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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.14112 |
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| _version_ | 1866911361631518720 |
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| author | Ge, Jianquan Tao, Ya Zhou, Yi |
| author_facet | Ge, Jianquan Tao, Ya Zhou, Yi |
| contents | For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $α(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a topological sphere for the maximal bound $α(n,[\frac{n}{2}],H,c)$, or has up to $k$-th homology groups vanishing. This gives an almost complete (except for the differentiable sphere theorem) characterization of compact submanifolds with pinched Ricci curvature, generalizing celebrated rigidity results obtained by Ejiri, Xu-Tian, Xu-Gu, Xu-Leng-Gu, Vlachos, Dajczer-Vlachos. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14112 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms Ge, Jianquan Tao, Ya Zhou, Yi Differential Geometry 53C20, 53C24, 53C40 For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $α(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a topological sphere for the maximal bound $α(n,[\frac{n}{2}],H,c)$, or has up to $k$-th homology groups vanishing. This gives an almost complete (except for the differentiable sphere theorem) characterization of compact submanifolds with pinched Ricci curvature, generalizing celebrated rigidity results obtained by Ejiri, Xu-Tian, Xu-Gu, Xu-Leng-Gu, Vlachos, Dajczer-Vlachos. |
| title | Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms |
| topic | Differential Geometry 53C20, 53C24, 53C40 |
| url | https://arxiv.org/abs/2411.14112 |