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STUDY ON THE GROUPS OF DIFFEOMORPHISMS

Research Project

Project/Area Number 09440028
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTHE UNIVERSITY OF TOKYO

Principal Investigator

TSUBOI Takashi  University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40114566)

Co-Investigator(Kenkyū-buntansha) MINAKAWA Hiroyuki  Hokkaido University, Graduate School of Sciences, Assistant, 大学院・理学研究科, 助手 (30241300)
KAMISHIMA Yoshinobu  Kumamoto University, Faculty of Science, Associate Professor, 理学部, 助教授 (10125304)
OHSHIKA Ken'ichi  University of Tokyo, Graduate School of Mathematical Sciences, Associate Profess, 大学院・数理科学研究科, 助教授 (70183225)
SHISHIKURA Mitsuhiro  University of Tokyo, Graduate School of Mathematical Sciences, Associate Profess, 大学院・数理科学研究科, 助教授 (70192606)
MORITA Shigeyuki  University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
中山 裕道  広島大学, 総合科学部, 講師 (30227970)
水谷 忠良  埼玉大学, 理学部, 教授 (20080492)
矢野 公一  東京大学, 大学院・数理科学研究科, 助教授 (60114691)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥7,900,000 (Direct Cost: ¥7,900,000)
Fiscal Year 1998: ¥4,100,000 (Direct Cost: ¥4,100,000)
Fiscal Year 1997: ¥3,800,000 (Direct Cost: ¥3,800,000)
Keywordsdiffeomorphisms / volume preserving / Lipschitz / boundary / contact structure / Anosov / foliation / 微分同相群 / リプシッツ同相 / 力学系 / 区分線形同相 / 特性類 / 多様体 / 葉層構造
Research Abstract

We studied the groups of diffeomorphisms of the circle and the disk. In particular, we found an important relationship between the group of diffeomorphisms of the circle and the group of the area preserving diffeomorphisms of the disk. Namely using tha fact that any diffeomorphism of the boundary circle of the disk has an extention to an area preserving diffeomorphism of the disk, we wrote down the relationship between the Euler class for the group of diffeomorphisms of the circle and the Calabi invariant for the group of the area preserving diffeomorphisms of the disk. Moreover we showed the sane result for the groups of Lipschitz homeomorphisms.
We studied similar relationship between the group of diffeomorphisms of 2-sphere and that of 3-ball, between the group of diffeomorphisms of 3-sphere and that of 4-ball, or more generally, the group of diffeonorphisms of the boundary of a compact manifold and that of the manifold. We also studied the group of Lipschitz homeomorphisms and we obtained a new result on the perfectness of such groups.
Relating to the contact structure on the 3-manifolds, we investigated the generalization of the notion of the projectively Anosov flows in higher dimensions. We studied the diffeomorphism classes of such objects. We look at the algebraic models in detail. We also studied Anosov flows and found a classification for regular projectively Anosov flows without compact leaves. We also studied complex vector fields and complex contact structures.
We studied the characteristic classes for foliations and the SC^-*S algebra for the foliations. In particular we investigated the case where the foliation has transversely piecewise linear structure. We also looked at the finitely presented simple groups.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (19 results)

All Other

All Publications (19 results)

  • [Publications] TSUBOI,Takashi: "The Calabi invariant and the Euler class" Transactions Amer,Math.Soc. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] MORITA,Shigeyuki: "Casson invariant, signature defect of manifolds and the secondary characteristic classes of surface burdlic" Journal of Diff. Geom.47. 560-599 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] SHISHIKURA,Mitsuhiro: "The Haucdary.diucysion of the baundary of the Mandilbret set and Julis sats" Aun,of Math.147. 225-267 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] OHSHIKA,Kenichi: "Accnvrtyeuca theorem for Kleinian groups uclich are free products" Math.Aun.309. 53-70 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] KAMISHIMA,Yoshinobu: "Locally confoumally Kahlerian structures and unifouizaban" Proc, 1st Brazil-USA Workshop in Campinas. 174-190 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] MINAKAWA, Hiroyuki: "Classification of exotic circles of PL_ナ (S^1)" Hokkaido Math.J.26. 685-697 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] TSUBOI,Takashi: "The Calabi invariant and the Euler class" Transactions A.M.S.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] MORITA,Shigeyuki: "Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles" Journal of Differential Geometry. 47. 560-599 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] SHISHIKURA,Mitsuhiro: "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets." Ann.of Math.(2) 147, no.2. 225-267 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] OHSHIKA,Ken'ichi: "A convergence theorem for Kleinian groups which are free products." Math.Ann.309, no.1. 53-70 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] KAMISHIMA,Yoshinobu: "Locally conformally K\"ahlerian structures and uniformization" Proc.1-st Brazil-USA Workshop in Campinas. 174-190 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] MINAKAWA,Hiroyuki: "Classification of exotic circles of SPL_- (S^l) S" Hokkaido Math.J.Vol.26, No.3. 685-697 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takashi TSUBOI: "Irrational foliations of S^3×S^3" Hokkaido Math.J.27. 605-623 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Takashi TSUBOI: "The Calabi invariant and the Euler class." Transactions.Amer.Math.Soc. to appear.

    • Related Report
      1998 Annual Research Report
  • [Publications] Shigeyuki Morita: "Casson invariant Signeture defect of fromed man fold and the Secondary characteristic classes of surface bondles." J.of DiH.Geam.47. 560-599 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] Mitsuhiro Shishikura: "The Hausdorff dimension of the boundary of the Mandel brotset and Julie Sets" Ann.of Math. 147. 225-267 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiroyuki Minakawa: "Classification of exotic circles of PL_+ (S')" Hokkaido Math.J.26. 685-697 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] TSUBOI,Takeshi: "Irrational foliations of S^3×S^3" Hokkaido Math,J. (発行予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] TSUBOI,Takeshi: "The Calabi invariant and the Euler class" Transactions.A.M.S.(発行予定).

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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