On the average indices of closed geodesics on positively curved Finsler spheres

TL;DR

This paper investigates the existence and properties of closed geodesics on positively curved Finsler spheres, establishing conditions under which infinitely many or specific irrational average index geodesics exist.

Contribution

It proves new existence results for closed geodesics on Finsler spheres with curvature constraints, including conditions for infinitely many or irrational index geodesics.

Findings

Either infinitely many prime closed geodesics exist or a certain number have irrational average indices.

For bumpy metrics with finite geodesics, at least n-3 geodesics have irrational average indices.

Results apply to Finsler spheres with curvature bounds and reversibility conditions.

Abstract

In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\ge 6$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many prime closed geodesics or there exist $[\frac{n}{2}]-2$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist $n-3$ closed geodesics possessing irrational average indices provided the number of closed geodesics is finite.