The Study of geometry of localry conformally flat Riemannian manifolds

在这个项目中，我们主要研究了完全局部共形平坦黎曼流形的几何和子流形的几何。用多种不同的方法研究多种流形的拓扑结构和曲率结构的几何问题是我们的目的。我们研究了(1)共形平面流形的几何，(2)空间形式超曲面的几何，(3)子流形的几何，(4)球面定理的几何，(5)Alexandrov空间的几何，并得到了许多新的结果。在(1)的研究基础上，我们对具有非负常数标量曲率和里奇张量常数范数的三维完全局部共形平坦黎曼流形进行了分类。我们还给出了这类具有负常数曲率的流形的某些特征。在(2)的研究基础上，证明了具有恒定标量曲率和两个不同主曲率的欧几里得空间中的完全超曲面与旋转的完全非紧超曲面是等距的。在此基础上，我们得到了常标量曲率欧几里得空间中完全局部共形平面超曲面的分类，并给出了Yau猜想的部分答案。在(3)的研究基础上，我们将在具有常平均曲率的三维欧几里得空间完全曲面上的Klotz和Osserman的结果推广到任意高维和任意高余维。在常标量曲率的欧氏空间中，我们也得到了完全子流形的重要结果。在研究式(4)的基础上，证明了在径向曲率条件下，最大直径定理成立。在研究式(5)的基础上，我们得到了一个新版本的Alexandrov-Topologov定理。

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.