Multiple closed geodesics on bumpy Finsler $n$-spheres

TL;DR

This paper proves that on any bumpy Finsler metric on a rational homological sphere, there are always at least two distinct prime closed geodesics, extending understanding of geodesic multiplicity in Finsler geometry.

Contribution

The paper establishes the existence of at least two prime closed geodesics on bumpy Finsler spheres, a new result in the study of geodesic multiplicity.

Findings

At least two distinct prime closed geodesics exist on bumpy Finsler spheres.

The result applies to all rational homological spheres of dimension n ≥ 2.

The proof extends previous results in Finsler geometry and geodesic theory.

Abstract

In this paper we prove that for every bumpy Finsler metric $F$ on every rationally homological $n$-dimensional sphere $S^n$ with $n\ge 2$, there exist always at least two distinct prime closed geodesics.