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

TL;DR

This paper establishes rigidity results for compact submanifolds in Euclidean and spherical spaces with Ricci curvature bounded below by a specific function, characterizing their geometric and topological structure.

Contribution

It generalizes existing rigidity theorems by providing new bounds and characterizations for submanifolds with pinched Ricci curvature, including topological and isometric classifications.

Findings

Submanifolds are either isometric to the Einstein Clifford torus or topological spheres under maximal bounds.

Certain homology groups vanish up to the k-th level for these submanifolds.

The results extend and unify previous rigidity theorems in the literature.

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.