The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form $S^{2n+1}/ \Gamma$

TL;DR

This paper establishes the optimal lower bound for the number of closed geodesics on Finsler compact space forms, extending Katok's example and applying resonance identities and index jump theorems.

Contribution

It provides the first proof of the optimal lower bound for closed geodesics on Finsler space forms, combining topological, variational, and symplectic methods.

Findings

At least 2n+2 distinct closed geodesics exist on certain Finsler space forms.

Resonance identity links closed geodesics to topological invariants.

The bound matches the example by Katok, confirming optimality.

Abstract

Let $M=S^{2n+1}/ \Gamma$, $\Gamma$ is a finite group which acts freely and isometrically on the $(2n+1)$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we first investigate Katok's famous example about irreversible Finsler metrics on the spheres to study the topological structure of the contractible component of the free loop space on the compact space form $M$, then we apply the result to establish the resonance identity for homologically visible contractible minimal closed geodesics on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct contractible minimal closed geodesics on $(M, F)$. As its applications, using this identity and the enhanced common index jump theorem for symplectic paths proved by Duan, Long and Wang in \cite{DLW2}, we show that there exist at least $2n+2$ distinct closed geodesics on every compact space form $S^{2n+1}/ \Gamma$ with a bumpy irreversible Finsler metric $F$ under some natural curvature condition, which is the optimal lower bound due to Katok's example.