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

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Let M=Sⁿ/ Γ and h be a nontrivial element of finite order p in π₁(M), where the integer n≥2, Γ 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 RPⁿ in [arXiv:1607.02746] to general compact space forms.