典型黎曼流形与子流形的分类研究

It is the core subject of the global and local differential geometric theory that by employing the basic geometric invariants to give characterization or to carry out the classification of those Riemannian manifolds and/or submanifolds with typical properties, which was always intensely focused by geometers and that a large number of classical deep results had been established. In recent years, along with the successful applications of methods in those subjects like geometric analysis, theory of critical points on functionals, as well as the differential topology, particularly with the emergence of new research methods itself, remarkable progress has been made in research of this field. In this proposal, relying on the accumulation of knowledge and the achievement by our project team during the research and experiments in past several years, we will engaged in research related to the classification problem of Riemannian manifolds and submanifolds with special geometric and topological properties, especially focus on making the innovative research results in the study includes the following topics: the classification of submanifolds in space forms with special properties under the conformal transformation group or the Lie sphere transformation group, especially the open Cecil-Chi-Jensen problem about Dupin hypersurfaces; the classification of hypersurfaces closely related to the affine hyperspheres in affine differential geometry; the geometric classification of parallel submanifolds and real hypersurfaces in complex space forms with special properties; the analytic and geometric characterizations of canonical Riemannian metrics closely related to the critical points of important Riemannian functionals.

利用基本几何不变量对典型黎曼流形与子流形进行刻画和分类，是整体与局部微分几何理论的核心课题，历来为几何学家高度关注，涌现了大批经典的深刻结果。近年来，伴随着几何分析方法、泛函的临界点研究技巧、微分拓扑学研究成果等在该课题研究上的成功运用和其自身新的研究方法的涌现，该领域的研究又取得了显著进展。本项目将依托课题团队多年来从事相关研究的知识积累和成果基础，集中开展具有特殊几何与拓扑性质的典型黎曼流形与子流形的分类研究，将特别立足于在包括下列课题的研究上做出创新性研究成果：空间形式中特殊性质子流形在共形变换群或Lie球变换群下的分类，特别是关于Dupin超曲面悬而未决的Cecil-Chi-Jensen问题；仿射微分几何中与仿射球面紧密相关的超曲面的分类；复空间形式中平行子流形和特殊性质实超曲面的几何分类；与重要黎曼泛函临界点相关的典型度量的几何与分析刻画。

本项目的研究严格按计划围绕典型黎曼流形与子流形的分类进行，在一系列课题研究中实现突破并取得重要成果，共正式发表SCI论文20篇，出版学术著作一部。主要业绩体现在以下七大方面：（1）在球面子流形研究中，在解决余维2的特殊情形基础上，进而彻底完成了具有平行Moebius第二基本形式的子流形的完全分类。（2）在等积仿射微分几何中，完成了四维Einstein仿射球面的分类并发现了一般维数Einstein仿射球面最佳的刚性现象。（3）在中心仿射微分几何中，限定局部严格凸中心仿射超曲面的条件下，完成了具有平行三次形式的超曲面、具有平行迹零三次形式的超曲面、迷向仿射超曲面等多项重要分类。（4）在六维近凯勒流形S^3*S^3的子流形研究中，完成了包括其近复曲面的刚性研究、其具有平行第二基本形式拉格朗日子流形的完全分类、其迷向拉格朗日子流形的完全分类、其具有典型性质的超曲面的完全分类等多项工作。（5）在复射影空间CP^n子流形研究中，对具有等变CR极小三维球面的浸入进行了完全分类；特别地，对复射影空间CP^3中具有等变极小性质的三维球面进行了刻划。（6）在黎曼流形研究中，对广义m-拟爱因斯坦流形在平行利奇张量或常数量曲率条件下进行了分类、研究了平行Cotton张量流形和一些特殊的“收缩型孤子”流形的刚性等。（7）本项目执行期间，主持人在李安民院士领导下，会同Udo Simon教授和赵国松教授完成了学术著作《Global affine differential geometry of hypersurfaces》（第二版）并由De Gruyter出版社于2015年7月正式出版发行，此书的出版适时总结了最近二十年来仿射微分几何领域的重要进展，对国际上仿射微分几何的研究起到重要的推动作用。