Existence of closed geodesics on certain non-compact Riemannian manifolds

2023/08/01 by Akashdeep Dey, Dey, Akashdeep
Biochemistry, Genetics and Molecular Biology · Mathematics · #Analytic and geometric function theory #Bone Metabolism and Diseases #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2308.00217

openalex publication_date 2023/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a complete Riemannian manifold. Suppose M contains a bounded, concave, connected open set U with C⁰ boundary and M∖ U is connected. We assume that either the relative homotopy set π₁(M,M∖ U)=0 or the union of all the conjugate subgroups of the image of the homomorphism π₁(M∖ U)→ π₁(M) (induced by the inclusion M∖ U\hookrightarrow M) is a proper subset of π₁(M). (The first condition is equivalent to π₁(M∖ U)→ π₁(M) is surjective; the second condition is satisfied if the relative homology group H₁(M,M∖ U)≠ 0.) Then there exists a non-trivial closed geodesic on M. This partially proves a conjecture of Chambers, Liokumovich, Nabutovsky and Rotman.

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