伪黎曼空间中2-调和类空子流形的研究

J.Eells和J.H.Sampson在上世纪60年代提出了2-调和映照的概念[1]，受此启发，若欧氏空间中子流形M的平均曲率向量场为调和的，B.Y. Chen称之为2-调和子流形[2],这等价于M的位置向量场是M到欧氏空间的2-调和映照.伴随着2-调和映照研究的深入，黎曼空间中2-调和子流形的研究得到蓬勃的发展[3-13].然而与之相对照的是,伪黎曼空间中的2-调和子流形的研究尚处在亟待发展的阶段.本项目拟就伪黎曼空间中2-调和类空子流形的几何进行研究,主要内容包括刻画非调和2-调和类空子流形- - -即称之为正常2-调和类空子流形的存在性及其几何与拓扑特征、实现对伪黎曼空间形式中正常2-调和类空超曲面的较为完全的刚性分类.本项目的实施可以进一步理解伪黎曼度量对子流形几何的影响，预期研究成果将可以从几何的角度解释某些物理现象，同时将在计算机辅助几何设计方面有一定的应用前景.

Following the concept of biharmonic maps suggested by J.Eells and J.H.Sampson [1] in the last decade, in the different setting, motivated by that works, B.Y.Chen [2] defined biharmonic submanifolds in Euclidean space as those with harmonic mean curvature vector field, equivalently, the position vector field is a biharmonic map. Then the study on biharmonic submanifolds in Riemannian space has been developed vigorously [3- - 13] encouraged by that of biharmonic maps. In contrast with that, though there are a few results on biharmonic submanifolds in pseudo-Riemannian spaces, the study of the corresponding problems in pseudo-Riemannian spaces is still in need of development. In this project, we plan to study the geometry of spacelike biharmonic submanifolds in pseudo-Riemannian spaces, the main content includs characterizing the geometry and topology of non-harmonic biharmonic spacelike submanifolds - - - called proper biharmonic spacelike submanifolds, and classifying proper biharmonic spacelike hypersurfaces in the sense of rigidity. The research results can help one to understand the geometry of submanifolds affected with pseudo-Riemannian metric, and can explain some physical phenomenon from the geometric point of view. It also can be applied to the field of boundary based surface design in computational geometry.