Topological and geometric rigidity of nonnegatively curved submanifolds

Abstract

We investigate the topology and geometry of compact submanifolds in space forms of nonnegative curvature satisfying a lower bound on the sectional curvature that depends only on the length of the mean curvature vector of the immersion. We show that this condition imposes strong constraints on either the topology or geometry of the submanifold. More precisely, we prove that, for dimensions $n\ge5$, the manifold is homeomorphic to a sphere, whereas in dimension four it is either diffeomorphic to $\mathbb S^4$ or isometric to $\mathbb CP^2$, in which case the immersion is the standard embedding into $\mathbb S^7$ followed by an umbilical inclusion. Additionally, we provide examples showing that the spherical conclusion cannot in general be strengthened to an extrinsic rigidity statement, since there exist many geometrically distinct spheres satisfying the pinching condition.