Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

Abstract

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if Mn is simply connected and the k-th Ricci curvature of Mn is bounded below by a quantity involving the mean curvature of Mn and the curvature of the ambient manifold, then Mn is diffeomorphic to the standard sphere \({\mathbb{S}^n}\). For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold Mn is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.