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arXiv · 2411.14112

Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms

Jianquan Ge·Ya Tao·Yi Zhou
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Abstract

For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $\alpha(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a topological sphere for the maximal bound $\alpha(n,[\frac{n}{2}],H,c)$, 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.

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Jianquan Ge, Ya Tao, Yi Zhou. 2024-11-21. Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms. https://arxiv.org/abs/2411.14112

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