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

Abstract

Let $M=S^n/\hat{I}“$ and $h$ be a nontrivial element of finite order $p$ in $\ddot{I}{€}_1(M)$, where the integer $n\ge 2$, $\hat{I}“$ 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.

33 pages, All comments are welcome. arXiv admin note: substantial text overlap with arXiv:1607.02746