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Mutual Invariance between Geometric Structures and Toplogical Structures on Manifolds

Research Project

Project/Area Number 06640161
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionKumamoto University

Principal Investigator

KAMISHIMA Yoshinobu  Departement of Mathematics, Kumamoto Universisty, Associate Professor of Mathematics, 理学部, 助教授 (10125304)

Co-Investigator(Kenkyū-buntansha) OKA Yukimasa  KUMAMOTO,University Associate Professor of Mathematics, 理学部, 助教授 (50089140)
YOSHIDA Tomoyukii  HOKKAIDO,University Professor of Mathematics, 理学部, 教授 (30002265)
Project Period (FY) 1994 – 1995
Project Status Completed (Fiscal Year 1995)
Budget Amount *help
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1995: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1994: ¥1,400,000 (Direct Cost: ¥1,400,000)
KeywordsWeyl Curvature tensor / Symplectic structure / Pseoudo-Hermiyian structure / Uniformization / Geometric structure / Vanishing curvature / CR-structure / Conformal atructure / Contactization / CR-structure / Contactization / pseudo-Hermitian構造 / Contact form / Weyl曲率tensor / ローレンツ計量 / モデル空間(G,X) / 曲率関数Λ / uniformization / CR構造
Research Abstract

We have observed the ralation between geometric and topological structures on smooth manifolds. H.Weyl has introduced the notion of conformal structure from the viewpoint of the Gauge theory. And he found an invariant on conformal structure, which is now called Weyl Conformal Curvature Tensor. It is the fundamental result in differential geometry that the Weyl conformal curvature tensor of an n-dimensional Riemannian manifold M^n (n>3) vanishes if and only if M^n is locally conformally equivalent to the flat euclidean space. Along this line but only Riemannian geometry, we have examined the invariance of conformal sturusture to other geometric structures. More precisely, as a geometric structure and a conformal invariant to even (resp. odd) dimensional manifolds, we brougth into focus Kahler manifolds for which the Bochner curvature tensor has been defined and CR-manifolds for which the Chern-Moser-Webster curvature tensor has been defined respectively. We shall define a conformal equivalence to the given geometric structure, and then construct a conformal invariant (tensor) on it.When that invariant vanishes, we observed what kind of new (or classical) geometry (G,X) comes out. Similtaneously, we have obtained a classification theorerm that such a manifold with vanishing invariant tensor can be uniformized with respect to the model space (G,X).

Report

(3 results)
  • 1995 Annual Research Report   Final Research Report Summary
  • 1994 Annual Research Report
  • Research Products

    (25 results)

All Other

All Publications (25 results)

  • [Publications] 神島芳宣: "On the 3-dimensional pseudo-Hermitian space forms and other geometric structures" Kumamoto J.of Math.7. 71-93 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 神島芳宣: "Standard pseudo-Hermitian structure and Seifert fibration on CR manifold" Annals of Global Analysis and Geometry. 12. 261-289 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 神島芳宣: "Uniformization of Kahler manifolds with vanishing Bochner tensor" Acta Mathematica. 172. 299-308 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 神島芳宣: "Pseudo-Hermitian structure on manifolds from Riemannian geometry" Differential Geometry and Related Topics Proceedings of Workshops in Pure Math. Part III. 13. 165-213 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 神島芳宣: "Geometric flows on compact manifolds and Global rigidity" to appear in Topology. (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 神島芳宣: "Topolgy of CR-manifolds and Kahler manifolds" Symplectic Geometry and Related Topics to appear in Proceedings of Workshops in Pure Math.(1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 神島芳宣: "to appear in Kumamoto J. of Math." Transformation groups on Heisenbery geometry,(1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y.Kamishima: "On the 3-dimensional pseudo-Hermitian space forms and other geometric structures" Kumamoto J.of Math. 7. 71-93 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y.Kamishima: "Standard pseudo-Hermitian structure and Seifert fibration on CR manifold" Annals of Global Analysis and Geometry. 12. 261-289 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y.Kamishima: "Uniformization of Kahler manifolds with vanishing Bochner tensor" Acta Mathematica. 172. 299-308 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y.Kamishima: "Pseudo-Hermitian structure on manifolds from Riemannian geometry, , Differential Geometry and Related Topics, Proceedings, Seoul Natinal University in Korea, Hong Jong (ed.)" Proceedings of Workshops in Pure Math. 13, Part III. 165-213 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y.Kamishima: "Geometric flows on compact manifolds and Global rigidity" Topology. (to appear in). (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y.Kamishima: "Topolgy of CR-manifolds and Kahler manifolds, Symplectic Geometry and Related Topics, Proceedings, Kyowon University in Korea, Jeong Seog Ryu (ed.)" Proceedings of Workshops in Pure Math. (to appear in). (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] "Transformation groups on Heisenberg geometry" Kumamoto J.of Math. (to appear in). (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 神島 芳宣: "On the 3-dimensional pseudo-Hermitian space forms and other geometric structures" Kumamoto J.of Math.7. 71-93 (1994)

    • Related Report
      1995 Annual Research Report
  • [Publications] 神島 芳宣: "Standard pseudo-Hermitian structure and Seifert fibration on CR manifold" Annals of Global Analysis and Geometry. 12. 261-289 (1994)

    • Related Report
      1995 Annual Research Report
  • [Publications] 神島 芳宣: "Uniformization of Kahler manifolds with vanishing Bochner tensor" Acta Mathematica. 172. 299-308 (1994)

    • Related Report
      1995 Annual Research Report
  • [Publications] 神島 芳宣: "Pseudo-Hermitian structure on manifolds from Riemannian geometry" Differential Geometry and Related Topics Proceedings of Workshops in Pure Math.Part III. 13. 165-213 (1994)

    • Related Report
      1995 Annual Research Report
  • [Publications] 神島 芳宣: "Geometric flows on compact manifolds and Global rigidity" to appear in Topology. (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] 神島 芳宣: "Topolgy of CR-manifolds and Kahler manifolds" Symplectic Geometry and Related Topics to appear in Proceedings of Workshops in Pure Math.(1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] 神島 芳宣: "Transformation groups on Heisenberg geometry," to appear in Kumamoto J.of Math.(1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] 神島芳宜: "Pseudo-Hermitian structure on manifolds form Riemannian geometry" Proceedings,Seoul National University in Korea,Hong Jong(ed.),Proceedings of Workshops in Pure Math.13. 165-215 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 神島芳宜: "On the 3-dimensional pseudo-Hermitian space forms and other geometric structures" Kumamoto J.of Math.7. 71-93 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 神島芳宜: "Uniformization of Kahler manifolds with vanishing Bochner tensor" Acta Mathematica.172. 299-308 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 神島芳宜: "Standard pseudo-Hermitian structure and Seifert fibration on CR manifold" Annals of Global Analysis and Geometry.12. 261-289 (1994)

    • Related Report
      1994 Annual Research Report

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Published: 1994-04-01   Modified: 2016-04-21  

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