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

Abstract

Let M = S n /Γ and h be a nontrivial element of finite order p in π 1 (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 [39] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on RP n in [25] to general compact space forms.

Multiplicity of non-contractible closed geodesics on Finsler compact space forms

“…In [26], Liu and Xiao established the resonance identity for the non-contractible closed geodesics on RP n , and together with [12] and [41] proved the existence of at least two distinct non-contractible closed geodesics on every bumpy RP n with n ≥ 2. Furthermore, Liu, Long and Xiao [25] established the resonance identity for noncontractible closed geodesics of class [h] on compact space form M = S n /Γ and obtained at least two non-contractible closed geodesics of class [h] provided Γ is abelian and h is nontrivial in π 1 (M ). Recently, Liu [27] obtained an optimal lower bound estimation of the number of contractible closed geodesics on bumpy Finsler compact space form S 2n+1 /Γ with reversibility λ and flag curvature K satisfying…”

The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form $S^{2n+1}/ Γ$

“…Let M = S 2n+1 /Γ, Γ is a finite group which acts freely and isometrically on the (2n + 1)-sphere and therefore M is diffeomorphic to a compact space form which is typically a non-simply connected manifold. In papers [44] and [27], the existence of at least two distinct non-contractible closed geodesics on every bumpy S n /Γ with n ≥ 2 was proved, however, this paper is concerned with the total number of closed geodesics on Finsler S 2n+1 /Γ, the main ingredients are the investigations of the topological structure of the contractible component of the free loop space on S 2n+1 /Γ which yields a new resonance identity for homologically visible contractible minimal closed geodesics on Finsler S 2n+1 /Γ , and the enhanced common index jump theorem for symplectic paths discovered by Duan, Long and Wang in [13].…”

“…In [28], Liu and Xiao established the resonance identity for the non-contractible closed geodesics on RP n , and together with [14] and [44] proved the existence of at least two distinct non-contractible closed geodesics on every bumpy RP n with n ≥ 2. Furthermore, Liu, Long and Xiao [27] proved that every bumpy Finsler compact space form S n /Γ possesses two distinct closed geodesics in each of its nontrivial classes.…”

“…Based on the works of [13] and [27], it is natural to ask whether we can improve the lower bound of the total number of closed geodesics on bumpy Finsler compact space forms under some natural conditions. Note that the only non-trivial group which acts freely on S 2n is Z 2 and S 2n /Z 2 = RP 2n (cf.…”