question of whether or not any Landsberg metric is a Berwald metric.

This problem, presented by D. Bao, is called the "unicorn problem" because it is one of the most important unsolved problems in Finsler geometry. There are some structures of Berwald metrics studied by Z. I. Szabó ^[18,19,20]. While some geometers got other results ^{[2,5,7,11,16,17,21,24,26]}. However, the "unicorn problem" is still open. Meanwhile, there is another related problem in Finsler geometry:

The question of whether or not any weakly Landsberg metric is a Landsberg metric.

Clearly, in the two-dimensional case, any weakly Landsberg metric is a Landsberg metric. Some rigidity problems are related to weakly Landsberg metrics ^[3,6,15]. T. Aikou studied the weakly Landsberg spaces, which he referred to as the generalized Landsberg ^[1]. B. Li and Z. Shen studied weakly Landsberg $(\alpha, \beta)$-metrics and found that there exists weakly Landsberg metrics, which are not Landsberg metrics ^[9]. Thus, it is natural to study more general metrics than ever before. In this paper, we mainly study the general $(\alpha, \beta)$-metric that was first introduced by C. Yu and H. Zhu ^[22]. A general $(\alpha, \beta)$-metric is a Finsler metric expressed in the following form,

where $\phi(b^2, s)$ is a $C^{\infty}$ function, $\alpha = \sqrt{a_{ij}(x)y^iy^j}$ is a Riemannian metric and $\beta = b_{i}(x)y^{i}$ is a $1$-form, $b: = \|\beta_x\|_\alpha$. Specially, $F$ is the spherically symmetric metric when $\alpha = |y|$ is the Euclidean metric and $\beta = \langle x, y\rangle$ is the Euclidean inner production. It is easy to see that $\beta$ is closed and conformal to $\alpha$ for all spherically symmetric metrics, i.e.,

where $c = c(x)$ is a $C^\infty$ scalar function and the subscript $"|"$ denotes the covariant derivative with respect to the Riemannian metric $\alpha$. When $c = 0$, then $\beta$ is said to be parallel to $\alpha$. Some papers studied general $(\alpha, \beta)$-metrics under this condition ^[8,22,23,25] and many interesting properties were found. Thus, it is meaningful to study the problems when $\beta$ is closed and conformal to $\alpha$.

In this paper, we first study weakly Landsberg $(\alpha, \beta)$-metrics satisfying (1.1). We make an assumption that the expression of $\phi = \phi(b^2, s)$ does not depend on $n$, and this assumption is used to prove the following theorem. After getting the formula of the mean Landsberg curvature of these metrics, the following theorem is obtained.

Theorem 1.1. Let $F = \alpha\phi(b^2, s)$ be a general $(\alpha, \beta)$-metric on an n-dimensional manifold ($n\geq 2$). Suppose that the expression of the function $\phi$ does not depend on the dimension $n$ and $\beta$ is closed and conformal to $\alpha$ satisfying (1.1) with $c \neq0$, then $F$ is a weakly Landsberg metric if, and only if, it is a Landsberg metric.

In the above theorem, the expression of $\phi$ does not depend on $n$, which means that $\phi$ is only a function of $b^2$ and $s$. For instance, it does depend on $n$ if $\phi = n + s$, while it does not if $\phi = 1 + s$. Of course, $F$ is still related to the dimension $n$ when we express $F = \alpha \phi$ in $\alpha$, $\beta$ and $b^2$. $F$ will change with the change of dimension, but the variable $n$ will not appear in the expression of $\phi = \phi(b^2, s)$. Then the expression of $X$ and $H$ in (2.13) are just functions of $b^2$ and $s$. The dimension $n$ will not be a variable in their expressions. Then the expression of Berwald curvature and Landsberg curvature do not depend on $n$ in this case. For example, all of the spherically symmetric Finsler metrics satisfy this condition. This condition is used in getting (3.23) and (3.24) by (3.22) in proving Theorem 1.1 in Section 3.

When $\phi$ depends on $n$, we guess Theorem 1.1 is also correct, although we haven't found a workable way to prove it. Here $c = c(x)\neq0$ is natural because $F$ must be trivial Berwaldian if $c = 0$ (i.e., $\beta$ is parallel to $\alpha$).

S. Zhou and B. Li proved that the almost regular Landsberg general $(\alpha, \beta)$-metric must be Berwaldian when the dimension $n\geq3$ ^[25]. However, the two-dimensional case has not been solved. In Theorem 1.2, we give the equivalent conditions of Berwald metric in the two-dimensional case, then the expression of $\phi$ can also be given.

Theorem 1.2. Let $F = \alpha\phi(b^2, s)$ be an almost regular general $(\alpha, \beta)$-metric on a $2$-dimensional manifold. Suppose that $\beta$ is closed and conformal to $\alpha$ satisfying (1.1) with $c \neq0$. Thus, $F$ is a Berwald metric if, and only if, $\phi$ is given by

where

and $c_i = c_i(b^2), (i = 1, 2, ..., 7)$ are $C^\infty$ functions of $b^2$ satisfying the following condition,

Two explicit examples of two-dimensional Berwald metrics are given in Section 4 (Examples 4.1 and 4.2).

2.   Preliminaries

In this section, some definitions and lemmas needed are introduced. There are most of the notations^[4,14]. A Finsler metric on a manifold $M$ is a function $F:TM\rightarrow [0, +\infty)$, which satisfies

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

(ii) $F(x, \lambda y) = \lambda F(x, y)$, for any $\lambda > 0$;

(iii) $g_{ij}(x, y) = \frac{1}{2}\frac{\partial^2F^2}{\partial y^i\partial y^j}$ is positive definitely, for any $y\neq0$.

Specially, $F$ is called a Riemannian metric if $g_{ij} = g_{ij}(x)$. $F$ is called a Minkowskian metric if $g_{ij} = g_{ij}(y)$. A class of computable Finsler metrics called $(\alpha, \beta)$-metric is defined by

where $\phi$ is a $C^\infty$ function, $\alpha = \sqrt{a_{ij}(x)y^i y^j}$ is a Riemannian metric and $\beta = b_i(x)y^i$ is a 1-form. When $\phi = 1$, $F = \alpha$ is a Riemannian metric. When $\phi = 1+s$, $F = \alpha+\beta$ is a Randers metric ^[13]. Further, when $\phi = \phi(b^2, s)$, another class of Finsler metric called general $(\alpha, \beta)$-metric can be given as follows

When $\alpha = |y|$ is the Euclidean metric and $\beta = \langle x, y\rangle$ is the Euclidean inner production of $x$ and $y$, $F = |y|\phi({\vert{x}\vert}, \frac{\langle{x, y}\rangle}{|y|})$ is called a spherically symmetric metric. There are some results on these metrics ^[12,23,24]. In this case, (1.1) is always satisfied and $c = 1$.

By a direct computation, the $g_{ij}$ of $F$ in (2.2) can be obtained.

where

In this paper, the subscript "1" means partial derivative with respect to $b^2$, the subscript "2" means partial derivative with respect to $s$, such as $\phi_1 = \frac{\partial \phi}{\partial (b^2)}$, $\phi_2 = \frac{\partial \phi}{\partial s}$.

The positivity of general $(\alpha, \beta)$-metric can be ensured by the following lemma.

Lemma 2.1. Let $M$ be an n-dimensional manifold ^[22]. $F = \alpha \phi(b^2, \frac{\beta}{\alpha})$ 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 positive $C^{\infty}$ function satisfying

when $n\geq3$ or

when $n = 2$, where $s$ and $b$ are arbitrary numbers with $|s|\leq b < b_0.$

In order to obtain geodesic coefficients and mean Landsberg curvature of $F$, the first step is to compute the inverse of $(g_{ij})$.

where $a^{ij} = (a_{ij})^{-1}$, $b^i = a^{ij}b_j$,

In this paper, the subscript $"|"$ denotes the covariant derivative with respect to the Riemannian metric $\alpha$, such as $b_{i|j}$. In order to simplify the computation, we use the following expression

The spray $G = y^i\frac{\partial}{\partial x^i}-2G^i\frac{\partial}{\partial y^i}$ of Finsler metric $F$ is a vector field on $TM$, where $G^i = G^i(x, y)$ are called geodesic coefficients (spray coefficients) and defined by

where $(g^{ij}) = (g_{ij})^{-1}$.

The following lemma gives $G^i$ of general $(\alpha, \beta)$-metrics.

Lemma 2.2. The geodesic coefficients $G^i$ of a general $(\alpha, \beta)$-metric $F = \alpha \phi(b^2, \frac{\beta}{\alpha})$ ^[22] are given as

where

Here when (1.1) is satisfied, the expression of $G^i$ can be simplified. By (1.1),

Then the geodesic coefficients $G^i$ can be rewritten as

where

and

Non-Riemannian quantities play an important role in Finsler geometry, such as the Berwald curvature, the mean Berwald curvature, the Landsberg curvature and the mean Landsberg curvature. The Berwald curvature of $F$ is a tensor defined by

where

and

A Finsler metric is called a Berwald metric if and only if $B^i_{jkl} = 0$. Obviously, $G^i$ are quadratic in $y$ if and only if $F$ is a Berwald metric. By the definition of Berwald curvature, S. Zhou and B. Li obtained the expression of $B^i_{jkl}$ ^[25]. We found there were some typing errors of $B^i_{jkl}$, and the expression of $B^i_{jkl}$ under condition (1.1) can be given by

where

The mean Berwald curvature of $F$ is a tensor which can be defined by

where

A Finsler metric is called a weakly Berwald metric if and only if $B_{mjk}^m = 0$.

The Landsberg curvature of $F$ is a tensor defined by

where

and

A Finsler metric is called a Landsberg metric if and only if $L_{jkl} = 0$.

Based on (2.18), the expression of $L_{jkl}$ is given ^[25] as follows

where

The following lemma gives the equivalent conditions of Berwald metric in two-dimensional case.

Lemma 2.3. Let $F$ be a two-dimensional Finsler metric. $F$ is Berwaldian if, and only if, it is Landsbergian and weakly Berwaldian ^[14].

The definitions of mean Landsberg curvature and weakly Landsberg metric are given in next section.

3.   Weakly Landsberg general $(\alpha, \beta)$-metric

In this section, the expression of $J_j$ is given and Theorem 1.1 is proved. The mean Landsberg curvature is a tensor defined by

where

and

A Finsler metric is called a weakly Landsberg metric if and only if $J_j = 0$. Here we give the formula of $J_j$ directly by (2.6), (2.23) and (3.2).

Lemma 3.1. Let $F = \alpha \phi(b^2, s)$ be a general $(\alpha, \beta)$-metric on an $n$-dimensional manifold, then the mean Landsberg curvature $J = J_j dx^j$ is given by

where

Proof. By direct contractions, we get

Substituting (2.6) and (2.23) into (3.2) yields

The second equality in the above equation is given by (3.6), and the other equalities are given by (3.7) and (3.8).

By the above results, (3.3) can be given.

The equivalent conditions of Landsberg metric are given ^[25]; in this section, the following conditions will be used in our proof of Theorem 1.1.

Lemma 3.2. Let $\phi = \phi(b^2, s)$ be a positive $C^{\infty}$ function ^[25], then $\phi$ is a Landsberg metric if, and only if, the function $\phi = \phi(b^2, s)$ satisfies

where $c_{1} = c_{1}(b^2)$ is a $C^{\infty}$ function of $b^2$.

For the convenience of calculation, we must first prove the following lemma, which will be used in Theorem 1.1.

Lemma 3.3. Let $F = \alpha \phi(b^2, \frac{\beta}{\alpha})$ be a non-Riemannian general $(\alpha, \beta)$-metric on an n-dimensional manifold $n\geq2$, then $\phi$ is a $C^\infty$ function satisfies

Proof. If $s\phi+(b^2-s^2)\phi_2 = 0$. Integrating the above equation with respect to $s$ yields

where $\bar c_1 = \bar c_1(b^2)$ is $C^{\infty}$ function of $b^2$. Unquestionably, $F = \alpha\phi$ is a Riemannian metric. If $(\phi-s\phi_2)\phi_2-\, s\phi\, \phi_{22} = 0$, then integrating the above equation with respect to $s$ yields

where $\bar c_2 = \bar c_2(b^2), \bar c_3 = \bar c_3(b^2)$ are $C^{\infty}$ functions of $b^2$. Since $F = \alpha\phi$ is a Riemannian metric, this is a contradiction to the conditions.

Now by Lemmas 3.1–3.3, Theorem 1.1 can be proved.

Proof of Theorem 1.1.

${\bf “Necessity”}$ Let

Differentiating the first equation of (3.14) with respect to $s$ yields

Differentiating (3.15) with respect to $s$ yields

Differentiating the second equation of (3.14) with respect to $s$ yields

By rewriting (3.14)–(3.17), we get

To get the expression of $J_j$ by substituting the above identities into (3.4) and by $(3.4)\times s^2$, we get

The expression of the right hand of Eq (3.4) is a fraction by substituting identities (3.18) and (3.19) into it, where $s$ is the denominator. According to the rationality of the equation, the denominator is $s\neq0$, but by definition, $s = 0$ is reasonable. In fact, the $s$ in the denominator can be eliminated by the numerator. In order to avoid confusion, here we consider (3.4) $\times s^2$, and the result is not affected. Substituting (3.18) and (3.19) into (3.5) yields

In order to give the proof more conveniently, (3.3) can be rewritten as

Based on the assumption that $\phi$ does not depend on the dimension $n$, there is no $n$ in the expressions of $C$, $E$ and $\eta$. By (3.22), $J_j = 0$ is equivalent to

and

It is a fact that $\alpha$ is a Riemannian metric, so $\alpha\neq0$. By (2.19), if $h_j = 0$, $\alpha^2$ can be divided exactly by $\beta y_j$, it is impossible, thus $h_j\neq0$. By (3.21), (3.23) is equivalent to

If $c = 0$, by (2.23) we obtain $L_{jkl} = 0$ and Theorem 1.1 is proved.

If $c\neq0$. By Lemma 3.3, we get

Then $\bar H$ can be solved from (3.25),

Differentiating (3.27) with respect to s yields

By the same reason before (3.25), (3.24) is equivalent to

Plugging (3.20) and (3.21) into (3.29)$\times$$s^2$ yields

Plugging (3.27) and (3.28) into (3.30) yields

By Lemma 2.1,

By Lemma 3.3,

Meanwhile, by the definition of $b$ and $s$, we get

Hence, (3.31) is equivalent to

Because $c\neq0$, (3.35) is equivalent to

Integrating the above equation with respect to $s$ yields

where $c_1 = c_1(b^2)$ is a $C^\infty$ function of $b^2$. Differentiating (3.36) with respect to s yields

Plugging (3.36) and (3.37) into (3.27) yields

By Lemma 3.2, $F$ is a Landsberg metric.

"Sufficiency" By definition, all Landsberg metrics are weakly Landsberg metrics.

As spherically symmetric metrics compose a special class in general $(\alpha, \beta)$-metrics and satisfy (1.1). Thus, the following Corollary 3.4 can be obtained as a special case of Theorem 1.1.

Corollary 3.4. Let $F = |y|\phi({\vert{x}\vert}, \frac{\langle{x, y}\rangle}{|y|})$ be a spherically symmetric metric on an n-dimensional manifold ($n\geq 2$). Suppose $n$ vanishes in the expression of $\phi$. Then $F$ is a weakly Landsberg metric if, and only if, it is a Landsberg metric.

Proof. When $F$ is a spherically symmetric metric, then

Obviously, Theorem 1.1 is satisfied in spherically symmetric case.

4.   Berwald general $(\alpha, \beta)$-metric

S. Zhou and B. Li studied the Berwald general $(\alpha, \beta)$-metric on an $n$-dimensional manifold $n\geq3$ and gave the classification by obtaining the equivalent equations of the Berwald metric when $n\geq3$^[25]. In this section, by Lemma 2.3 and based on the equivalent conditions of the Berwald metric, we get Lemmas 4.2 and 4.3. From these, we get two equivalent equations in Proposition 4.4, then Theorem 1.2 can be proved.

The following lemma^[25] is obtained, which is needed in the proof of Lemma 4.2.

Lemma 4.1. Let $F = \alpha \phi(b^2, s)$ be a non-Riemannian general $(\alpha, \beta)$-metric on an n-dimensional manifold $n\geq2$ ^[25]. Suppose $I = I(b^2, s)$ and $N = N(b^2, s)$ are arbitrary $C^{\infty}$ functions, then the following facts hold:

$(i)$ $(n\geq3)$ $h_{j}h_{k}h_{l}I+h_{jk}h_{l}N+h_{jl}h_{k}N+h_{kl}h_{j}N = 0$ if and only if $I = 0$ and $N = 0$;

$(ii)$ $(n = 2)$ $h_{j}h_{k}h_{l}I+h_{jk}h_{l}N+h_{jl}h_{k}N+h_{kl}h_{j}N = 0$ if and only if $(b^2-s^2)I+3N = 0$.

Based on Lemma 4.1, in order to prove Proposition 4.4, the Landsberg curvature of the general $(\alpha, \beta)$-metric can be rewritten in two-dimensional case as follows.

Lemma 4.2. Let $F = \alpha \phi(b^2, s)$ be a non-Riemannian general $(\alpha, \beta)$-metric on a two-dimensional manifold. The Landsberg curvature $L_{jkl}$ can be given as

where

Proof. Contracting (2.18) with $b_i$ and $y_i$ respectively yields

and

where

From $F = \alpha \phi(b^2, s)$, we get

By (2.22),

Plugging (4.2) and (4.3) into (4.7) yields

where

By Lemma 2.3, the mean Berwald curvature is essential in our proof. In the following lemma, we present it in a two-dimensional case.

Lemma 4.3. Let $F = \alpha \phi(b^2, s)$ be a non-Riemannian general $(\alpha, \beta)$-metric on a two-dimensional manifold. The mean Berwald curvature $B_{mjk}^m$ is given by

Proof. By contracting (2.18) for $i$ and $l$, we get

When $n = 2$, we have

where

By the assumption that the dimension is two, $h_ih_j$ can be expressed as follows

By the above expressions,

Therefore, (4.10) is equivalent to

By Lemmas 2.3, 4.2 and 4.3, two equivalent equations of the Berwald general $(\alpha, \beta)$-metric can be obtained as follows.

Proposition 4.4. Let $F = \alpha \phi(b^2, s)$ be a non-Riemannian general $(\alpha, \beta)$-metric on a $2$-dimensional manifold, then $F$ is a Berwald metric if, and only if, it satisfies

and

Proof. By Lemma 4.1 and (4.1) in Lemma 4.2 when $n = 2$, $L_{jkl} = 0$ if, and only if,

Which is equivalent to

By (4.11), $b^2-s^2\neq0$ and $h_j\neq0$, we get $h_{jk}\neq0$. By Lemma 4.3, $B_{mjk}^m = 0$ if, and only if,

By above results and Lemma 2.3, $F$ is a Berwald metric if, and only if, it satisfies (4.13) and (4.14).

By solving (4.13) and (4.14), $\phi$ can be given.

Proof of Theorem 1.2.

By Proposition 4.4 and considering $(4.13)\times(b^2-s^2)-(4.14)\times[s\phi+(b^2-s^2)\phi_2]$, we get

By (3.18), the above equation can be rewritten as

Integrating the above equation twice yields

where $\gamma_2 = \gamma_2(b^2), \gamma_3 = \gamma_3(b^2)$ are $C^\infty$ functions of $b^2$. By plugging (4.20) into the first equation of (3.14) and solving $X$, we get

where $c_1 = c_1(b^2), c_2 = -\gamma_2(b^2), c_3 = \frac{b^2\gamma_2(b^2) +2\gamma_3(b^2)}{b^4}$ are $C^\infty$ functions of $b^2$. Differentiating (4.20) with respect to $s$ yields

Differentiating (4.22) with respect to $s$ yields

Plugging (3.19), (4.21)–(4.23) into (4.14) yields

By solving the above equation, we get

where $c_4 = c_4(b^2)$ is a $C^\infty$ function of $b^2$. By (4.25) and the second equation of (3.14), $H$ can be solved as

where $c_5 = c_5(b^2), c_6 = c_6(b^2)$ are $C^\infty$ functions of $b^2$. Differentiating (4.26) with respect to $s$ yields

By (2.14) and (2.15), we have

and

Differentiating (4.28) with respect to $s$ yields

By $(4.30) \; \rm and \; (4.29)$, we get

By $(4.31)\times s-(4.28)$, we get

Plugging (4.20) and (4.22), (4.25) and (4.27) into (3.18) yields

which is equivalent to

Integrating the above equation, (1.2) is obtained, where $c_7 = c_7(b^2)$ is $C^\infty$ function of $b^2$. To prove the sufficiency, by plugging (4.21), (4.26) and (1.2) into (4.28), we get the condition (1.4). Plugging (4.21) and (4.26) into (2.13) yields

where

The following steps are to prove $G^i$ are quadratic in $y$. In fact, in the two-dimensional case, $\sqrt{{b}^{2}-{s}^{2}}$ can be simplified. By assumption,

Thus,

where $\det a = a_{11}a_{22}-(a_{12})^2$ is the determinant of $a_{ij}$. By (4.35) and (4.37),

which are quadratic in $y$, and then by (4.36) and (4.37), $\Gamma^1$ can be expressed as follows

$\Gamma^2$ can be given in the same method

Undoubtedly, $\Gamma^i$ is quadratic in $y$, and it is proved that $G^i$ is quadratic in $y$. Therefore, $F$ is a Berwald metric.

As a special class in general $(\alpha, \beta)$-metric, for spherically symmetric metric, the following corollary can be easily proved.

Corollary 4.5. Let $F = |y|\phi({\vert{x}\vert}, \frac{\langle{x, y}\rangle}{|y|})$ be a spherically symmetric metric on a two-dimensional manifold. Then $F$ is a Berwald metric if, and only if, $\phi$ is given by (1.2), and $c_1 = c_1(b^2), c_2 = c_2(b^2), c_3 = c_3(b^2), c_4 = c_4(b^2), c_6 = c_6(b^2), c_7 = c_7(b^2)$ are $C^\infty$ functions of $b^2$ satisfy (1.4).

Proof. As $F$ is a spherically symmetric metric, we get

Therefore, Theorem 1.2 is satisfied in spherically symmetric case. □

Next, by choosing suitable $c_1, c_2, c_3, c_4, c_6, c_7$, some explicit examples can be given as follows.

Example 4.1. Let $c_1 = 0$, $c_2 = 0$, $c_3 = 1$, $c_4 = b^2$, $c_6 = 0$ and $c_7 = c_7(b^2) > 0$ in (1.3), then

By (1.2), $\phi$ can be given as

By a direct computation,

This proves $\phi$ satisfies Lemma 2.1. Therefore, $F = \alpha\phi$ is a Berwald metric.

Example 4.2. Let $c_1 = 0$, $c_2 = b^2$, $c_3 = 2$, $c_4 = 0$, $c_6 = 0$ and $c_7 = c_7(b^2) > 0$ in (1.3), then

By (1.2), $\phi$ can be given as

It is easy to verify that $\phi$ satisfies Lemma 2.1. Therefore, $F = \alpha\phi$ is a Berwald metric.

5.   Conclusions

In this paper, we firstly get the expression for the mean Landsberg curvature of general $(\alpha, \beta)$-metric under the condition that $\beta$ is closed and conformal to $\alpha$. When the expression of the function $\phi$ does not depend on $n$, the equivalent relationship between Landsberg metric and weakly Landsberg metric is obtained. This is Theorem 1.1. Secondly, we get the expressions of mean Berwald curvature and Landsberg curvature in two-dimensional case. Base when based on Lemma 2.3, the classification of two-dimensional Berwald general $(\alpha, \beta)$-metrics under the condition (1.1) with $c \neq0$. This is Theorem 1.2. Hence, these results can be used in studying (weakly) Landsberg metrics and Berwald metrics in future.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This research is partly supported by the ZPNSFC (Y23A010060), NNSF (2021J110) and K. C. Wong Magna Fund in Ningbo University.

Conflict of interest

There is no conflict of interest.