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A study on foliations, contactstrucures, and symplectic styructures on 3 and 4 dimensional manifolds

Research Project

Project/Area Number 16540080
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

MITSUMATSU Yoshihiko  Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70190725)

Co-Investigator(Kenkyū-buntansha) MIYOSHI Shigeaki  Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (60166212)
TAKAKURA Tatsuru  Chuo University, Faculty of Science and Engineering, Associate Professor, 理工学部, 助教授 (30268974)
MATSUMOTO Shigenori  Nihon University, Faculty of Science and Technology, Professor, 理工学部, 教授 (80060143)
TSUBOI Takashi  University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40114566)
ONO Kaoru  Hokkaido University, Graduate School Science, Professor, 大学院・理学研究科, 教授 (20204232)
森吉 仁志  慶應義塾大学, 理工学部, 助教授 (00239708)
Project Period (FY) 2004 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2004: ¥2,000,000 (Direct Cost: ¥2,000,000)
Keywordsfoliation / contact structure / Anosov flow / Stein filling / spinnable structure / symplectic structure / Reeb vector field / Hamiltonian dynamical system / symplectic構造 / 射影的Anosov流 / open book分解 / Thurston(Bennequin)の不等式
Research Abstract

The head Mitsumatsu and an investigator Miyoshi colaborated with others to study the euler class of tangent bundles to foliations and (so called Thurston-Winkelnkemper's) contact structures which are associate with spinnable structures, as a typical class of convergences of contact structures to foliations. Especially they studied the (non-)vanishing of the euler class and the violation of Thurston's inequality and Bennequin's inequlity, from the topological view point of monodromies. As a consequence, a certain class of mapping classes of a surface with boundary can be presented neither as a product of only right-handed Dehn twists nor as that of only right-handed ones. This result was presented in several symposiums including the annual meeting of MSJ in March 2006 as a special invited talk by Miyoshi. The paper is under submission.
The investigator Ono studied the symplectec homology from Floer theory as well as from Seiberg-Witten theory. Including the solution to the Flux conjecture as well as the detemination of the symplectic filling of the link of simple singularities, his contributions to this area are profound.
The investigator Tsuboi studied the relationship between foliation theory and that of contact structures from the view point of the group of contact diffeomorphisms. The investigator Matsumoto stepped further to the foliation theory and studied the ends of Lie foliations.
The head investigator also studied the incompressible fluid dynamics in the framework of the geometry of volume preserving diffeomorphisms and infinite dimensional Hamiltonian systems. He proved that looking from the point of view of such global differential geometry is still valid even to viscous fluides with dissipations. This study is presented in many symposiums, especially as a special project talk in the annual meeting of MSJ in September 2005.

Report

(3 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • Research Products

    (16 results)

All 2006 2005 2004 Other

All Journal Article (16 results)

  • [Journal Article] Convergence of contact structures to foliations, with an Appendex : On Bennequin's isotopy lemma by Y. MITSUMATSU and Atsuhide MORI2006

    • Author(s)
      Yoshihiko MITSUMATSU
    • Journal Title

      Proceeding of Foliation 2005 to appear

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Convergence of contact structures to foliations, with an Appendix : On Bennequin's isotopy lemma by Y.MITSUMATSU and Atsuhide MORI2006

    • Author(s)
      Yoshihiko MITSUMATSU
    • Journal Title

      Proceedings of Foliation 2005 (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Ends of leaves of lie foliations2005

    • Author(s)
      G.Hector, S.Matsumoto, G.Meigniez
    • Journal Title

      J. Math. Soc. Japan 57

      Pages: 753-779

    • NAID

      10017177824

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Simple singularities and symplectic fillings2005

    • Author(s)
      Hiroshi Ohta, Kaoru Ono
    • Journal Title

      J. Differential Geom. 69

      Pages: 1-42

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Symplectic 4-manifolds contaning singular rational curves with (2,3)-cusp2005

    • Author(s)
      Hiroshi Ohta, Kaoru Ono
    • Journal Title

      Seminaire et Congres, Societe Mathematique de France 5

      Pages: 233-241

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A short note on symplectic Floer theory2005

    • Author(s)
      Kaoru Ono
    • Journal Title

      ``Noncommutative Geometry and Physics'', World Soientific

      Pages: 301-319

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Ends of leaves of Lie foliations2005

    • Author(s)
      G.Hector, S.Matsumoto, G.Meignie
    • Journal Title

      J.Math.Soc.Japan 57

      Pages: 753-779

    • NAID

      10017177824

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Simple singularities and symplectic fillings2005

    • Author(s)
      Hiroshi Ohta, Kaoru Ono
    • Journal Title

      J.Differential Geom. 69

      Pages: 1-42

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] A short note on symplectic Floer theory2005

    • Author(s)
      Kaoru Ono
    • Journal Title

      Noncommutative Geometry and Physics(World Scientific)

      Pages: 301-319

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Ends of leaves of Lie foliations2005

    • Author(s)
      G.Hector, S.Matsumoto, G.Meigniez
    • Journal Title

      J.Math.Soc.Japan 57

      Pages: 753-779

    • NAID

      10017177824

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Symplectic 4-manifolds contaning singular rational curves with (2,3)-cusp2005

    • Author(s)
      Hiroshi Ohta, Kaoru Ono
    • Journal Title

      Seminaire et Congres, Societe Mathematique de France 10

      Pages: 233-241

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A short note on symplectic Floer theory2005

    • Author(s)
      Kaoru Ono
    • Journal Title

      "Noncommutative Geometry and Physics", World Scientific

      Pages: 301-319

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On representability of the smooth Euler class2004

    • Author(s)
      Miyoshi, Shigeaki
    • Journal Title

      Tohoku Math.J. 56-4

      Pages: 523-530

    • NAID

      110001043907

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Regular projectively Anosov flows on the Seifert fibered 3-manifolds2004

    • Author(s)
      Tsuboi, Takashi
    • Journal Title

      Journal of the Mathematical Society of Japan 56-4

      Pages: 1233-1253

    • NAID

      10013753590

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Appendix to "Restrictions on symplectic fibrations"2004

    • Author(s)
      Kcedra, Jaroslaw, Kaoru Ono
    • Journal Title

      Differential Geom.Appl. 21-1

      Pages: 108-112

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Convergence of contact structures to foliations, with an Appendix : On Bennequin' s isotopy lemma by Y.MITSUMATSU and Atsuhide MORI

    • Author(s)
      Yoshihiko MITSUMATSU
    • Journal Title

      to appear in Proceedings of Foliation 2005, World Scientific

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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