LetMbe a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphereS n+p(n≥2 ,p≥1). By using the Sobolev inequalities of P. Li (1980) toLpestimate for the square lengthσof the second fundamental form and the norm of a tensorΦ, related to the second fundamental form, we set up some rigidity theorems. Denote by‖σ‖ptheLpnorm ofσandHthe constant mean curvature ofM. It is shown that there is a constantCdepending only onn,H, andkwhere(n−1) kis the

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... lower bound of Ricci curvature such that if‖σ‖ n/2<C, thenMis a totally umbilic hypersurface in the sphereS n+1.