Submanifold In Sphere And Locally Symmetric And Comformally Flat Manifold

Content: Some curvature problems about the compact minimal submanifold in sphere, It is usually called intrinstic rigidity ,papers[1],[2],[3],[4] have obtained many good results. Those results are usually described with the square of the length of the second fundamental form ,scalar curvature ,Ricci curvature.At first, this paper considers submanifold with pararrel mean curvature vector and positive scalar curvature in a sphere, make use of the sphere of the length of the Riemanian curvature tensor, some intrinstic rigidity results are obtained .Second,the author considers the compact pseudoumbilical submanifold with pararrel mean curvature vector in sphere. We use the minimal eigenvalue of a operator ,we estimates the sphere of length of the second fundamental form. At last ,the author explodes the ambient space ,considers the pseudoumbilical submanifold with pararrel mean curvature vector in locally symmetric and comformally flat manifold,so the correspond results due to Xu Zhao-di[8] are generalized.