Finsler metrics of weakly isotropic flag curvature

TL;DR

This paper investigates Finsler metrics with scalar flag curvature, establishing that under weakly isotropic conditions, such metrics are necessarily Randers metrics in dimensions three or higher, and also in the projectively flat case.

Contribution

It proves that Finsler metrics with weakly isotropic flag curvature are Randers metrics in dimensions ≥3 and when projectively flat, expanding understanding of their geometric structure.

Findings

Finsler metrics with weakly isotropic flag curvature are Randers metrics in dimension ≥3.

Projectively flat Finsler metrics with such curvature are also Randers metrics.

Derived equations that scalar flag curvature must satisfy in these contexts.

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.