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2001 Fiscal Year Final Research Report Summary

Global Research of Geometry related with Poisson and Contact Manifolds.

Research Project

Project/Area Number 11640060
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionSaitama University

Principal Investigator

MIZUTANI Tadayoshi  Saitama University, Dept. of Math., Professor, 理学部, 教授 (20080492)

Co-Investigator(Kenkyū-buntansha) NAGASE Masayoshi  Saitama Univ., Dept. of Math., Professor, 理学部, 教授 (30175509)
SAKAMOTO Kunio  Saitama Univ., Dept. of Math., Professor, 理学部, 教授 (70089829)
OKUMURA Masafumi  Saitama Univ., Dept. of Math., Professor, 理学部, 教授 (60016053)
FUKUI Toshizumi  Saitama Univ., Dept. of Math., Associate Professor, 理学部, 助教授 (90218892)
TAKEUCHI Kisao  Saitama Univ., Dept. of Math., Professor, 理学部, 教授 (00011560)
Project Period (FY) 1999 – 2001
KeywordsNambu-Poisson manifold / Leibniz algebra / Leibniz cohomology / central extension / singular foilation / Pfaff system
Research Abstract

In the first year of the term of the project, we investigated Nambu-Jacobi manifolds and gave a characterization of such manifolds interms of multi-vector fields. This result is written in the preprint Foliations assocaited with Nambu-Jacobi structures which is a joint paper with K. Mikami(Akita University).
In the second and the third year of the project, we were concerned with two topics. The one is the Leibniz algebra associated with a Nambu-Poisson manifold. We first observed that given a decomposable integrable p-form, the space of p+1-vector fields on the manifold have a structure of Leibniz algebra. Further we observed that this algebra structure depends only on the diffeomorphism class of the foliation defined by the p-form. Also, there is a natural Leibniz homomorphism from this algebra to the Lie algebra which is formed by the vector fields tangent to the foliation. As in the case of Lie algebras, this extension of algebra corresponds to a 2-dimensional cocycle of a Leibniz cohomology. These results are contained in the paper Y. Hagiwara-Tmizutani "Leibniz algebras associated with foliations" The other is study of the Pfaff system regarding it as a submanifold of the symplectic manifold T^*M. A. typical result of this direction is that the Pfaff system is completely integrable if it is a coisotropic submanifold of T^*M. From this vie point we described the Godbillon-Vey class as a intersection of certain naturally defined multi-vector fields.

  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] Tadayoshi Mizutani: "On Exact Poisson Manifolds of Dimension 3"Proceedings of FOLIATIONS : GEOMETRY AND DYNAMICS (ed. by P. Walczak, et al.). 371-386 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Fakui, T-C.Kuo, L.Paunescu: "Constructing Blow-analytic Isomorphisins"Ann. Inst. Fourier, Grenoble. 51. 1071-1087 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T Fukui, L.Paunescu: "Stratification Theory from the Weighted Point of View"Canadian Journal of Mathematics. 53. 73-97 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masayoshi Nagase: "Twistor space and the Seiberg-Witten equation"Saitama Mathematical Journal. 18. 39-60 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masayoshi Nagase: "The adiabatic limits of signature operators for Sping manifolds"Osaka Journal of Mathematics. 38. 541-564 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masayoshi Nagase: "Twistor spaces and the adiabatic limits of Dirac operators"Nagoya Mathematical Journal. 164. 53-73 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 徳永浩雄, 島田伊知朗, 石川剛郎, 齋藤幸子, 福井敏純: "特異点の数理4 代数曲線と特異点"共立出版. 384 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Mizutani: "On Exact Poisson Manifolds of Dimension 3"Proc. of FOLIATIONS : GEOMETRY AND DYNAMICS (ed. by P. Walczak et al) World Scientific. 371-386 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Nagase: "Twistor space and the Seiberg-Witten equation"Saitama Math. J.. 18. 39-60 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Nagase: "Twistor spaces and the adiabatic limits of Dirac operators"Nagoya Math. J.. 164. 53-73 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Nagase: "The adiabatic limits of signature operators for Spin manifolds"Osaka J. of Math.. 38. 541-564 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Djoric, M. Okumura: "0n contact submanifolds in complex projective space"Math. Nachr.. 202. 17-23 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Okumura: "CR submanifolds of maximal CR dimension of complex projective space"Bull. of the Greek Math. Soc.. 44. 31-39 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Fukui, J. Weyman: "Cohen-Macauley properties of Thom-Boardman strata I : Morin's ideal"Proc. London Math. Soc.. 80. 257-303 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Fukui, L. Paunescu: "Stratification theory from the weighted point of view"Canadian J. of Math.. 53. 73-97 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Fukui, T-C. Kuo, L. Paunescu: "Constructing Blow-analytic Isomorphisms"Ann. Inst. Fourier, Grenoble. 51. 1071-1087 (2001)

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

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Published: 2003-09-17  

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