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

Study of smooth non-compact Lie group actions on compact smooth manifolds

紧光滑流形上的光滑非紧李群作用研究

基本信息

批准号：
09640140
负责人：
MUKOYAMA Kazuo
金额：
$ 1.02万
依托单位：
Tokyo Metropolitan College of Aeronautical Engineering
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640140/
关键词：
smooth action non-compact Lie group orbit space

项目摘要

It is important to study non-compact Lie group actions on compact smooth manifolds.In this note, we classify smooth SU(p, 1)-actions on the (2p+2q-1)-sphere and on the complex projective (p+q-1)-space whose restriction to the maximal compact subgroup S(U (p) X U (q)) is standard.The results are as follows.Let U, V be the set of smooth SU(p, q)-actions on P_<p+q-1> (C), S^<2p+2q-1> whose restricted S(U(p) X U(q))-action is standard, respectively.1. There is a one-to-one correspondence between U and the set of pairs (phi', f' ), where phi' is a smooth N'(p, q)=SU(1,1)-action on P_1(C) whose restriction on S(U (p) X U (q))*N'(p, q) is standard and f : P_1(C)*P_1(C) is a smooth N'(p, q)-equivariant map satisfying one condition.From this fact we see that there exist infinitely many smooth actions on P_<p+q-1>(C) which are mutually distinct.2. There is a one-to-one correspondence between V and the set of pairs (phi, f) , where phi is a smooth N(p, q)=U(1, 1)-action on S^3 whose restriction on S(U(p) X U(q)) *N(p, q) is standard and f : S^3*P_1(C) is a smooth N(p, q)-equivariant map satisfying one condition.3. There exists a mapping rho : V*U which is onto but not one-to-one.

研究紧光滑流形上的非紧李群作用具有重要意义。本文对（2p+2q-1）球面上的光滑SU(p, 1)-作用和复射光(p+q-1)-空间上的光滑SU(p, 1)-作用进行了分类，该空间对最大紧子群S(U （p) X U(q)）的限制是标准的。结果如下：设U， V分别为P_<p+q-1> (C), S^<2p+2q-1>上的光滑SU(p, q)-作用的集合，它们的受限S(U(p) X U(q))-作用是标准的。U与对集（phi', f‘）之间存在一一对应关系，其中phi’是一个光滑的N'(p, q)=SU（1,1）作用于P_1(C)，其对S(U (p) X U(q))*N‘（p, q）的约束是标准的，f: P_1(C)*P_1(C)是一个满足一个条件的光滑的N’（p, q）等变映射。由此可见，在P_<p+q-1>(C)上存在无穷多个相互不同的光滑作用。在V和对集合（,f）之间存在一一对应关系，其中，是一个光滑的N(p, q)=U（1,1）作用于S^3，其对S(U(p) X U(q)) *N（p, q）的限制是标准的，f: S^3*P_1(C)是一个光滑的N(p, q)-满足一个条件的等变映射。存在一个映射V*U它是映上的，但不是一对一的。