The spaces of non-contractible closed curves in compact space forms

Abstract

In this article we demonstrate how to use a result from [3,14] for calculating the rational equivariant cohomology of non-contractible loop spaces for the compact space forms. We also show how to use these calculations for establishing the existence of closed geodesics. The path spacesLet M n be a closed Riemannian manifold. Let us denote by Λ(M n ) = H 1 (S 1 , M ) the space of H 1 -maps γ : [0, 1] → M n , f (0) = f (1), of a circle S 1 = R/Z into M n , by Ω x (M n ) the subspace of Λ(M n ) formed by loops starting and ending at γ(0) = γ(1) = x ∈ M n , and by Π + (M n ) and Π(M n ) the quotients of Λ(M n ) with respect to the SO(2)(= S 1 )-action: ϕ · γ(t) = γ(t + ϕ), ϕ ∈ S 1 = R/Z, and the O(2)-action respectively. Here the O(2) action is the extension of the SO(2)-action by the involution σ · f (t) = f (−t).

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

“…In [39], Taimanov calculated the rational equivariant cohomology of the spaces of non-contractible loops in compact space forms which is crucial for us to prove Theorem 1.1 and can be stated as follows.…”

“…In particular, if Γ = Z 2 , then S n /Γ is the n-dimensional real projective space RP n . Motivated by the works [44], [12] and [25] about closed geodesics on Finsler RP n , and based on Taimanov's work [39] on rational equivariant cohomology of non-contractible loops on S n /Γ, this paper is concerned with the multiplicity of closed geodesics on Finsler S n /Γ. Let (M, F ) be a Finsler manifold and ΛM be the free loop space on M defined by ΛM = γ : S 1 → M | γ is absolutely continuous and 1 0 F (γ,γ) 2 dt < +∞ , endowed with a natural structure of Riemannian Hilbert manifold on which the group S 1 = R/Z acts continuously by isometries (cf.…”

“…Then in [24], Liu combined Fadell-Rabinowitz index theory with Taimanov's topological results to get many multiplicity results of non-contractible closed geodesics on positively curved Finsler RP n . Very recently, Liu and Xiao [25] established the resonance identity for the non-contractible closed geodesics on RP n , and together with [12] and [39] proved the existence of at least two distinct non-contractible closed geodesics on every bumpy RP n with n ≥ 2.…”

“…Note that the only non-trivial group which acts freely on S 2n is Z 2 and S 2n /Z 2 = RP 2n (cf. P.5 of [39]), then we only need to prove Theorem 1.1 for the case when n is odd.…”

“…Then every bumpy Finsler metric F on M has at least two distinct non-contractible closed geodesics of the class [h].Note that the only non-trivial group which acts freely on S 2n is Z 2 and S 2n /Z 2 = RP 2n (cf. P.5 of[39]). Since we have proved the same result as the above Theorem 1.2 for RP 2n in Theorem 1.2…”