黎曼流形与黎曼子流形的刚性及分类问题研究

The investigations on rigidity and classification of Riemannian manifolds and their submanifolds are the core subjects of global and local differential geometric theory, and have always been intensely focused by geometricians since its beginning, and that a large number of classical impressive results have been established. In recent years, along with the successful applications of methods in those subjects like geometric analysis, theory of critical points on geometric functionals, as well as the development of differential topology, particularly with the emergence of new research methods in Riemannian manifolds and their submanifolds themselves, significant breakthrough and amazing achievements have been made in research for a series of issues. In this project, relying on our more than 20 years of accumulated knowledge and the achievements of ourselves investigating the geometric problems, with employing the local moving frame and exterior differential method, the special successful techniques developed with dealing with various geometric flows, as well as the effective methods in geometric analysis and a prior estimation, we will concentrate and engage in studying problems related to rigidity and classification of Riemannian manifolds and submanifolds which possess canonical geometric topological properties. Especially focus on innovative achievements in study of the following topics: the rigidity and classification of affine hypersurfaces satisfying canonical properties in equiaffine/centroaffine differential geometry; the rigidity and classification of canonical submanifolds (including real hypersurfaces) in well known Kaehler and nearly Kaehler manifolds; the analytic and geometric characterizations of canonical Riemannian metrics (resp. submanifolds) closely related to the critical points of important functionals of Riemannian manifolds (resp. submanifolds).

黎曼流形与黎曼子流形的刚性及分类问题研究，是整体与局部微分几何理论的核心课题，历来受到几何学家高度关注，涌现出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，该领域在一系列课题研究上取得了重大突破和惊人的成就。本项目将依托课题团队二十多年来从事相关研究的知识积累和成果基础，综合运用局部活动标架与外微分方法、针对各种曲率流的研究而发展起来的特殊技巧、几何分析与先验估计等行之有效的研究方法，集中开展具有特殊几何与拓扑性质的典型黎曼流形与黎曼子流形的刚性及分类问题研究，将特别专注在下列课题的研究上取得创新性成果：等积仿射微分几何和中心仿射微分几何中具典型性质超曲面的刚性及分类；凯勒和近凯勒流形中典型子流形（实超曲面）的刚性及分类；与重要黎曼泛函或子流形泛函的临界点相关的极值度量与极值子流形的刚性及分类。

本项目的研究严格按计划围绕黎曼流形与黎曼子流形的刚性及分类进行，在一系列课题研究中实现突破并取得重要成果，共发表论文26篇，其中SCI论文25篇。项目重要结果包括以下八个方面：（1）在球面的子流形研究中，对具有平行Moebius第二基本形式的子流形给出了完全分类；对仿Blaschke张量具有常特征值的超曲面进行了定性刻画。（2）在等积仿射超曲面研究中，发现了一般维数Einstein仿射球面和局部共形平坦仿射球面的最佳刚性现象和椭球面新的整体特征刻画；建立了新的一类仿射球面的局部分类。（3）在中心仿射超曲面研究中，给出了局部严格凸的迷向中心仿射超曲面的完全分类；证明每个Tchebychev卵形面都是椭球面，从而解决了这一长期未决问题；建立了卵形面关于中心仿射不变量的最佳整体刚性和椭球面关于中心仿射不变量的新刻画。（4）在六维近凯勒流形S^3*S^3的子流形研究中，关于P-不变曲面、Hopf超曲面、Einstein超曲面和局部共形平坦超曲面的分类取得一批重要成果。（5）对复射影空间CP^n中具有等变CR极小的三维球面浸入给出了完全分类；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画；分类了复射影空间CP^n中两个常曲率乘积型的拉格朗日子流形。（6）对六维近凯勒流形S^6的拉格朗日子流形，分别建立了关于第二基本形式模长平方和Ricci曲率的最佳刚性定理，给出了最接近于全测地的非全测地拉格朗日子流形的特征刻画。（7）建立了Sasaki空间形式中C-全实子流形的刚性定理；给出了Sasaki空间形式中具有常截面曲率C-全实子流形的完全分类。（8）研究复二次空间中实超曲面，发现了其中典型超曲面的新刻画；证明了其中不存在局部共形平坦或循环Ricci张量的实超曲面。上述研究成果极大丰富了黎曼子流形的几何理论，对于现代等仿射超曲面几何和中心仿射超曲面几何都具有重要的科学意义。