Finsler metrics of scalar flag curvature play an important role to show the complexity and richness of general Finsler metrics. In this paper, on an $n$-dimensional manifold $M$ we study the Finsler metric $F=F(x,y)$ of scalar flag curvature ${\bf K} = {\bf K}(x,y)$ and discover some equations ${\bf K}$ should be satisfied. As an application, we mainly study the metric $F$ of weakly isotropic flag curvature ${\bf K} = \frac{3 \theta}{F} + \sigma$, where $\theta=\theta_i(x) y^i \neq 0$ is a $1$-form and $\sigma =\sigma(x)$ is a scalar function. We prove that in this case, $F$ must be a Randers metric when $dim(M) \geq 3$. Further, without the restriction on the dimension we prove that projectively flat Finsler metrics of such weakly isotropic flag curvature are Randers metrics too.

中文翻译：

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弱各向同性旗曲率的 Finsler 度量

标量旗曲率的 Finsler 度量在展示一般 Finsler 度量的复杂性和丰富性方面发挥着重要作用。在本文中，在 $n$ 维流形 $M$ 上，我们研究了标量旗曲率 ${\bf K} = {\bf K}(x,y)$ 的 Finsler 度量 $F=F(x,y)$ )$ 并发现一些方程 ${\bf K}$ 应该是满足的。作为应用，我们主要研究弱各向同性旗曲率的度量$F$${\bf K} = \frac{3 \theta}{F} + \sigma$，其中$\theta=\theta_i(x) y ^i \neq 0$ 是 $1$ 形式，$\sigma =\sigma(x)$ 是标量函数。我们证明，在这种情况下，当 $dim(M) \geq 3$ 时，$F$ 必须是 Randers 度量。此外，在没有维度限制的情况下，我们证明这种弱各向同性标志曲率的投影平坦 Finsler 度量也是 Randers 度量。