Geometric reserch of closed differential forms on manifolds

流形上闭微分形式的几何研究

基本信息

批准号：
09640101
负责人：
ABE Kojun
金额：
$ 0.96万
依托单位：
SHINSHU UNIVERSITY
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

项目摘要

The purpose of this research project is to study geometric properties of closed differential forms on manifolds. First, for each generator x of the singular cohomology group of a symmetric space M, we find a closed differential form which corresponds x under the de Rham isomorphism. As a result of the study we can determine the structure of the cohomology ring of M and also investigate the global geometric structure using phi.In the case when M is the complex projective space or the quaternionic projective space, it is well known that the corresponding geometric structures are Kaler structure and quaternionic Kahler structure respectively, In the term of this project we have studied the following :(1) Compute the volumes of the symmetric structures,(2) Study the 8-form on Cayley projective space and exceptional symmetric space EIII which corresponding to the generator of the cohomology group,(3) Compute the 4-forms on quaternionic Kahler symmetric space which correspond to the first Pontrjagin classes.Next we studied the calibration on R^n. The classifications of the calibrations on R^n are given for n <less than or equal> 8 but the problem is difficult for n <greater than or equal> 9. Because the most useful calibrations on are highly symmetric we calculate the invariant calibrations on R^9 and R^<10> under the orthogonal groups.The above problems are motivated by studing the differentiable structure of-the orbit structure of C-manifolds, In the term of this project we also studied the structure of the equivariant diffeomorphism groups of a C-manifold with codimension one orbit. I collaborated with Fukui in this point of view.The research project supported the following works by the investigators :(1) Asada studied the global structure of the loop groups,(2) Mukai and Kachi computed the homotopy groups of the projective spaces,(3) Tamaki studied the spectral sequences.

该研究项目的目的是研究流形上闭微分形式的几何性质。首先，对于对称空间 M 的奇异上同调群的每个生成元 x，我们找到一个闭微分形式，它对应于 de Rham 同构下的 x。通过研究，我们可以确定 M 的上同调环的结构，并利用 phi 研究全局几何结构。当 M 为复射影空间或四元射影空间时，众所周知，相应的几何结构分别是卡勒结构和四元数卡勒结构，在本项目期间我们研究了以下内容：（1）计算对称结构的体积，（2）研究凯莱射影空间上的8-形式和对应于上同调群的生成元的例外对称空间EIII，(3)计算对应于第一类Pontrjagin类的四元数Kahler对称空间上的4-形式。接下来我们研究了R^n的校准。 R^n 上的校准分类是针对 n <小于或等于> 8 给出的，但对于 n <大于或等于> 9 来说，问题很困难。因为最有用的校准是高度对称的，所以我们计算了不变校准正交群下的R^9和R^<10>。上述问题是通过研究C-流形的轨道结构的可微结构引发的，在本项目期间我们还研究了余维一轨道的 C 流形的等变微分同胚群。在这个观点上我与 Fukui 合作。该研究项目支持研究人员的以下工作：（1）Asada 研究了环群的全局结构，（2）Mukai 和 Kachi 计算了射影空间的同伦群，（ 3) Tamaki 研究了光谱序列。