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维平滑歧管上的CR结构由一对（Omega，j）组成，满足（i）Omega是M的接触形式，并且（ii）让Null Omega = {x * tm * tm * omega（x）= 0}，这是一个a sodimension 1 sobimension 1 subimension 1 subimension 1 sub-bundle the Pttent bundle tm bundle tm bundle。然后在Null Omega上有一个复杂的结构J。即J是零欧米茄上的几乎复杂的结构，当Null Omega <Cross Product> C = T^<1,0> + T^<0,1>是规范分裂时，因此[T^<1,0>，T^<1,0>] * T^<1,0>。另外，如果Hermitian配对Q：T^<10> Xt^<1,0> - > C（由Q（x，y）= d-omega（x，jy）定义为正定义，则严格是伪convex。 1958年，布斯比（Boothby）和王（Wang）在光滑的歧管上引入了常规接触结构，并建立了振动定理。常规的CR结构将定义为CR结构，其基础接触结构是常规的。然后，布斯比和王的结果将被概括如下：M^<2n+1>在且仅当M是主要圆圈bundle pi：m-> n时，在Kahler歧管上，其基本2形式的Omega满足以下属性；（1）捆绑包的欧拉类由整体共生[Omega] * H^2（N; Z）表示。（2）deta = pi^*欧米茄，其中eta是M的连接形式。