Local Symmetry Of The Negative Curvature Manifold Submanifold In Rigid Geometry,

In the present thesis, we mainly study geometric rigidity problems of the complete submanifold with parallel mean curvature in the negative pinched lo-cally symmetric manifold, and generalize the theorems of H. W. Xu, X.A.Ren[7].Firstly, we prove the rigidity theorem of complete hypersurfaces with con-stant mean curvature in a negative symmetric manifold. We use the basic func-tions and locally symmetric of the ambient space, and get the following result:Let Nn+1 be an n+1-dimensional simply connected symmetric Riemannian manifold with negative pinching curvature, i.e., -1≤KN≤δ(δ=δ(n,p)<0), and Mn be a complete hypersurface with constant mean curvature H(H>1) in Nn+1. If and if supM S<α(n, H), then M is congruent to totally umbilical hypersurface.Secondly, we further study the rigidity theorem of the complete subman-ifold with parallel mean curvature in above ambient space and generalize the conclusion to higher dimensions, we obtain the following:Let Nn+p(p≥2) be an n+p-dimensional simply connected symmetric Rie-mannian manifold with pinching curvature, i.e., -1≤KN≤δ(δ=δ(n,p)<0), and Mn be a complete submanifold with parallel mean curvature H(H>1) in Nn+p.If and if supM S<α(n,H), then M is congruent to one of the following:(1) totally umbilical submanifold;(2) Nn+p totally isometric to Hn+p(-1), and M is congruent to the Veronese surface in S4(?).