Remark on non-contractible closed geodesics and homotopy groups

2023/07/26 by Egor Shelukhin, Jun Zhang, Shelukhin, Egor +1 · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2307.13877

Abstract

We prove that if the m-th homotopy group for m ≥ 2 of a closed manifold has non-trivial invariants or coinvariants under the action of the fundamental group, then there exist infinitely many geometrically distinct closed geodesics for a C⁴-generic Riemannian metric. If moreover there are infinitely many conjugacy classes in the fundamental group, then the same holds for every Riemannian metric.

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