Geometric and topological rigidity of pinched submanifolds

Abstract

We investigate the geometry and topology of compact submanifolds of arbitrary codimension in space forms satisfying a certain pinching condition involving the length of the second fundamental form and the mean curvature. We prove that this pinching condition either forces homology to vanish in a range of intermediate dimensions, or completely determines the submanifold up to congruence. The results are sharp and extend previous results due to several authors without imposing any further assumption on the mean curvature.