Multiple closed geodesics on positively curved Finsler manifolds

TL;DR

This paper proves the existence of multiple closed geodesics on certain positively curved Finsler manifolds, establishing lower bounds on their number and properties under specific curvature conditions.

Contribution

It provides new existence results for multiple closed geodesics on Finsler manifolds with curvature bounds, including the case of exactly three geodesics in three dimensions.

Findings

At least [ (dim M + 1)/2 ] closed geodesics exist under given curvature conditions.

If finite, the number of non-hyperbolic closed geodesics is at least [ dim M / 2 ].

Exactly three closed geodesics exist on 3-dimensional manifolds satisfying the curvature pinching condition.

Abstract

In this paper, we prove that on every Finsler manifold $(M,\,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\left(\frac{\lambda}{\lambda+1}\right)^2<K\le 1$, there exist $[\frac{\dim M+1}{2}]$ closed geodesics. If the number of closed geodesics is finite, then there exist $[\frac{\dim M}{2}]$ non-hyperbolic closed geodesics. Moreover, there are 3 closed geodesics on $(M,\,F)$ satisfying the above pinching condition when $\dim M=3$.