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

Abstract

For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function

\alpha (n,k,H,c)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha (n,k,H,c)$$\end{document}

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,\left[\frac{n}{2}\right],H,c)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha (n,\left[ \frac{n}{2}\right] ,H,c)$$\end{document}

, 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.

Keywords

Pinched Ricci curvature / Rigidity theorem / Einstein / 53C20 / 53C24 / 53C40

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Funding

National Natural Science Foundation of China(No. 12171037)

Fundamental Research Funds for the Central Universities(No. 12271040)

China Postdoctoral Science Foundation(BX20230018)

National Key R&D Program of China(2020YFA0712800)

RIGHTS & PERMISSIONS

Peking University