Geometric reserch of closed differential forms on manifolds

本课题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的奇异上同群的每个生成子x，我们找到了一个在de Rham同构下对应x的闭微分形式。通过研究，我们可以确定M的上同环的结构，并利用φ来研究其整体几何结构。当M为复射影空间或四元数射影空间时，众所周知，对应的几何结构分别为Kaler结构和四元数Kahler结构；本课题研究了以下内容：(1)计算对称结构的体积；(2)研究了对应于上同调群生成的Cayley投影空间和例外对称空间EIII上的8型；(3)计算了对应于第一类Pontrjagin类的四元Kahler对称空间上的4型。接下来我们研究了对R^n的标定。给出了n <小于等于bbbb8时对R^n标定的分类，但当n <大于等于bbbb9时，问题比较困难。由于在高度对称上最有用的标定，我们在正交群下计算了R^9和R^<10>上的不变标定。上述问题的动机是研究c流形的轨道结构的可微结构，本课题还研究了余维为1轨道的c流形的等变微分同构群的结构。在这一点上，我与福井进行了合作。本项目支持研究者的工作：(1)Asada研究了环群的整体结构，(2)Mukai和Kachi计算了射影空间的同伦群，(3)Tamaki研究了谱序列。

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.