The Study of geometry of localry conformally flat Riemannian manifolds

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In this project, we mainly investigated the geometry of complete locally conformally flat Riemannian manifolds and the geometry of submanifolds. It is our purpose to research geometric problems on topoiogical structures and curvature structures of many kinds of manifolds by many different methods. We studied (1) the geometry of conformally flat manifolds, (2) the geometry of hypersurfaces in space forms, (3) the geometry of submanifolds, (4) the geometry of sphere theorems and (5) the geometry of Alexandrov spaces and obtained many new results.On the research of (1), we classified 3-dimensional complete locally conformally flat Riemannian manifolds with non-negative constant scalar curvature and constant norm of the Ricci tensor. We also gave certain characterizations of such manifolds with negative constant scalar curvature.On the research of (2), we proved that complete hypersurfaces in a Euclidean space with constant scalar curvature and two distinct principal curvatures are isometric to complete and noncompact hypersurfaces of revolution. From this result, we obtained a classification of complete locally conformally fiat hypersurfaces in a Euclidean space with constant scalar curvature and gave a partial answer of Yau's conjecture.On the research of (3), we extended the result due to Klotz and Osserman on complete surfaces in the 3dimensional Euclidean space with constant mean Curvature to any higher dimension and any higher co-dimension. We also obtained important result on complete submanifolds in the Euclidean space with constant scalar curvature.On the research of (4), we proved that under condition of radial curvature, maximal diameter theorem holds.On the research of (5), we obtained a new version of Alexandrov-Topologov theorem.