On Finsler Surfaces with Isotropic Main Scalar

Abstract

Let $(M,F)$ be a Finsler surface with the isotropic main scalar $\mathcal{I}=\mathcal{I}\left(x\right)$. The well-known Berwald’s theorem states that F is a Berwald metric if and only if it has a constant main scalar $\mathcal{I}=constant$. This ensures a kind of equality of two non-Riemannian quantities for Finsler surfaces. In this paper, we consider a positively curved Finsler surface and show that $\mathbf{H}=0$ if and only if $\mathcal{I}=0$. This provides an extension of Berwald’s theorem. It follows that F has an isotropic scalar flag curvature if and only if it is Riemannian. Our results yield an infrastructural development of some equalities for two-dimensional Finsler manifolds.

1. Introduction

By an investigation of the class of two-dimensional Finsler spaces, one can find that these spaces have special geometric properties that separate them from the spaces of higher dimensions [1,2,3,4,5]. For example, all two-dimensional Finsler metrics are C-reducible and do not need to be Randers or Kropina metrics. This fact means that the well-known conclusive theorem of Matsumoto-Hōjō does not work properly for two-dimensional Finsler spaces. Also, all two-dimensional Finsler spaces are relatively Landsberg spaces, implying that the relative rate of change in the Landsberg curvature of Finsler surfaces along geodesics is a scalar function on the tangent bundle. As another interesting subject, the study of the flag curvature of two-dimensional Finsler spaces, shows that these spaces inherently possess scalar flag curvature. Namely, the flag curvature is independent of the flag. It was Ludwig Berwald who systemically studied two-dimensional spaces and made very important pioneering works in this class of Finsler spaces [6].

Finsler surfaces have many geometric properties that are different from higher-dimensional Finsler spaces. In particular, for two-dimensional Finsler metrics, there exists a notion of the main scalar, denoted by $\mathcal{I}=\mathcal{I}(x,y)$, which is characterized by the following properties. The main scalar $\mathcal{I}=\mathcal{I}(x,y)$ of the two-dimensional Finsler manifold $(M,F)$ vanishes identically if and only if the metric reduces to a Riemannian metric. Among the class of two-dimensional Finsler metrics, those with an isotropic main scalar $\mathcal{I}=\mathcal{I}\left(x\right)$ have interesting curvature properties. In [6], Berwald proved that a two-dimensional Finsler metric with an isotropic main scalar $\mathcal{I}=\mathcal{I}\left(x\right)$ is a Berwald metric or Landsberg metric if and only if $\mathcal{I}\left(x\right)=constant$. In [7], Matsumoto showed that a two-dimensional one-form metric is a Landsberg metric if and only if it is a Berwald metric. Berwald metrics are divided into two classes: either a T-Minkowskian metric or $\mathcal{I}\left(x\right)=constant$. Recently, Yang-Cheng proved that a Finsler surface with an isotropic main scalar is locally conformally flat if and only if $\mathcal{I}\left(x\right)=constant$ [8].

In [6], Berwald studied Finsler surfaces with the isotropic main scalar $\mathcal{I}=\mathcal{I}\left(x\right)$ and showed that all such Finsler metrics can be expressed as one of the following cases:

$$\begin{array}{ccc}& & F={\theta }^s{\gamma }^{1-s};(s=s\left(x\right)\ne 0,s\left(x\right)\ne 1);\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & F=\sqrt{({\theta }^2+{\gamma }^2)e^{2rarctan\left(\frac{\theta }{\gamma }\right)}},r=r\left(x\right),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & F=\theta e^{\frac{\gamma }{\theta }},\hfill \end{array}$$

(3)

where $\theta ={\theta }_i\left(x\right)y^i$ and $\gamma ={\gamma }_i\left(x\right)y^i$ are two independent one-forms. Their main scalars are given, respectively, by

$$\begin{array}{ccc}& & \mathcal{I}\left(x\right)=\pm \sqrt{\frac{{(2s\left(x\right)-1)}^2}{ϵs\left(x\right)\left(s\right(x)-1)}},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & \mathcal{I}\left(x\right)=\pm \sqrt{\frac{4r{\left(x\right)}^2}{1+r{\left(x\right)}^2}},\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & \mathcal{I}=\pm 2,\hfill \end{array}$$

(6)

where $ϵ$ in (4) is the index of F satisfying $ϵ=1$ if $s\left(x\right)>1$ or $s\left(x\right)<0$, and $ϵ=-1$ if $0<s\left(x\right)<1$. If the main scalar satisfies $\mathcal{I}=0$, which is equivalent to $s=1/2$ in (1) or $r=0$ in (2), then F reduces to a Riemannian metric. Based on the classification of Finsler surfaces with an isotropic main scalar, as above, Berwald classified projectively flat Finsler surfaces with an isotropic main scalar [6]. Moreover, Berwald proved the following rigidity result.

Theorem 1

([6] Berwald Theorem). Let $(M,F)$ be a Finsler surface. Suppose that F has an isotropic main scalar. Then, F is a Berwald metric if and only if it has a constant main scalar.

The E-curvature is obtained by taking a trace of the B-curvature. Also, the H-curvature is obtained by taking a horizontal derivation of the E-curvature along Finslerian geodesics.

Theorem 2.

Let $(M,F)$ be a positively curved Finsler surface. Suppose that F has an isotropic main scalar. Then, $\mathbf{H}=0$ if and only if F is Riemannian. Equivalently, $\mathbf{H}=0$ if and only if F has a vanishing main scalar.

Here, we remark upon two important subjects. First, every Berwald metric satisfies $\mathbf{H}=0$. Then, Theorem 2 implies that every positively curved Berwald surface with an isotropic main scalar is Riemannian and then $\mathcal{I}=0$. Obviously, Theorem 2 is an extension of the Berwald Theorem for positively curved surfaces. Second, the positive-definiteness of F in Theorem 2 is necessary. For an example, see the following.

Example 1.

Consider the manifold $M:={\mathbb{R}}^2$ with the local coordinates $(x^1,x^2)\in {\mathbb{R}}^2$. Then, the coordinate basis $(\partial /\partial x^i)$ induces the global coordinates $(y^1,y^2)\in T{\mathbb{R}}^2$. Let us put $a:=x^1$, $b:=x^2$, $p:=y^1$, $q:=y^2$, and $r:=p/q$. The Berwald–Rund metric $F(x,y):=q{(\epsilon +r)}^2$ is a solution of the PDE: $\epsilon \partial \epsilon /\partial a=\partial \epsilon /\partial b$, where $\epsilon =\epsilon (a,b)$ is a given function [9]. One can see that if $q\le 0$, then $F\le 0$. Also, for $q>0$, $F=0$ along the ray $r=-\epsilon$. Thus, F is a y-local Finsler metric. This shows that F is only positive and strongly convex on the upper half (and with a ray excluded) of each tangent plane $T_x{\mathbb{R}}^2$. A simple calculation shows that F is a non-Riemannian Berwald metric. This implies that $\mathbf{H}=0$ while $\mathcal{I}=3/\sqrt{2}$.

F has an isotropic mean Berwald curvature if its mean Berwald curvature satisfies

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

where $\kappa =\kappa \left(x\right)$ is a scalar function on M and $\mathbf{h}:=h_{ij}d{x}^{i}d{x}^j$ denotes the angular metric. For $y\in T_{x}M$, define the Landsberg curvature by ${\mathbf{L}}_y(u,v,w):=-1/2{\mathbf{g}}_y\left({\mathbf{B}}_y(u,v,w),y\right).$ The quantity $\mathbf{L}$ is called the Landsberg curvature. The Finsler metric F is called a relatively isotropic Landsberg metric if it satisfies

$$\mathbf{L}=\nu F\mathbf{C},$$

where $\nu =\nu \left(x\right)$ is a scalar function on M and $\mathbf{C}$ denotes the mean Cartan torsion of F. Also, the Finsler metric is called an isotropic Berwald metric if its Berwald curvature can be written as follows

$$\begin{array}{c}\hfill {\mathbf{B}}_y(u,v,w)=\frac{\upsilon }{F}\{\mathbf{h}(u,v)\left(w-{\mathbf{g}}_y(w,\ell )\ell \right)+\mathbf{h}(v,w)\left(u-{\mathbf{g}}_y(u,\ell )\ell \right)+\mathbf{h}(w,u)\left(v-{\mathbf{g}}_y(v,\ell )\ell \right)\\ \hfill +2F{\mathbf{C}}_y(u,v,w)\ell \},\end{array}$$

where $\upsilon =\upsilon \left(x\right)$ is a scalar function on M. There is a generalization of isotropic Berwald metrics, namely, Douglas metrics.

Theorem 3.

Let $(M,F)$ be Finsler surface with an isotropic main scalar. Then, the following holds:

(i) 

F has a relatively isotropic Landsberg curvature and an isotropic mean Berwald curvature;

(ii) 

F is a Douglas metric if and only if it is an isotropic Berwald metric. In this case, F has an isotropic S-curvature.

A Finsler manifold $(M,F)$ is called homogeneous if its group of isometries acts transitively on the manifold. For a homogeneous Douglas surface, we obtain the following.

Corollary 1.

Every homogeneous Douglas surface with an isotropic main scalar is Riemannian or locally Minkowskian.

Finally, we consider a regular Finsler surface with a constant main scalar and prove the following.

Corollary 2.

Every Finsler surface with a constant main scalar is Riemannian or locally Minkowskian.

2. Preliminaries

Let $(M,F)$ be a Finsler metric and the following be the fundamental tensor of F:

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

The Cartan torsion of F is defined by

$$C_{ijk}:=\frac{1}{2}\frac{\partial g_{ij}}{\partial y^k}.$$

The tensor $C_{ijk}$ is called the Cartan torsion.

For $y\in T_x{M}_0$, define ${\mathbf{I}}_y:T_{x}M\to \mathbb{R}$ by

$${\mathbf{I}}_y\left(u\right)=\sum _{i=1}^n{g}^{ij}\left(y\right){\mathbf{C}}_y(u,{\partial }_i,{\partial }_j),$$

where $\left\{{\partial }_i\right\}$ is a basis for $T_{x}M$ at $x\in M$. The family $\mathbf{I}:={\left\{{\mathbf{I}}_y\right\}}_{y\in T{M}_0}$ is called the mean Cartan torsion. By Diecke’s theorem, a positive-definite Finsler metric F is Riemannian if and only if ${\mathbf{I}}_y=0$.

For a Finsler manifold $(M,F)$, its induced spray on $TM$ is denoted by $\mathbf{G}=\mathbf{G}(x,y)$ which, in a standard coordinate $(x^i,y^i)$ for $T{M}_0$, is given by $\mathbf{G}=y^i\partial /\partial x^i-2G^i(x,y)\partial /\partial y^i$, where

$$G^i:=\frac{1}{4}{g}^{il}\left[\frac{{\partial }^2{F}^2}{\partial x^k\partial y^l}{y}^k-\frac{\partial F^2}{\partial x^l}\right],$$

where $G^i$ represents the geodesic coefficients of F in the same local coordinate system.

The Busemann–Hausdorff volume form $d{V}_f:={\sigma }_F\left(x\right)d{x}^1\cdots d{x}^n$ on an n-dimensional Finsler manifold $(M,F)$ can be written as follows:

$${\sigma }_F\left(x\right):=\frac{\mathrm{Vol}\left({\mathbb{B}}^n\left(1\right)\right)}{\mathrm{Vol}\left\{\left(y^i\right)\in {\mathbb{R}}^n|F\left(y^i\frac{\partial }{\partial x^i}{|}_x\right)<1\right\}}.$$

The S-curvature of F is given by

$$\mathbf{S}(x,y):=\frac{\partial G^i}{\partial y^i}-y^i\frac{\partial }{\partial x^i}\left[ln{\sigma }_F\right].$$

(7)

The distorsion of F is given by

$$\tau (x,y)=ln\frac{\sqrt{det\left(g_{ij}(x,y)\right)}}{\sigma \left(x\right)}.$$

By definition, the distortion $\tau$ is homogeneous of a degree of one with respect to y.

For a vector $y\in T_x{M}_0$, the Berwald curvature ${\mathbf{B}}_y:T_{x}M\times T_{x}M\times T_{x}M\to T_{x}M$ is defined by ${\mathbf{B}}_y(u,v,w):=B_{jkl}^i\left(y\right)u^j{v}^k{w}^l\partial /\partial x^i{|}_x$, where

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

F is called a Berwald metric if $\mathbf{B}=\mathbf{0}$. Every Berwald metric satisfies $\mathbf{S}=0$ (see [10]).

For $y\in T_{x}M$, define the Landsberg curvature ${\mathbf{L}}_y:T_{x}M\times T_{x}M\times T_{x}M\to \mathbb{R}$ by

$${\mathbf{L}}_y(u,v,w):=-\frac{1}{2}{\mathbf{g}}_y\left({\mathbf{B}}_y(u,v,w),y\right).$$

A Finsler metric F is called a Landsberg metric if $\mathbf{L}=0$.

The E-curvature of F is defined by

$${\mathbf{E}}_y(u,v):=\frac{1}{2}\sum _{i=1}^n{g}^{ij}\left(y\right){\mathbf{g}}_y\left({\mathbf{B}}_y(u,v,{\partial }_i),{\partial }_j\right),$$

(8)

where $\left\{{\partial }_i\right\}$ is a basis for $T_{x}M$ at $x\in M$. In the local coordinates, ${\mathbf{E}}_y(u,v):=E_{ij}\left(y\right)u^i{v}^j$, where

$$E_{ij}:=\frac{1}{2}{B}_{mij}^m.$$

Using a horizontal derivation of the mean of the Berwald curvature $\mathbf{E}$ along Finslerian geodesics give us the H-curvature $\mathbf{H}=\mathbf{H}(x,y)$, which is defined by ${\mathbf{H}}_y=H_{ij}d{x}^i\otimes d{x}^j$, where

$$H_{ij}:=E_{ij|m}{y}^m.$$

The notation “|” denotes the horizontal derivation with respect to the Berwald connection.

The Douglas curvature is defined by

$$D_{jkl}^i:=\frac{{\partial }^3{G}^i}{\partial y^j\partial y^k\partial y^l}-\frac{2}{n+1}\frac{{\partial }^3}{\partial y^j\partial y^k\partial y^l}\left[\frac{\partial G^m}{\partial y^m}{y}^i\right].$$

(9)

If $\mathbf{D}=0$, then F is a Douglas metric.

For a non-zero vector $y\in T_x{M}_0$, the Riemann curvature is a family of linear transformations ${\mathbf{R}}_y:T_{x}M\to T_{x}M$, which is defined by ${\mathbf{R}}_y\left(u\right):=R_k^i\left(y\right)u^k\partial /\partial x^i$, where

$$R_k^i\left(y\right)=2\frac{\partial G^i}{\partial x^k}-\frac{{\partial }^2{G}^i}{\partial x^j\partial y^k}{y}^j+2G^j\frac{{\partial }^2{G}^i}{\partial y^j\partial y^k}-\frac{\partial G^i}{\partial y^j}\frac{\partial G^j}{\partial y^k}.$$

(10)

The family $\mathbf{R}:={\left\{{\mathbf{R}}_y\right\}}_{y\in T{M}_0}$ is called the Riemann curvature.

The flag curvature is denoted by $\mathbf{K}=\mathbf{K}(y,P)$ with the flag $P:=span\{y,u\}$ and given by

$$\mathbf{K}(x,y,P):=\frac{{\mathbf{g}}_y\left(u,{\mathbf{R}}_y\left(u\right)\right)}{{\mathbf{g}}_y(y,y){\mathbf{g}}_y(u,u)-{\mathbf{g}}_y{(y,u)}^2}.$$

(11)

F is of the scalar flag curvature $\mathbf{K}=\mathbf{K}(x,y)$ if its Riemannian curvature satisfies the following

$$R_j^i=\mathbf{K}{F}^2{h}_j^i.$$

Also, F is of isotropic and constant flag curvature if $\mathbf{K}=\mathbf{K}\left(x\right)$ and $\mathbf{K}=constant$, respectively.

Theorem 4

([11] Akbar-Zadeh Theorem). Let F be a Finsler metric of the scalar flag curvature $\mathbf{K}=\mathbf{K}(x,y)$ on a manifold M. Then, F has the isotropic flag curvature $\mathbf{K}=\mathbf{K}\left(x\right)$ if and only if $\mathbf{H}=0$.

In [12], Najafi, Shen, and Tayebi generalized the Akbar-Zadeh Theorem and proved the following.

Theorem 5

([12] Najafi–Shen–Tayebi Theorem). Let F be a Finsler metric of a scalar flag curvature on an n-dimensional manifold M. Let θ be an arbitrary one-form on M. Then

$$\mathbf{H}=\frac{(n^2-1)\theta }{2F}$$

(12)

if and only if F is of a weakly isotropic flag curvature

$$\mathbf{K}=\frac{3\theta }{F}+\sigma ,$$

(13)

where $\sigma =\sigma \left(x\right)$ is a scalar function on M.

Throughout this paper, we use the Berwald connection on Finsler manifolds. Let $\left\{e_j\right\}$ denote the local frame for ${\pi }^{*}TM$. Suppose that $\{{\omega }^i,{\omega }^{n+i}\}$ denotes the corresponding local co-frame for $T\ast \left(T{M}_0\right)$. Let the set $\left\{{\omega }_j^i\right\}$ denote the local Berwald connection forms with respect to $\left\{e_j\right\}$. In this case, the connection forms of the Berwald connection satisfy the following structure equations:

• Torsion freeness

  $$\begin{array}{c}\hfill d{\omega }^i={\omega }^j\wedge {\omega }_j^i;\end{array}$$

  (14)
• Almost metric compatibility
  $$\begin{array}{c}\hfill d{g}_{ij}-g_{kj}{\omega }_i^k-g_{ik}{\omega }_j^k=-2L_{ijk}{\omega }^k+2C_{ijk}{\omega }^{n+k},\end{array}$$

  (15)

  where

  $${\omega }^i:=d{x}^i,{\omega }^{n+k}:=d{y}^k+y^j{\omega }_j^k.$$

Then, the following holds

$$g_{ij|k}=-2L_{ijk},g_{ij,k}=2C_{ijk}.$$

3. Proof of Theorem 2

For any Minkowskian plane $(V,\mathcal{F})$ and any vector $v\in V$ with $\mathcal{F}\left(v\right)\ne 0$, there is a non-zero vector $w\in V$ such that it is orthogonal to v with respect to the fundamental tensor raised by the Minkowski functional $\mathcal{F}$. The special and useful Berwald frame was founded and developed by L. Berwald to study Finsler surfaces [6]. It works under the assumption that the fundamental tensor is positive definite. Let $(M,F)$ be a two-dimensional Finsler manifold. It is easy to see that for every $\mathbf{y}\in T_{x}M$, $x\in M$, there is a vector ${\mathbf{y}}^{\perp }\in T_x{M}_0$ such that

$${\mathbf{g}}_{\mathbf{y}}(\mathbf{y},{\mathbf{y}}^{\perp })=0,{\mathbf{g}}_{\mathbf{y}}({\mathbf{y}}^{\perp },{\mathbf{y}}^{\perp })=F\left(\mathbf{y}\right).$$

The pair $\{\mathbf{y},{\mathbf{y}}^{\perp }\}$ is called the Berwald frame at $\mathbf{y}$.

Based on the Berwald frame, the Cartan torsion can be determined by a scalar function on the slit tangent bundle. Let us define

$$\mathcal{I}\left(\mathbf{y}\right):=\frac{{\mathbf{C}}_{\mathbf{y}}({\mathbf{y}}^{\perp },{\mathbf{y}}^{\perp },{\mathbf{y}}^{\perp })}{F\left(\mathbf{y}\right)}.$$

One can see that $\mathcal{I}\left(\lambda \mathbf{y}\right)=\mathcal{I}\left(\mathbf{y}\right)$ holds for $\forall \lambda >0$ and $\forall \mathbf{y}\in T_x{M}_0$. We call $\mathcal{I}$ the main scalar of the Finsler metric F.

In most of the literature regarding Finsler geometry, the special notion $(\ell ,m)$ was used instead of $\{\mathbf{y},{\mathbf{y}}^{\perp }\}$. By considering this notion, for a scalar $T=T(x,y)$, we define the horizontal scalar derivatives $(T_{|1},T_{|2})$ and vertical scalar derivatives $(T_{,1},T_{,2})$ as follows

$$T_{|i}:=T_{|1}{\ell }_i+T_{|2}{m}_i,F{T}_{,i}:=T_{,1}{\ell }_i+T_{,2}{m}_i,$$

(16)

where

$$T_{|i}:=\frac{\partial T}{\partial x^i}-G_i^j\frac{\partial T}{\partial y^j},F{T}_{,i}:=F\frac{\partial T}{\partial y^i}$$

denote the horizontal and vertical derivations with respect to the Berwald connection of F.

To prove Theorem 2, we need to the explicit construction of the flag curvature of a two-dimensional Finsler manifold. Then, we prove the following.

Lemma 1.

The flag curvature of every two-dimensional Finsler manifold $(M,F)$ satisfies the following

$${\mathbf{K}}_{y^i}+T{\tau }_{y^i}=0,$$

(17)

where

$$\mu :=-\frac{2}{\mathcal{I}}{\mathcal{I}}_{|1},T(x,y):=\frac{1}{4}\left(4\mathbf{K}+{\mu }^2-\frac{2}{F}{\mu }_{|k}{y}^k\right).$$

Proof. 

The curvature form of the Berwald connection is given by

$${\Omega }_j^i=d{\omega }_j^i-{\omega }_j^k\wedge {\omega }_k^i=\frac{1}{2}{R}_{jkl}^i{\omega }^k\wedge {\omega }^l-B_{jkl}^i{\omega }^k\wedge {\omega }^{n+l}.$$

(18)

Differentiating (15) implies that

$$\begin{array}{ccc}\hfill g_{pj}{\Omega }_i^p+g_{ip}{\Omega }_j^p=& -& 2L_{ijk|s}{\omega }^k\wedge {\omega }^s-2L_{ijk,s}{\omega }^k\wedge {\omega }^{n+s}-2C_{ijs|k}{\omega }^k\wedge {\omega }^{n+s}-2C_{ijs}{\Omega }^s\hfill \\ \hfill & & -2C_{ijs,k}{\omega }^{n+k}\wedge {\omega }^{n+s}.\hfill \end{array}$$

(19)

Putting (18) into (19) gives us the following

$$\begin{array}{c}\hfill g_{pj}{B}_{ikl}^p+g_{ip}{B}_{jkl}^p=2C_{ijl|k}+2L_{ijk,l}.\end{array}$$

(20)

By (20), we have

$$\begin{array}{ccc}& & g_{pk}{B}_{ijl}^p+g_{jp}{B}_{ikl}^p=2C_{jkl|i}+2L_{ijk,l},\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& & g_{pi}{B}_{jkl}^p+g_{kp}{B}_{ijl}^p=2C_{ikl|j}+2L_{ijk,l}.\hfill \end{array}$$

(22)

From (20)+(21)–(22), we obtain

$$B_{jkl}^p=g^{ip}\left\{C_{ijl|k}+C_{ikl|j}-C_{jkl|i}+L_{ijk,l}\right\}.$$

(23)

On the other hand, the Cartan torsion of a Finsler manifold $(M,F)$ has no components in the direction ${\ell }^i$, i.e., $C_{ijk}{y}^i=0$. Then, it can be written in the Berwald frame $(\ell ,m)$ as follows

$$F{C}_{ijk}=\mathcal{I}{m}_i{m}_j{m}_k.$$

(24)

Using a horizontal derivation of (24) implies that

$$F{C}_{ijk|s}=\left({\mathcal{I}}_{|1}{\ell }_s+{\mathcal{I}}_{|2}{m}_s\right)m_i{m}_j{m}_k.$$

(25)

Contracting (25) with $y^s$ yields

$$F{L}_{ijk}={\mathcal{I}}_{|1}{m}_i{m}_j{m}_k.$$

(26)

By putting (25) and (26) in (23), we obtain

$$F{B}_{jkl}^i=\left\{-2{\mathcal{I}}_{|1}{\ell }^i+\left({\mathcal{I}}_{|1,2}+{\mathcal{I}}_{|2}\right)m^i\right\}{m}_j{m}_k{m}_l.$$

(27)

Let us put

$${\mathcal{I}}_2:={\mathcal{I}}_{|1,2}+{\mathcal{I}}_{|2}.$$

(28)

Thus, the Berwald curvature of Finsler surfaces is given by

$$B_{jkl}^i=\frac{1}{F}\left({\mathcal{I}}_2{m}^i-2{\mathcal{I}}_{|1}{\ell }^i\right)m_j{m}_k{m}_l.$$

(29)

By (24) and (29), we have

$$\begin{array}{c}\hfill B_{jkl}^i=-\frac{2{\mathcal{I}}_{|1}}{F\mathcal{I}}{C}_{jkl}{y}^i+\frac{{\mathcal{I}}_2}{3F}\left\{h_{jk}{h}_l^i+h_{kl}{h}_j^i+h_{lj}{h}_k^i\right\},\end{array}$$

(30)

where $\mathbf{h}=h_{ij}d{x}^{i}d{x}^j$ denotes the angular metric and is defined by the following

$$h_{ij}=g_{ij}-F^{-2}{y}_i{y}_j.$$

Then, for a Finsler surface, the Berwald curvature can be written as follows

$$\begin{array}{c}\hfill B_{jkl}^i=-\frac{1}{F}\mu C_{jkl}{y}^i+\lambda \left\{h_{jk}{h}_l^i+h_{kl}{h}_j^i+h_{lj}{h}_k^i\right\},\end{array}$$

(31)

where

$$\mu :=-\frac{2}{\mathcal{I}}{\mathcal{I}}_{|1},\lambda :=\frac{1}{3}{\mathcal{I}}_2.$$

(32)

Taking a trace of (31) yields

$$E_{ij}=\frac{3}{2}\lambda F^{-1}{h}_{ij}.$$

(33)

Contracting (31) with $y_i$ implies that

$$L_{ijk}+\frac{1}{2}\mu F{C}_{ijk}=0.$$

(34)

For the Finsler metrics of the scalar flag curvature, the following holds

$$L_{ijk|m}{y}^m=-\frac{1}{3}{F}^2\left\{{\mathbf{K}}_{·i}{h}_{jk}+{\mathbf{K}}_{·j}{h}_{ik}+{\mathbf{K}}_{·k}{h}_{ij}+3\mathbf{K}{C}_{ijk}\right\}$$

(35)

and

$$J_{k|m}{y}^m=-\frac{1}{3}{F}^2\left\{(n+1){\mathbf{K}}_{·k}+3\mathbf{K}{I}_k\right\}.$$

(36)

Taking a trace of (34) implies that

$$J_k=-\frac{1}{2}\mu F{I}_k.$$

(37)

Taking a horizontal derivation of (37) along Finslerian geodesics yields

$$J_{k|m}{y}^m=-\frac{1}{2}F\left({\mu }_0{I}_k+\mu J_k\right)=-\frac{1}{4}F\left(2{\mu }_0-{\mu }^{2}F\right)I_k.$$

(38)

By (36), (38), and $I_k={\tau }_{·k}$, we obtain (17). □

Now, we are ready to present the key lemma that enables us to prove Theorem 2. More precisely, we prove the following.

Lemma 2.

Let $(M,F)$ be a positively curved Finsler surface. Suppose that F has an isotropic main scalar. Then F has an isotropic Gaussian curvature if and only if it is Riemannian.

Proof. 

Let $\mathcal{I}=\mathcal{I}\left(x\right)$. Then, by (28) and (32), we obtain

$$\lambda =\frac{1}{3}{\mathcal{I}}_2=\frac{1}{3}{\mathcal{I}}_{|2}.$$

(39)

Equation (39) shows that if $\mathcal{I}=\mathcal{I}\left(x\right)$, then $\lambda =\lambda \left(x\right)$. Then, (33) and (39) imply that F has an isotropic mean Berwald curvature

$$E_{ij}=\frac{3}{2}\rho F^{-1}{h}_{ij}.$$

(40)

where $\rho :=1/3{\mathcal{I}}_{|2}$. Taking a horizontal derivation of (40) along Finslerian geodesics gives us

$$H_{ij}=\frac{3}{2F}{\rho }_{x^m}\left(x\right)y^m{h}_{ij}.$$

(41)

Every Finsler surface has a scalar flag curvature $\mathbf{K}=\mathbf{K}(x,y)$. According to Theorem 5, we find that F is of a weakly isotropic flag curvature

$$\mathbf{K}=\frac{3{\rho }_{x^m}\left(x\right)y^m}{F}+\sigma ,$$

(42)

where $\sigma =\sigma \left(x\right)$ is a scalar function on M. On the other hand, by (17), we have

$${\mathbf{K}}_{y^i}=\left(\frac{1}{2F}{\mu }_{x^k}\left(x\right)y^k-\mathbf{K}-\frac{1}{4}{\mu }^2\left(x\right)\right){\tau }_{y^i}.$$

(43)

Putting (42) in (43) gives us

$${\mathbf{K}}_{y^i}=\left(\frac{1}{2F}{\mu }_{x^k}\left(x\right)y^k-\frac{3}{F}{\rho }_{x^k}\left(x\right)y^k-\sigma -\frac{1}{4}{\mu }^2\left(x\right)\right){\tau }_{y^i}.$$

(44)

If F is Riemannian, then ${\tau }_{·i}=I_i=0$, and (44) implies that ${\mathbf{K}}_{y^i}=0$. It follows that $\mathbf{K}=\mathbf{K}\left(x\right)$.

Conversely, suppose that $\mathbf{K}=\mathbf{K}\left(x\right)$. We prove that F is Riemannian. First, we show that $\mathbf{K}=\sigma$. On the contrary, suppose that $\mathbf{K}\ne \sigma$. Then, by (42), we obtain

$$F=\frac{3}{\mathbf{K}-\sigma }{\rho }_{x^m}\left(x\right)y^m$$

(45)

which contradicts the positive-definiteness of the Finsler metric F. Then, ${\rho }_{x^m}\left(x\right)y^m=0$, and we obtain

$$\mathbf{K}=\sigma .$$

(46)

Therefore, (44) reduces to the following

$$\left(\frac{1}{2F}{\mu }_{x^k}\left(x\right)y^k-\sigma -\frac{1}{4}{\mu }^2\left(x\right)\right){\tau }_{y^i}=0.$$

(47)

We claim that F is a Riemannian metric. On the contrary, suppose that F is not Riemannian. According to (47), we have two main cases as follows:

• Case (i): ${\mu }^2\left(x\right)\ne -4\sigma$. Since F is not Riemannian, there is an open subset $\mathcal{U}$ such that ${\tau }_{y^i}(x,y)\ne 0$ for any $x\in \mathcal{U}$. Then, (47) implies that

$${\mu }_{x^k}\left(x\right)y^k-2\left(\sigma \left(x\right)+\frac{1}{4}{\mu }^2\left(x\right)\right)F=0,$$

(48)

which is equal to

$$F=-\frac{2}{4\sigma \left(x\right)+{\mu }^2\left(x\right)}{\mu }_{x^m}\left(x\right)y^m.$$

(49)

Equation (49) contradicts the positive-definiteness of F. Then, this case does not hold.

• Case (ii): Suppose that ${\mu }^2\left(x\right)=-4\sigma$. By assuming that $\mathbf{K}>0$ and by (46), we have $\sigma >0$. This is a contradiction. Thus, this case does not hold either. □

Proof of Theorem 2.

By Theorem 4, a Finsler metric of the scalar flag curvature $\mathbf{K}=\mathbf{K}(x,y)$ has an isotropic flag curvature $\mathbf{K}=\mathbf{K}\left(x\right)$ if and only if $\mathbf{H}=\mathbf{0}$. By assumption, we obtain $\mathbf{K}=\mathbf{K}\left(x\right)$. By Lemma 2, F reduces to a Riemannian metric. If F is Riemannian, then by (24), we obtain $\mathcal{I}=0$. This completes the proof. □

Proof of Theorem 3.

Let F be a two-dimensional Finsler metric with an isotropic main scalar $\mathcal{I}=\mathcal{I}\left(x\right)$. We prove Theorem 3, part by part, as follows.

• Proof of Part (i): By (32), we have

$$\mu :=-\frac{2}{\mathcal{I}\left(x\right)}{\mathcal{I}}_{|1}\left(x\right),\lambda :=\frac{1}{3}{\mathcal{I}}_2=\frac{1}{3}{\mathcal{I}}_{|2}\left(x\right).$$

(50)

In this case, (33) and (34) imply that F has an isotropic mean Berwald curvature and a relatively isotropic Landsberg curvature.

• Proof of Part (ii): By part (i), F has an isotropic mean Berwald curvature. In [13], it is proved that every Douglas metric with an isotropic mean Berwald curvature is an isotropic Berwald metric. On the other hand, every isotropic Berwald metric is a Douglas metric. In [10], it is shown that every isotropic Berwald metric has an isotropic S-curvature. This completes the proof. □

Proof of Corollary 1.

By part (ii) of Theorem 3, F has an isotropic S-curvature, $\mathbf{S}=3cF$, where $c=c\left(x\right)$ is a scalar function on M. In [14], it is proved that every homogeneous Finsler metric with an isotropic S-curvature satisfies $\mathbf{S}=0$, i.e., $c=0$. In this case, F reduces to a Berwald metric. By Szabó’s Theorem in [15], F is Riemannian or locally Minkowskian. □

Proof of Corollary 2.

Let F be a two-dimensional Finsler metric with a constant main scalar $\mathcal{I}=constant$. In this case, (50) implies that $\lambda =\mu =0$, and then, F satisfies $\mathbf{E}=0$ and $\mathbf{L}=0$. Then, F reduces to a Berwald metric and Szabó’s Theorem completes the proof. □