Project/Area Number  09640101 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  SHINSHU UNIVERSITY 
Principal Investigator 
ABE Kojun Shinshu University, Faculty of Science, Professor, 理学部, 教授 (30021231)

CoInvestigator(Kenkyūbuntansha) 
MATSUDA Toshimitsu Shinshu University, Faculty of Science, Associate Professor, 理学部, 助教授 (70020667)
KACHI Hideyuki Shinshu University, Faculty of Science, Associate Professor, 理学部, 助教授 (50020657)
二宮 晏 信州大学, 理学部, 教授 (40092887)
MUKAI Juno Shinshu University, Faculty of Science, Professor, 理学部, 教授 (50029675)
ASADA Akira Shinshu University, Faculty of Science, Professor, 理学部, 教授 (00020652)
TAMAKI Dai Shinshu University, Faculty of Science, Lecturer, 理学部, 講師 (10252058)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  manifold / differential form / Cayley projective space / calibration / transformation group / 多様体 / 微分形式 / Caybey射影空間 / 変換群 / Cayley射影空間 / ポントリャギン形式 / G多様体 
Research Abstract 
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 8form on Cayley projective space and exceptional symmetric space EIII which corresponding to the generator of the cohomology group, (3) Compute the 4forms 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 ofthe orbit structure of Cmanifolds, In the term of this project we also studied the structure of the equivariant diffeomorphism groups of a Cmanifold 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.
