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

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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