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

黎曼流形中子流形的曲率结构和拓扑结构研究

基本信息

批准号：
14540085
负责人：
CHENG Qing-ming
金额：
$ 2.18万
依托单位：
Saga University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540085/
关键词：
complete submanifod mean curvature scalar curvature eigen values of Laplacian Ricci curvature sectional curvature Euclidean space and sphere homogeneous manifold Laplacian eigenvalue complete minimail hypersuface harmonic stability /

项目摘要

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.

本课题主要研究黎曼流形中子流形的曲率结构和拓扑结构以及黎曼流形上拉普拉斯特征值和特征函数的几何结构。我们的目的是通过多种不同的方法来研究多种流形的拓扑性质和曲率等几何问题。我们研究了（1）欧几里得空间和球面中的拓扑结构和子流形曲率结构的几何。 (2)球内无限基本群的紧超曲面几何。 (3)4维空间形式的完全超曲面的曲率结构几何。 (4)洛伦兹空间中完全类空间超曲面的曲率结构几何。 (5)黎曼流形上拉普拉斯算子特征值和特征函数的几何。 (6)球面几何定理。 (7)统计流形共形射影结构的几何。 (8)流形径向曲率和拓扑几何学，取得了许多重要成果。在黎曼流形中完全子流形的拓扑结构和曲率结构研究方面取得了重要贡献，是本课题的表征。此外，Cheng和Yang在黎曼流形上拉普拉斯算子特征值的研究方面取得了重要成果。