若干典型空间的结构及子流形的刻画与分类研究

子流形几何是微分几何的一个重要分支，它与分析、拓扑、代数等学科密切相关。一些典型的外围空间（如实的或复的空间形式等）的子流形是该分支的重要研究对象。 在这方面，熟知的主要研究对象有（Lagrangian）极小子流形、平均曲率流的自收缩子解、等参超曲面及一般高余维子流形的共形几何等。本项目主要研究一类典型外围空间的几何或代数结构及其在某些特殊子空间上所诱导的几何结构，重点讨论这些诱导结构本身的几何特性及相互之间的内在联系，同时关注部分特殊子流形（如自收缩解及 $\lambda$-子流形等）的分类问题和刚性问题。在前期工作的基础上，项目组成员将对这些自然诱导的几何或代数结构（如仿射结构、共形结构、李代数结构等）、以及部分重要的不变量（如相应的仿射或共形黎曼度量、f-Ricci张量及Blaschke张量等）进行深入系统的研究。任务目标是要建立一些具有鲜明特色的特征刻画定理、分类定理或刚性定理。

The geometry of submanifolds is an important branch of the differential geometry, closely related to analysis, topology and algebra. In this area, submanifolds of some canonical ambient spaces (say real or complex space forms etc) are always the interesting objects for study. To mention a few, we have, as is known,the (Lagrangian) minimal submanifolds, self-shrinkers of the mean curvature flow, the isoparametric hypersurfaces and the conformal geometry of higher codimensional submanifolds in the real (Lorentzian-) space forms. The present project mainly studies the geometric or algebraic structures of a class of canonical ambient spaces and the induced geometric structures on some special subspaces, paying more attention on the characteristic features of these induced structures and their interactions, and also on the classification and rigidity problems for some special submanifolds (such as the self-shrinkers and their generalization--$\lambda$-submanifolds, etc). On the basis of the previous achievement, our group members are going to do the research systematically on the geometric or algebraic structures (say, the affine structure, the conformal structure and the structure of Lie algebras etc), and on some important invariants (the affine or conformal/Moebius metrics, the f-Ricci tensors and the Blaschke tensors).The goal of this project is to obtain some characterization theorems of specific favor, some classification theorems and rigidity theorems.