Geometric Structures on Manifolds and Representations of Fundamental Group

流形上的几何结构和基本群的表示

基本信息

批准号：
01540001
负责人：
KAMISHIMA Yoshinobu
金额：
$ 1.41万
依托单位：
Department of mathematics, Kumamoto University (1990)Hokkaido University (1989)
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1989
资助国家：
日本
起止时间：
1989 至 1990
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-01540001/
关键词：
CR-structure Lorentz structure Holonomy group Fundamental group Spherical CR structure Conformally flat structure One parameter group Topological rigidity ロ-レンツ構造ホロノミ-群共形平坦構造 oneパラメ-タ群 Topological viyidity CR-構造 CR-多様体 Scho

项目摘要

A CR-structure on a 2n + 1 dimensional smooth manifold M consists of a pair (omega, J) satisfying that (i) omega is a contact form of M , and (ii) Let Null omega = {X * TM* omega (X) = 0} which is a codimension 1 sub-bundle of the tangent bundle TM. Then there is a complex structure J on Null omega. Namely J is an almost complex structure on Null omega and when Null omega <cross product> C = T^<1,0> + T^<0,1> is the canonical splitting, it follows that [T^<1,0>, T^<1,0>] * T^<1,0>. In addition a CR-structure is strictly pseudo-convex if the Hermitian pairing Q : T^<10>XT^<1,0> -> C defined by Q (X, Y) = d-omega (X, JY) is positive definite. In 1958 Boothby and Wang introduced the regular contact structure on smooth manifolds and they have established the vibration theorem. A regular CR-structure will be defined as a CR-structure whose underlying contact structure is regular. Then the Boothby and Wang's result will be generalized as follows : M^<2n+1> admits a regular CR-structure if and only if M is a principal circle bundle pi : M -> N over a Kahler manifold N whose fundamental 2-form OMEGA satisfies the following properties ; (1) the euler class of the bundle is represented by an integral cocycle [OMEGA] * H^2 (N ; Z). (2) deta = pi^*OMEGA where eta is a connection form of M.

2n + 1 维光滑流形 M 上的 CR 结构由一对 (omega, J) 组成，满足 (i) omega 是 M 的接触形式，并且 (ii) 令 Null omega = {X * TM* omega ( X) = 0} 是切丛 TM 的余维 1 子丛。那么Null omega上有一个复杂的结构J。即 J 是 Null omega 上的近复结构，当 Null omega <叉积> C = T^<1,0> + T^<0,1> 是规范分裂时，可以得出 [T^<1,0 >, T^<1,0>] * T^<1,0>。此外，如果由 Q (X, Y) = d-omega (X, JY) 定义的埃尔米特配对 Q : T^<10>XT^<1,0> -> C 为正定的。 1958年Boothby和Wang引入了光滑流形上的正则接触结构，并建立了振动定理。规则的CR结构将被定义为其底层接触结构是规则的CR结构。那么 Boothby 和 Wang 的结果将推广如下： M^<2n+1> 承认正则 CR 结构当且仅当 M 是主圆丛 pi : M -> N 在卡勒流形 N 上，其基本 2- OMEGA 型满足以下特性； (1) 丛的欧拉类由积分余环 [OMEGA] * H^2 (N ; Z) 表示。 (2) deta = pi^*OMEGA 其中eta是M的连接形式。