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

2024/11/21 by Jianquan Ge, Ge, Jianquan, Yu Tao +3
Engineering · Mathematics · Physics and Astronomy · #3D Shape Modeling and Analysis #53C20 #53C24 #53C40 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2411.14112

openalex publication_date 2024/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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,[(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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• Topological and differentiable rigidity of submanifolds in space forms
• The topological rigidity theorem for submanifolds in space forms
• Ricci pinched compact submanifolds in spheres
• Rigidity of submanifolds with parallel mean curvature in space froms
• Ricci pinched compact submanifolds in space forms