Non-hyperbolic closed geodesics on positively curved Finsler spheres

TL;DR

This paper proves the existence of multiple non-hyperbolic closed geodesics on positively curved Finsler spheres under certain curvature conditions, extending understanding of geodesic multiplicity and stability.

Contribution

It establishes the existence of at least three distinct closed geodesics on Finsler spheres with specific curvature bounds, with at least two being elliptic, and all three non-hyperbolic when dimension is at least six.

Findings

At least three distinct closed geodesics exist under given curvature conditions.

At least two of these geodesics are elliptic.

All three are non-hyperbolic when the dimension is at least six.

Abstract

In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^n,F), n\ge 3$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$, there exist at least three distinct closed geodesics and at least two of them are elliptic if the number of prime closed geodesics is finite. When $n\ge 6$, these three distinct closed geodesics are non-hyperbolic.