The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

TL;DR

This paper proves the existence of at least two non-contractible closed geodesics on every bumpy Finsler compact space form, using resonance identities and extending previous results to more general settings.

Contribution

It establishes a resonance identity for non-contractible geodesics and proves the existence of multiple such geodesics on bumpy Finsler space forms, improving prior results.

Findings

At least two non-contractible closed geodesics exist on bumpy Finsler compact space forms.

Resonance identity for non-contractible geodesics is established.

Results extend previous work on real projective spaces to general space forms.

Abstract

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n\geq2$, $\Gamma$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [Taimanov 2016] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathcal{R}P^n$ in [arXiv:1607.02746] to general compact space forms.