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封闭差异形式的几何特性。首先，对于对称空间M的单数共同体学组的每个发电机X，我们找到了在De Rham同构下对应X的封闭差分形式。研究的结果是，我们可以确定M的共同体结构的结构，并使用Phi进行研究。在M是复杂的投影空间或Quaternionic投射空间时，众所周知，相应的几何结构是Kaler结构和Quaternionic Kahler结构，分别为Kahler结构，在此项目中，我们已经对以下批准（1）进行了调查（1）（1）（1）（1）（1）（1）（1）（1）（1）（1）结构，（2）研究了与同类群的发电机相对应的凯利投影空间和特殊的对称空间EIII的8型，（3）计算Quaternionic Kahler对称空间的4型，该空间对应于第一个Pontrjagin类。对r^n进行校准的分类为n <少于或等于> 8，但是n <大于或等于> 9的问题很难。由于最有用的校准是高度对称的，因此我们计算了r^9和r^9和r^<10>的不变校准。与编成一个轨道的C-Manifold的diffeeMorigant群的结构。我在这个角度与福生合作。研究项目支持研究人员的以下作品：（1）Asada研究了Loop组的全球结构，（2）Mukai和Kachi计算了投射空间的同型组，（3）Tamaki研究了光谱序列。