Geometric and topological rigidity of pinched submanifolds II

TL;DR

This paper investigates the geometric and topological properties of four-dimensional compact submanifolds in space forms under a pinching condition, employing advanced techniques from four-dimensional geometry and curvature theory.

Contribution

It extends previous results by providing sharp conditions for rigidity of four-dimensional submanifolds without extra assumptions on mean curvature or fundamental group.

Findings

Results are sharp and extend prior work.

Focus on four-dimensional submanifolds with intricate methods.

Utilizes concepts from nonnegative isotropic curvature and Bochner technique.

Abstract

We continue the study of the geometry and topology of compact submanifolds of arbitrary codimension in space forms that satisfy a pinching condition involving the length of the second fundamental form and the mean curvature. Our primary focus is on four-dimensional submanifolds, where both the results obtained and the methods employed differ substantially and are considerably more intricate than in higher dimensions. This study relies critically on concepts from four-dimensional geometry, the theory of Riemannian manifolds with nonnegative isotropic curvature, and the Bochner technique, each playing an essential role. The results are sharp and extend previous work by several authors, without imposing additional assumptions on either the mean curvature or the fundamental group of the submanifold.