Hypersurfaces And Conformally Flat Riemannian Manifold With The Prescribing Curvature

This paper consists of two parts. In part /, I present the local classification of the hypersurface with parallel Ricci curvature in constantly curved manifold by using the theorey of holonomy group, additionally, if this hypersurface is minimal immersion, I also give the classification of this hypersurface, which generalize the results related to H.B.Lawson. In part II, I study conformally flat manifolds with prescribing curvature, I present the classification of conformally flat manifold with product metric and the classification of conformally flat manifold with parallel Ricci curvature. Under prescribing scalar curvature, I study the compact manifold, I obtain some results which generalized the result of Goidberg's; meanwhile, I use the self adjoint operator definited by Cheng & Yau to calculate r, I obtain some new pinching results, which generalize the results related to Tanni,Omori,Yau and so on.