Geometric and topological rigidity of pinched submanifolds in Riemannian manifolds

TL;DR

This paper investigates the rigidity of compact submanifolds in Riemannian manifolds under a sharp pinching condition involving curvature measures, leading to strong geometric and topological restrictions.

Contribution

It extends sphere theorems to submanifolds in general Riemannian manifolds without assuming space form conditions, under a new sharp pinching criterion.

Findings

Establishes sharp rigidity results for submanifolds with pinched curvature

Provides topological classifications under curvature pinching

Extends classical sphere theorems beyond space forms

Abstract

We study the rigidity of compact submanifolds of Riemannian manifolds of arbitrary codimension that satisfy a sharp pinching condition involving the norm of the second fundamental form and the mean curvature. Without assuming that the ambient manifold is a space form, we show that this condition imposes strong geometric and topological restrictions on the submanifold. The resulting theorems are sharp and provide extensions of several known results in the literature, particularly sphere theorems, without requiring additional assumptions.