THE EXISTENCE OF TWO NON-CONTRACTIBLE CLOSED GEODESICS ON EVERY BUMPY FINSLER COMPACT SPACE FORM

摘要

Let M = S-n/Gamma and h be a nontrivial element of finite order p in pi(1)(M), where the integer n >= 2, 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 [39] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on RPn in [25] to general compact space forms.