Generic Existence of Infinitely Many Non-contractible Closed Geodesics on Compact Space Forms

Let M = Sⁿ /Γ and h be a nontrivial element of finite order p in π₁(Μ), where the integers n, p ≥ 2, Γ 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 there are infinitely many non-contractible closed geodesics of class [h] on the compact space form with C^r-generic Finsler metrics, where 4 ≤ r ≤ ∞. The conclusion also holds for C^r-generic Riemannian metrics for 2 ≤ r ≤ ∞. The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.