📝 SummarXiv

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.