On the multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

TL;DR

This paper investigates the number of closed geodesics on Finsler spheres with irrationally elliptic geodesics, establishing conditions for either a finite set or infinitely many such geodesics, and applies these results to specific cases.

Contribution

It provides new results on the multiplicity of closed geodesics on Finsler spheres with irrationally elliptic geodesics, including exact counts and existence conditions.

Findings

Exactly 2[ (n+1)/2 ] or infinitely many closed geodesics exist under certain conditions.

Existence of three distinct closed geodesics on bumpy Finsler S^3 with non-zero Morse index.

Abstract

If all prime closed geodesics on $(S^n,F)$ with an irreversible Finsler metric $F$ are irrationally elliptic, there exist either exactly $2\left[\frac{n+1}{2}\right]$ or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler $(S^3,F)$ if any prime closed geodesic has non-zero Morse index.