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Research on Geometric Structures, Schwarzian Derivatives of Maps and Partial Differential Equations

Research Project

Project/Area Number 16540085
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

OZAWA Tetsuya  Meijo University, Faculty of Science and Technology, Professor (20169288)

Co-Investigator(Kenkyū-buntansha) OKAMOTO Kiyosato  Meijo University, Faculty of Science and Technology, Professor (60028115)
HASHIMOTO Hideya  Meijo University, Faculty of Science and Technology, Professor (60218419)
KATO Yoshifumi  Meijo University, Faculty of Science and Technology, Assistant Professor (40109278)
TSUKAMOTO Michiro  Meijo University, Faculty of Science and Technology, Lecturer (80076637)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥2,320,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥120,000)
Fiscal Year 2007: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2006: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
KeywordsContact Structure / CR structure / Conformal Structure / Schwarzian Derivative / Partial Differential Equation / Normal Connection / Heisenberg Frame Buncle / Zero Curvature Condition / 射影構造
Research Abstract

1. The existence and the uniqueness of the so-called Tanaka-Webster connection of a CR structure was reproved by using a Heisenberg frame bundle, in consideration that specific connections are expected to play an important role as the Schwarzian derivatives of transformations in various geometric structures.
2. On the underlying contact structure of a CR structure, a necessary and sufficient condition of a Hamiltonian function to generate a contact transformation that preserves the CR structure is established.
3. su_2 frame bundle was examined in detail, and shown to be useful in CR geometry, for example, in analyzing the Tanaka-Webster connections.
4. It is shown that the deformation parameter of CR structure is explained by complex valued functions, which was used to exhibit the Levi form and the connection coefficients in precise form.
5. The notion of Schwarzian derivative for higher dimensional contact transformations was established, which was used to solve an equivalence problem of a certain system of partial differential equations.
6. The notion of Schwarzian derivative in CR geometry for strictly contact transformations was established. As the result, the fundamental equation of 3-dimensional CR structure was obtained, and a natural Hermitian structure on the solution space was detected. Finally the equivalence problem of 3 dimensional CR structure was solved.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (7 results)

All 2007 2006

All Journal Article (2 results) (of which Peer Reviewed: 1 results) Presentation (4 results) Book (1 results)

  • [Journal Article] Differential Equations and Schwarzian Derivatives2007

    • Author(s)
      Hajime SATO, Tetsuya OZAWA, Hiroshi SUZUKI
    • Journal Title

      Noncommutative Geometry and Physics 2005, Proceedings of the International Sendai-Beijing Joint Workshop

      Pages: 129-149

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Differential Equations and Schwarzian Derivatives2007

    • Author(s)
      Hajime SATO, Tetsuya OZAWA, Hiroshi SUZUKI
    • Journal Title

      Noncommutative Geometry and Phsics 2005, Proceedings of the International Sendai-Beijing Joint Workshop

      Pages: 129-149

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 部分複素幾何学と基本方程式2007

    • Author(s)
      小沢 哲也
    • Organizer
      シンプレクティック幾何学とその周辺
    • Place of Presentation
      岐阜経済大学
    • Year and Date
      2007-11-12
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] bubcomplex structures and the tundamental equations II2007

    • Author(s)
      Tetsuya OZAWA
    • Organizer
      Symplectic geometry and the surrounaings
    • Place of Presentation
      Gifu Keizai University
    • Year and Date
      2007-11-12
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 部分複素幾何学と基本方程式2006

    • Author(s)
      小沢 哲也
    • Organizer
      シンプレクティック幾何学とその周辺
    • Place of Presentation
      秋田大学
    • Year and Date
      2006-11-15
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] ouncomplex structures and the tunaamental equations.2006

    • Author(s)
      Tetsuya OZAWA
    • Organizer
      Sympiectic geometry and the surrounaings
    • Place of Presentation
      Akita University
    • Year and Date
      2006-11-15
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] Noncommutative Geometry and Physics 2005 : Proceedings of Int'l Sendai-Beijing Joint Workshop,2005.2007

    • Author(s)
      Watamura et al.(eds.)
    • Total Pages
      350
    • Publisher
      World Scientific Pub.
    • Related Report
      2006 Annual Research Report

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

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