Existence of closed geodesics on Finsler $n$-spheres

TL;DR

This paper proves the existence of at least n prime closed geodesics on certain Finsler n-spheres under specific curvature and length conditions, and explores their stability.

Contribution

It establishes new existence results for closed geodesics on Finsler spheres with explicit geometric bounds and analyzes their stability properties.

Findings

At least n prime closed geodesics exist under given conditions.

Conditions relate reversibility, curvature, and shortest geodesic length.

Stability properties of these geodesics are analyzed.

Abstract

In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ with reversibility $\lambda$ satisfying $F^2<(\frac{\lambda+1}{\lambda})^2g_0$ and $l(S^n, F)\ge \pi(1+\frac{1}{\lambda})$, there always exist at least $n$ prime closed geodesics without self-intersections, where $g_0$ is the standard Riemannian metric on $S^n$ with constant curvature 1 and $l(S^n, F)$ is the length of a shortest geodesic loop on $(S^n, F)$. We also study the stability of these closed geodesics.