Geometric reserch of closed differential forms on manifolds

本研究计画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x下的de Rham同构。研究的结果是确定了M的上同调环的结构，并利用φ研究了M的整体几何结构，当M是复射影空间或四元数射影空间时，其对应的几何结构分别是Kaler结构和四元数Kahler结构。（1）计算对称结构的体积，（2）研究Cayley射影空间和例外对称空间EIII上对应于上同调群生成元的8-形式，（3）计算四元数Kahler对称空间上对应于第一类Pontrjagin类的4-形式。给出了n = 8时R ^n上标定的分类，但n = 9时标定的分类问题比较困难.<less than or equal><greater than or equal>由于上最有用的标度是高度对称的，所以我们计算了R ^9和R ^9上的正交群下的不变标度，上述问题是由研究C-流形的轨道结构的可微结构所激发的，在本项目中我们还研究了余维为1轨道的C-流形的等变同态群的结构。<10>我和福井在这一点上进行了合作。该研究项目支持了研究人员的以下工作：（1）浅田研究了循环群的整体结构，（2）Mukai和Kachi计算了投射空间的同伦群，（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.