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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-流形 M 上的可积非简并余维 3 -子丛 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 CR 结构的推广。我们在邻域 U 上挑选出一个 sp (1) 值 1-形式 ω，使得 Null ω= DIU 并构造 (M,ω) 上的曲率不变量，其消失给出了平坦四元数 CR 几何的均匀化。 M 上获得的不变量与伪四元数 Kaehler 4n 流形的公式相同。从这个角度来看，我们展示了 Chern-Moser CR 结构的四元数类似物。(2) Long 和 Reid 证明，每一个维数 n>2 的黎曼平流形的微分同胚类都是作为完整的有限体积实双曲轨道的一些尖点横截面而出现的。对于复双曲情况，D. B. McReynolds 证明了每个 3 维下流形都微分同胚于完整有限体积复双曲 2 轨道的尖点横截面。我们通过使用 Seifert 纤维来研究这个实现问题。令 π 为 n 维晶体群。那么就有一个忠实的表示B：πZ^n×GL(n,Z)。特别是，每个紧致黎曼平面轨道折叠 R^n/π 都可以实现为完整有限体积实双曲轨道折叠的尖点横截面。 (3) 我们证明了每个紧致非球面齐次流形都是以非正曲率局部对称轨道折叠为基底的纤维上具有求解几何的纤维振动的总空间。我们在这种纤维上构建了一个迭代的单射 Seifert 纤维结构，这可以证明这些流形之间的每个同伦等价都是由微分同胚引起的。特别地，两个紧齐次非球面流形是微分同胚的当且仅当它们的基本群是同构的。