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The Spectral Geometry on the submanifold of (pseudo-) Euclidean space

（伪）欧几里得空间子流形上的谱几何

基本信息

批准号：
09640119
负责人：
ISHIKAWA Susumu
金额：
$ 2.05万
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：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

We obtained 25 and more prints in this Project (See REFERENCES).In 19971. We obtained some new results about the classification of biharmonic submanifold in psudo-Euclidean space. In detail,(1) The classification problem of biharmonic curves in psudo-Euclidean space was completed.(2) It was proved that the bihaemonic surfaces do not exist in 3 dimensional psudo-Euclidean space.(3) Some classification theorems of biharmonic surfaces in 4 dimensional psudo-Euclidean space was obtained.2. About the spacelike maximal submanifolds with some conditions for Ricci curvature immersed in de-Sitter sphere in psudo-Euclidean space, the classification of them was discussed.3. About the hypersurfaces of constant scalar curvature immersed in de-Sitter sphere in psudo-Euclidean space, the sphere theorem was discussed.4. About the comformally flat 3 dimensional Riemannian manifolds under some conditions for Ricci curvature and scalar curvature, the classification problem of them was discussed.In 19981. … More We obtained, under some conditions for scalar curvature, that any compact submanifold immersed in de-Sitter sphere in psudo-Euclidean space is only a standard sphere.2. We obtained a characterization about the Clifford torus.3. (1) We discussed about 3-dimensional comformally flat Riemannian manifold with non negative constant scalar curvature and the constan norm of Ricci curvature.(2) We discussed about 3-dimensional comformally flat Riemnnian manifold with negative constant scalar curvature and the constan norm of Ricci curvature.In 19991. We discussed the classification problem about the minimal closed surfaces in unit sphere with bounded norm of Ricci curvature. This result is concerted with the famous theorem by S.S.Chern, do Carmo and S. Kobayashi that the Clifford torus is only minimal closed surfaces of S=n in unit sphere.2. We obtained some progress concerned with the third work of listed in 19983. We now investigate the following open problems proposed by Bang-yen Chen;(1) The classification problem of the finite type surface in 3 Euclidean space.(2) The classification problem of the biharmonic submanifolds in n-dimensional Euclidean space.(3) The classification problem of the biharmonic submanifolds in 4-dimensional Minkovski space. Less

我们在这个项目中获得了25个和更多的印刷品（见参考文献）。得到了伪欧氏空间中双调和子流形分类的一些新结果。具体而言，（1）完成了伪欧氏空间中双调和曲线的分类问题。(2)证明了在3维伪欧氏空间中不存在二重曲面。(3)得到了4维伪欧几里得空间中双调和曲面的一些分类定理。2.讨论了伪欧氏空间中满足一定条件的Ricci曲率浸入de-Sitter球面的类空极大子流形的分类.讨论了伪欧氏空间中浸入de-Sitter球面的常数量曲率超曲面的球面定理.在Ricci曲率和数量曲率满足一定条件的情况下，讨论了共形平坦的三维黎曼流形的分类问题。 ...更多信息在一定数量曲率条件下，证明了伪欧氏空间中任何浸入de-Sitter球面的紧致子流形都是标准球面.得到了Clifford环面的一个特征. (1)讨论了具有非负常数量曲率和Ricci曲率的常数范数的三维共形平坦黎曼流形。(2)1991年，我们讨论了具有负常数量曲率和Ricci曲率的常数范数的三维共形平坦黎曼流形。讨论了单位球面上Ricci曲率范数有界的极小闭曲面的分类问题。这个结果与陈省身、杜卡莫和S.小林证明了Clifford环面是单位球面上S=n的极小闭曲面. 19983年第三批工作取得了一定的进展。本文研究了以下由Bang-yen Chen提出的公开问题：（1）3欧氏空间中有限型曲面的分类问题。(2)n维欧氏空间中双调和子流形的分类问题。(3)四维Minkovski空间中双调和子流形的分类问题。少