In this talk, I will discuss the rigidity problem of the oriented compact submanifold M with parallel mean curvature in a complete simply connected Riemannian manifold with positive pinched curvature. We prove that there exists a constant δ(n,p) in the interval (0, 1) such that if the sectional curvature of N is pinched in [δ(n,p), 1], and if the Ricci curvature and the scalar curvature of M satisfy certain conditions, then N is isometric to Sn+p . Moreover, M can be completely classified. This is a joint work with Prof. Hongwei Xu and Dr. Li Lei.

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