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1998 Fiscal Year Final Research Report Summary

Geometric reserch of closed differential forms on manifolds

Research Project

Project/Area Number 09640101
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionSHINSHU UNIVERSITY

Principal Investigator

ABE Kojun  Shinshu University, Faculty of Science, Professor, 理学部, 教授 (30021231)

Co-Investigator(Kenkyū-buntansha) TAMAKI Dai  Shinshu University, Faculty of Science, Lecturer, 理学部, 講師 (10252058)
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)
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)
Project Period (FY) 1997 – 1998
Keywordsmanifold / differential form / Cayley projective space / calibration / transformation group
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 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.

  • Research Products

    (18 results)

All Other

All Publications (18 results)

  • [Publications] K.Abe: "Realization of spaces E_6/(U(1)Spin(10)),E_7/(U(1)E_6),E_8/(U(1)E_7)and their volumes." Tokyo Jour.Math.Vol20. 73-86 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Abe: "Volumes of compact symmetric spaces." Tokyo Jour.Math.Vol20. 87-105 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Abe: "Invariant forms on the exceptional symmetric spaces FII and EIII." Proceedings of 1996 Korea-Japan conference on transformation group theory. 1-15 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Asada: "Hodge operators of mapping spaces." Group21,Physical Applications and Mathematical Aspects of Geometry,Groups and Algebras,. 925-928 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Asada: "Hodge operators over mapping spaces,Local study." BSG Proc.Vol1. 11-20 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Asada: "A remark on infinite dimensional Gaussina integrsal on a Sobolev space." J.Fac.Sci.Shinshu univ.Vol32. 61-67 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] J.Mukai: "Some homotopy groups of the double suspension of the real projective space RP^6" Matematica Contemporanea. Vol13. 235-249 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] J.Mukai: "Cohomotopy sets of projective planes." Jour.Fac.Sci.Shinshu Univ.Vol33. 1-7 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Kachi: "Some homotopy groups of the rotation group R_n" to appear in Hiroshima Math.J.Vol29. (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Abe: "E_6/ (U (1) Spin (10)), E_7/ (U (1) E_6), E_8/ (U (1) E_7) and their volumes." Tokyo Jour.Math.vol 20. 73-86 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Abe: "Volumes of compact symmetric spaces." Tokyo Jour.Math.vol 20. 87-105 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Abe: "Invariant forms on the exceptional symmetric spaces FII and EIII." Proceedings of 1996 Korea-Japan conference on transformation group theory. 1-15 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Asada: "Hodge operators of mapping spaces." Group21, Physical Application-s and Mathematical Aspects of Geometry, Groups and Algebras, World Sci.925-928 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Asada: "Hodge operators over mapping spaces, Local study." BSG Proc.1. 11-20 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Asada: "A remark on infinite dimensional Gaussina integrsal on a Sobolev space." J.Fac.Sci.Shinshu univ.32. 61-67 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J.Mukai: "Some homotopy groups of the double suspension of the real projective space RP^6." Matematica Contemporanea. 13. 235-249 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J.Mukai: "Cohomotopy sets of projective planes." Jour.Fac.Sci.Shinshu Univ.33. 1-7 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Kachi: "Some homotopy groups of the rotation group R_n." to appear in Hiroshima Math.J.29. (1999)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1999-12-08  

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