Existence of closed geodesics on Finsler $n$-spheres

Abstract

In this paper, we prove that on every Finsler $n$-sphere $(S^n,F)$ with reversibility $\lambda$ satisfying $F^2<(\frac{\lambda +1}{\lambda }{)}^2{g}_0$ 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.

12 pages