黎曼子流形的分类及相关问题研究

The characterization and classification of submanifolds in typical Riemannian manifolds are core research topics of differential geometry, which has always been highly valued by geometrists. In recent years, along with the successful application of theories and methods in those subjects, such as differential topology theory, algebraic geometry theory, curvature flow method, the combination of moving frame and exterior differential method and algebraic skills, a large number of significant achievements were obtained in this field. At the same time, the establishment of new theories, the discovery of new methods and the constant emergence of new problems keep the field full of vitality. In this project, relying on the previous research foundation on related problems, we will focus on the classification of submanifolds with special geometric and topological properties. Specifically, we will pay attention to the selection of typical frames and the flexible application of algebraic techniques, give full play to the power of the moving frame and the external differential method, and by means of knowledge fusion and method transfer in different branches, we hope to achieve innovative results on the following topics: the relationship between different Moebius invariants of the submanifolds in the unit sphere, the construction of typical examples, the classification of Dupin hypersurface in the context of Lie sphere geometry, the exploration of new Lie invariants, the classification of isoparametric hypersurface.

对典型黎曼流形中的子流形进行刻画和分类是微分几何学的核心研究课题之一，历来为几何学家们所高度重视。近年来，随着微分拓扑学理论、代数几何学理论、曲率流方法在该课题研究上的成功运用，活动标架和外微分方法与代数技巧的巧妙结合，该领域取得了大批重要研究成果。同时，子流形研究自身新理论的建立、新方法的发现和新问题的不断涌现，又使该领域的研究充满活力。在申请者多年从事相关研究的基础上，本项目将集中开展具有特殊几何与拓扑性质的子流形的分类问题研究。具体地，我们将关注典型标架的选取和代数技巧的灵活运用，充分发挥活动标架与外微分方法的威力，并借助于不同分支领域的知识融合和方法迁移，预期在下列课题的研究上做出创新性研究成果：球面中子流形的 Moebius 不变量之间的关系和典型例子的构造，Lie 球几何框架下 Dupin 超曲面的分类，新的 Lie 不变量的探寻，球面中等参超曲面的分类。

对典型黎曼流形中的子流形进行分类是微分几何学的核心研究课题之一，历来为几何学家们所高度重视。围绕这一课题，我们结合典型标架的选取和代数技巧的灵活运用，以及不同几何框架下研究方法与分析思路的交融，取得了以下几个方面的进展：(1)研究S^5中的Moebius极小超曲面，得到Moebius迷向条件下此类超曲面的分类定理；(2)研究乘积空间S^m×S^n中的子流形，得到半平行条件下的分类定理； (3)研究秩为2复维数是2m(m≥3)的复Grassmannians流形中的实超曲面，证明不存在任何局部共形平坦实超曲面；(4)研究Sasakian空间形式中的Legendre子流形，得到S^9中具有非负截面曲率的紧致极小Legendre子流形的分类定理。这些创新性研究成果对丰富和发展子流形几何理论很有意义，证明中所使用的新方法和代数技巧对子流形中其他问题的解决有重要的借鉴和促进作用。