Multiplicity of non-contractible closed geodesics on Finsler compact space forms

TL;DR

This paper proves the existence of multiple non-contractible closed geodesics on Finsler compact space forms under certain curvature and metric conditions, extending known results and providing optimal bounds.

Contribution

It establishes new lower bounds for the number of non-contractible closed geodesics on Finsler space forms, including generic cases with infinitely many such geodesics.

Findings

At least n-1 non-contractible closed geodesics under specific curvature conditions.

At least 2[ (n+1)/2 ] non-contractible closed geodesics for bumpy metrics, matching optimal bounds.

Infinitely many non-contractible closed geodesics for generic metrics under certain conditions.

Abstract

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n, p\geq2$, $\Gamma$ is a finite abelian group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we prove that for every irreversible Finsler compact space form $(M,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying \[ \frac{4p^2}{(p+1)^2} \big(\frac{\lambda}{\lambda+1} \big)^2 < K \leq 1,\;\;\lambda< \frac{p+1}{p-1}, \] there exist at least $n-1$ non-contractible closed geodesics of class $[h]$. In addition, if the metric $F$ is bumpy and \[ (\frac{4p}{2p+1})^2 (\frac{\lambda}{\lambda+1})^2 < K \leq 1,\;\;\lambda<\frac{2p+1}{2p-1}, \] then there exist at least $2[\frac{n+1}{2}]$ non-contractible closed geodesics of class $[h]$, which is the optimal lower bound due to Katok's example. For $C^4$-generic Finsler metrics, there are infinitely many non-contractible closed geodesics of class $[h]$ on $(M, F)$ if $\frac{\lambda^2}{(\lambda+1)^2} < K \leq 1$ with $n$ being odd, or $\frac{\lambda^2}{(\lambda+1)^2}\frac{4}{(n-1)^2} < K \leq 1$ with $n$ being even.