Geometric reserch of closed differential forms on manifolds

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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.