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）我们研究了4N+3-歧管M上的可集成的，非构层的编成3-刺激性D，其纤维支持4n维Quaternionic载体矢量空间的结构。它被认为是Quaternionic CR结构的概括。我们在邻里u上挑出一个sp（1）值1形式的ω，以使nullΩ= diu并在（m，ω）上构建曲率不变性，其消失对平坦的quaternionic cr几何形状均匀化。在M上获得的不变式的公式与伪Qualtrnionic Kaehler 4n-Manifolds的公式相同。从这个角度来看，我们表现出了Chern-Moser Cr结构的四元离子类似物。（2）长和Reid表明，每一个riemannian平面differism n> 2的差异性类别是出现的，因为完整有限体积的实际体积实际超纤维孔的某些尖端横截面。对于复杂的双曲线病例，D。B。McReynolds证明，每个3维的基型基曼植物对完整有限体积复合物复合双波利2孔的尖端横截面具有差异性。我们通过使用Seifert纤维来研究这个实现问题。令π为N维晶体学组。然后是一个忠实的表示b：πz^n×gl（n，z）。特别是，每一个紧凑的riemannian平坦的圆柱r^n/π都可以作为尖端的横截面的横截面，是一个完整的有限体积真正的多毛孔。（3）我们规定，每个紧凑的非球形同质歧管都是在基础上进行纤维的整个元素的启发的总空间。我们在此类纤维上构造了一个迭代的注射式塞菲尔特纤维结构，这允许证明这种歧管之间的每个同型都由差异性诱导。特别是，当且仅当它们的基本组是同构时，两个紧凑的同质非球形歧管是差异的。