权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Research on Geometric invariant on Manifolds and Lie transformation groups

流形和李变换群几何不变量的研究

基本信息

批准号：
17340019
负责人：
KAMISHIMA Yoshinobu
金额：
$ 4.26万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

(1) We have studied an integrable, nondegenerate codimension 3 -subbundle D on a 4n+3- manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of quaternionic CR structure. We single out an sp (1)-valued 1-form ω loally on a neighborhood U such that Null ω= DIU and construct the curvature invariant on (M,ω) whose vanishing gives a uniformization to flat quaternionic CR geometry. The invariant obtained on M has the same formula as that of pseudo-quaternionic Kaehler 4n-manifolds. From this viewpoint, we have exhibited a quaternionic analogue of Chern-Moser's CR structure.(2) Long and Reid have shown that the diffeomorphism class of every Riemannian flat manifold of dimension n>2 arises as some cusp cross-section of a complete finite volume real hyperbolic orbifold. For the complex hyperbolic case, D. B. McReynolds proved that every 3-dimensional infranilmanifold is diffeomorphic to a cusp cross-section of a complete finite volume complex hyperbolic 2-orbifold. We study this realization problem by using Seifert fibration. Let π be an n-dimensional crystallographic group. Then there is a faithful representation B: π Z^n×GL (n, Z). In particular, every compact Riemannian flat orbifold R^n/π can be realized as a cusp cross-section of a complete finite volume real hyperbolic orbifold.(3) We have proved that every compact aspherical homogeneous manifold is the total space of a fibration with solv-geometry on the fibers over a base which is a locally symmetric orbifold of non-positive curvature. We construct an iterated injective Seifert fibered structure on such fibrations, and this allows to prove that every homotopy equivalence between such manifolds is induced by a diffeomorphism. In particular, two compact homogeneous aspherical manifolds are diffeomorphic if and only if their fundamental groups are isomorphic.

(1)研究了4n+3流形M上的可积非退化余维3子束D，其纤维支持4n维四元数向量空间的结构。它被认为是四元数CR结构的一种推广。我们在邻域U上局部选取一个sp(1)值的1型ω，使得Null ω= DIU，并构造（M,ω）上的曲率不变量，它的消失使平面四元数CR几何均匀化。在M上得到的不变量与伪四元数Kaehler 4n流形的不变量具有相同的公式。从这个观点出发，我们展示了一个四元数的chen - moser的CR结构的类似物。(2) Long和Reid证明了维数为n>2的每一个黎曼平面流形的微分同胚类是一个完全有限体积实双曲轨道的顶点截面。对于复双曲情况，D. B. McReynolds证明了每一个三维基础流形都是微分同态于一个完全有限体积复双曲2-轨道的顶点截面。我们用塞费特振动法研究了这个实现问题。设π是一个n维晶体群。然后有一个可靠的表示B: π Z^n×GL （n， Z）。特别地，每一个紧黎曼平面轨道R^n/π都可以被实现为一个完全有限体积实双曲轨道的顶点截面。(3)证明了每一个紧致非球面齐次流形都是在非正曲率的局部对称轨道上的纤维上的具有解几何的纤维的总空间。我们在这些纤维上构造了一个迭代内射塞弗特纤维结构，从而证明了这些流形之间的每一个同伦等价都是由一个微分同构诱导的。特别地，当且仅当两个紧致齐次非球流形的基群同构时，它们是微分同构的。