Finsler metrics of weakly isotropic flag curvature

Finsler metrics of scalar flag curvature play an important role to show the complexity and richness of Finsler geometry. In this paper, on an $n$-dimensional manifold $M$ we study the Finsler metric $F = F(x, y)$ of scalar flag curvature $\mathbf{K} = \mathbf{K} (x, y)$ and discover some equations $\mathbf{K}$ should be satisfied. As an application, we mainly study the metric $F$ of weakly isotropic flag curvature. We prove that in this case, $F$ must be a Randers metric when $\operatorname{dim} (M) \geq 3$, or be of constant flag curvature. Without the restriction on the dimension, the same result is obtained for projectively flat Finsler metrics of weakly isotropic flag curvature.

Full Text (PDF format)

Research supported by the National Science Foundation of China (11371209), National Science Foundation of Zhejiang Province (R18A010002, LY13A 010013), and the K.C. Wong Magna Fund in Ningbo University.

Received 17 May 2016

Accepted 10 October 2017

Published 12 March 2020