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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.00217 |
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| _version_ | 1866917857294548992 |
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| author | Dey, Akashdeep |
| author_facet | Dey, Akashdeep |
| contents | Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded, concave, connected open set $U$ with $C^0$ boundary and $M\setminus U$ is connected. We assume that either the relative homotopy set $π_1(M,M\setminus U)=0$ or the union of all the conjugate subgroups of the image of the homomorphism $π_1(M\setminus U)\rightarrow π_1(M)$ (induced by the inclusion $M\setminus U\hookrightarrow M$) is a proper subset of $π_1(M)$. (The first condition is equivalent to $π_1(M\setminus U)\rightarrow π_1(M)$ is surjective; the second condition is satisfied if the relative homology group $H_1(M,M\setminus U)\neq 0$.) Then there exists a non-trivial closed geodesic on $M$. This partially proves a conjecture of Chambers, Liokumovich, Nabutovsky and Rotman. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00217 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence of closed geodesics on certain non-compact Riemannian manifolds Dey, Akashdeep Differential Geometry Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded, concave, connected open set $U$ with $C^0$ boundary and $M\setminus U$ is connected. We assume that either the relative homotopy set $π_1(M,M\setminus U)=0$ or the union of all the conjugate subgroups of the image of the homomorphism $π_1(M\setminus U)\rightarrow π_1(M)$ (induced by the inclusion $M\setminus U\hookrightarrow M$) is a proper subset of $π_1(M)$. (The first condition is equivalent to $π_1(M\setminus U)\rightarrow π_1(M)$ is surjective; the second condition is satisfied if the relative homology group $H_1(M,M\setminus U)\neq 0$.) Then there exists a non-trivial closed geodesic on $M$. This partially proves a conjecture of Chambers, Liokumovich, Nabutovsky and Rotman. |
| title | Existence of closed geodesics on certain non-compact Riemannian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2308.00217 |