广义C-空间中g.o.流形的分类

Before 1983, it was generally believed that all g.o. manifolds are naturally reductive. However, in 1983 A. Kaplan constructed the first counter example of Riemannian g.o. manifold which is not naturally reductive. Since then, g.o. manifolds have been studying extensively by many mathematicians. There are two strains of thought about the research of g.o. manifolds in compact Riemannian manifolds. One is giving the classification of g.o. manifolds with the same dimension, the other is giving the classification of g.o. manifolds according to the type of manifolds. Until now, the classification of g.o. manifolds in compact Riemannian manifols also remains an open problem and is not much progress, we plan to give the classification of g.o. manifolds in generalized C-spaces. In addition, this project intends to carry out the rearch of the symplectic topology of the homogeneous spaces with symplectic structure, one can further understand the homogeneous space itself through this research, on the other hand it is also possible to promote the development of the symplectic topology through the study and calculation of such manifolds.

1983年以前，人们一直认为自然约化流形与g.o.流形是等价的。直到1983年A.Kaplan给出了一个非自然约化g.o.流形的例子，从此数学家门开启了对g.o.流形的研究。对于紧致黎曼流形中g.o.流形的分类，研究的思路主要有两种：1.按维数分类，2.按流形类别分类。目前，紧致黎曼流形中g.o.流形的分类仍然是是一个公开问题，且进展缓慢，本项目拟对广义C-空间中g.o.流形进行完全分类。另外，本项目拟开展对具有辛结构的齐性空间的辛拓扑的研究，一方面可以通过这样的研究更进一步了解此类齐性空间的本身的性质，另一方面也可以通过对此类流形的研究与计算推动辛拓扑的发展。

1958年, W.Ambrose与 I.Singer 给出自然约化黎曼流形与g.o.流形等价的证明. 但是直到1983年, A.Kaplan给出了第一个非自然约化g.o.流形的例子, 这是一个有2维中心的6维幂零黎曼流形, 从此人们对g.o.流形才有了更为广泛的研究. 但是直到目前黎曼流形中g.o.流形分类仍然是一个公开问题. 主要原因是针对研究的对象, 很难找到满足其为g.o.流形的或者充分、或者必要、或者充要条件. 本项目主要研究广义 C-空间中g.o.流形的分类. 我们分三种情况对C-空间中g.o.流形进行分类：1. 在第一种情况中给出了C-空间为g.o.流形的必要条件; 2. 在第二、三种情况中给出了C-空间为g.o.流形的或者必要条件, 或者充分条件, 或者充要条件, 并且给出了具体的分类结果.