Finsler metrics of weakly isotropic flag curvature

Abstract: 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.