Rigidity of closed submanifolds in a locally symmetric Riemannian manifold

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