On The Curvature,Topology And M(?)bius Characterization Of Riemannian Manifold

Riemannian geometry started from the famous speech "On the hypothesis as the foundation of geometry" of the German mathematician B. Riemann, it became into an important mathematics theory after the improvement and generalization of many famous mathematician. A.Einstein took the gravitation phenomenon as the curvature property of Riemannian space, the physics phenomenon may be considered as geometry phenomenon. Therefore, Riemannian geometry may be applied in general relativity theory and theoretical physics. The investigation of Riemannian geometry, from localization to globe, produced many important results, which can be used in many mathematical fields such as algebra topology, partial differential equation, multiple valued complex analysis as well as modern physics.Riemannian manifold is the studying object of Riemannian geometry. The investigation of the curvature and topology as well as Mobius characterization of Riemannian manifold is to investigate the curvature and analytic structure and the geometry phenomenon which included in the curvature and analytic structure. The Mobius invariants under the conformal transformation as well as the curvature and topological property which denoted in terms of Mobius invariants are also the very important studying object of conformal differential geometry.The aim of this paper is to study the curvature and topological property and the Mobius characterization of submanifolds, Mobius submanifolds, Willmore submanifolds, spacelike submanifolds as well as hypersurfaces with constant mean curvature in locally symmetric Riemannina manifold. The main results are listed in the following:1. The n dimensional compact hypersurfaces with constant scalar curvature in unit sphere Sn+1(1) are studied. A topological anwser is given to an important problem proposed by Q.M.Cheng. The submanifolds with high codimensional are also investigated. Some important classification theorems are obtained,which generalize and improve the results on compact submanifolds with constant scalar cuevature. For the hypersurfaces with two distinct principal curvatures, the the-orems of curvature and topological property are also given.2. The Mobius characterization of submanifolds with vanishing Mobius form and no umbilic points in unit sphere Sn+p(1) are investigated. By making use of the Mobius invariants-the Blaschke thensor and the Mobius sectional curvature and Mobius Ricci curvature, we obtain some important classification theorems. We also give some classification theorems of submanifold with flat Mobius normal connection.3.The curvature and topological property of Willmore submanifolds in unit sphere Sn+p(1) is studied. Some important integral equalities on Willmore submanifolds are obainted. Some rigidity theorems in terms of sectional curvature and Ricci curvature are obtained by making use of the integral equalities. Some classification theorems of Willmore submanifolds with flat normal connection are given.4. It is proved that the complete spacelike submanifolds with parallel mean curvature vector in de Sitter space Sn+P(c), if H2 > c, then the squared norm of the second fundamental form is bounded above.5. The n-dimensional complete spacelike hypersurfaces in a de Sitter space S₁ⁿ⁺¹ with constant scalar curvature and with two distinct principal curvatures are investigated, some characterization theorems are obtained. In particular, the complete spacelike hypersurfaces in S₁ⁿ⁺¹ with the scalar curvature and the mean curvature being linearly related are also studied.Some important classification theorems are obtainted.6.The complete minimal immersed hypersurfaces and complete hypersurfaces with constant mean curvature in locally symmetric manifold are investigated, Some characteristic theorems of complete hypersurfaces in locally symmetric manifold are obtained, which generalize and improve the results obtained by Shui N.Q. and Wu G.Q., Hlineva S.and Belchev E. as well as Alencar H. and do Carmo M.