Research on curvature structures and topological structures of submanifolds in Riemannian manifolds

本文主要研究黎曼流形上子流形的曲率结构和拓扑结构，以及黎曼流形上拉普拉斯算子的特征值和特征函数的几何性质。我们的目的是用许多不同的方法研究各种流形的拓扑和曲率性质的几何问题。我们研究了(1)欧氏空间和球面中子流形的拓扑结构和曲率结构的几何。(2)球面中具有无限基本群的紧致超曲面的几何。(3)4维空间形式中完备超曲面的曲率结构几何。(4)Lorentz空间中完备类空超曲面的曲率结构几何。(5)黎曼流形上拉普拉斯的特征值和特征函数的几何。(6)球面几何定理。(7)满足流形的共形射影结构的几何。(8)关于流形的径向曲率和拓扑的几何，得到了许多重要的结果。这是我们项目的特征，为研究黎曼流形中完备子流形的拓扑结构和曲率结构做出了重要贡献。此外，程和杨在研究黎曼流形上的拉普拉斯本征值方面得到了重要的结果。

In this project, we mainly investigated the curvature structures and topological structures of submanifolds in Riemannian manifolds and the geometry of eigenvalues and eigenfunctions of Laplacian on Riemannian manifolds. It is our purpose to research geometric problems on properties oftopology and curvatures of many kinds of manifolds by means of many different methods. We studied (1)the geometry of topological structures and curvature structures of submanifolds in Euclidean spaces and spheres. (2)the geometry of compact hypersurfaces with infinite fundamental group in spheres. (3)the geometry of curvature structures of complete hypersurfaces in 4-dimensional space forms. (4)the geometry of curvature structures of complete space-like hypersurfaces in Lorentz spaces. (5)the geometry ofeigenvalues and eigenfunctions of Laplacian on Riemannian manifolds. (6)the geometry of sphere theorems. (7)the geometry of conformally projective structures of satistical manifolds. (8)the geometry on radial curvature and topology of manifolds and obtained many important results.It is characterizations of our project that important contributions on the research of topological structures and curvature structures of complete submanifolds in Riemannian manifolds are obtained. Furthermore, Cheng and Yang obtained important results for the investigation on eigenvalues of Laplacian on Riemannian manifolds.