Rigidity of closed submanifolds in a locally symmetric Riemannian manifold

Abstract

Let Mn(n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an (n + p)-dimensional locally symmetric Riemannian manifold Nn+p. We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu [15].