高维非自然约化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. So far people have obtained the following results: Every simply-connected g.o. manifold of dimension n<6 is naturally reductive; All the cases for the six-dimensional simply connected Riemannian g.o. manifolds and seven-dimensional Riemannian g.o. manifolds which are not naturally reductive have been classified. But the problem of classification of non-naturally reductive g.o. manifolds of dimension n>7 remains open and is not much progress. In this project, we mainly study the problem of classification of non-naturally reductive g.o. manifolds for the flag manifolds G/H and the homogeneous manifolds G/H_1, here H=H_0*H_1, and H_0 is 1-dimensional center of H. When we study non-naturally reductive g.o. manifolds of dimension n>7, there are two difficulties, one of which is how to choose the metric on the manifolds which is convenient for calculation, the other is how to find a set of orthogonal basis on the tangent space of the manifolds which is convenient for calculation. We will study these two problems in this project. We will give a new method to determine whether a manifold is non-naturally reductive or not, and give the classification of non-naturally reductive g.o. manifolds for the flag manifolds G/H and the homogeneous manifolds G/H_1.

1983年以前，人们一直认为g.o.流形与自然约化流形等价。直到1983年，A.Kaplan给出了一个非自然约化g.o.流形的例子，从此数学家们对g.o.流形才有了更广泛的研究。目前已有的工作是：维数小于等于5的单连通g.o.流形均是自然约化流形，6维单连通非自然约化g.o.流形与7维非自然约化g.o.流形已经有了完全分类。但是维数大于7的非自然约化g.o.流形的分类一直是一个公开问题，且没有太大进展。本项目主要研究维数大于7的两类齐性流形：旗流形G/H与流形G/H_1，其中H=H_1*H_0，H_0为H的一维中心。针对这两类流形，本项目研究对维数大于7的非自然约化流形分类时遇到的两个难题：1.在流形M上定义一个便于计算的黎曼度量；2.在流形切空间上给出一组便于计算的正交基。拟给出新的方法来判断g.o.流形的非自然约化性，进而对这两类流形中非自然约化的g.o.流形进行分类。

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. 主要原因是很难找到满足研究对象为g.o.流形的或者充分、或者必要、或者充要条件的条件. 本项目主要研究M-流形中 g.o.流形的分类. 首先我们给出了M-流形迷向表示的不可约子模与旗流形迷向表示不可约子模之间的关系. 然后发现M-流形中g.o.流形的分类可根据广义旗流形迷向表示不可约子模个数 s 分三种情况来讨论: 1. s>2, 给出了M-流形为g.o.流形的必要条件; 2. s=2, 根据不同情况分别给出了M-流形为 g.o.流形或者必要条件, 或者充分条件, 或者充要条件; 3. s=1, 给出了M-流形为 g.o.流形的充要条件.