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

TL;DR

This work derives sharp Ricci-pinching rigidity for compact submanifolds $M^n$ in space forms, establishing that under $\operatorname{Ric}\ge\alpha(n,k,H,c)$ with $2\le k\le n/2$ and $n\ge5$, either $\pi_1(M)=0$ and low-degree homology vanishes or $M$ is isometric to an Einstein Clifford torus embedded in a totally umbilical sphere; at the maximal bound, the submanifold is a topological sphere or has vanishing homology up to level $k$. A central technical tool is the core Ricci-pinching lemma, which forces a rigid normal bundle structure and leads to two principal normals and, in equality cases, either a Dupin-type decomposition or an Einstein hypersurface realization via Onti18. The results extend and unify previous rigidity theorems (Ejiri, Xu-Tian, Xu-Gu, Xu-Leng-Gu, Vlachos, Dajczer-Vlachos) and provide a near-complete geometric classification under pinched Ricci curvature for both Euclidean and spherical space forms. The corollaries address the maximal $k=n/2$ and odd-dimensional cases, yielding explicit Clifford-type classifications and sphere-like rigidity.

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 $α(n,[\frac{n}{2}],H,c)$, 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.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5