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

我们在这个项目中获得了25张以上的印刷品（见参考文献）。在19971年。得到了拟欧几里德空间中双调和子流形分类的一些新结果。(1)完成了拟欧几里德空间中双调和曲线的分类问题。(2)证明了三维拟欧几里得空间中不存在双曲面。(3)得到了四维拟欧几里德空间中双调和曲面的若干分类定理。讨论了拟欧几里德空间中具有一定条件的Ricci曲率浸入de-Sitter球的类空间极大子流形的分类。讨论了拟欧几里德空间中浸入de-Sitter球中的常标量曲率超曲面的球定理。讨论了三维共形平坦黎曼流形在一定条件下的Ricci曲率和标量曲率的分类问题。在19981年。在一定的标量曲率条件下，我们得到了在拟欧几里德空间中，任何浸没在de-Sitter球中的紧化子流形都是一个标准球。我们得到了Clifford环的一个特征。(1)讨论了具有非负常数标量曲率和Ricci曲率常范数的三维共形平坦黎曼流形。(2)讨论了具有负常数标量曲率和Ricci曲率常范数的三维共形平坦黎曼流形。在19991年。讨论了具有Ricci曲率有界范数的单位球极小封闭曲面的分类问题。这一结果与S.S.Chern、do Carmo和s.kobayashi关于Clifford环面只是单位球面上S=n的最小闭曲面的著名定理相一致。1983年上市的第三项工作取得了一些进展。我们现在研究以下由陈邦彦提出的开放性问题：(1) 3欧几里德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

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