• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Geometry of Almost Complex Manifolds

Research Project

Project/Area Number 14540070
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNIIGATA UNIVERSITY

Principal Investigator

SEKIGAWA Kouei  NIIGATA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (60018661)

Co-Investigator(Kenkyū-buntansha) MATSUSHITA Yasuo  Shiga Prefectural University, Faculty of Technology, Professor, 工学部, 教授 (90144336)
HASEGAWA Keizo  NIIGATA UNIVERSITY, Facalty of Education and Human Science, Associate Professor, 教育人間科学部, 助教授 (00208480)
INNNAMI Nobuhiro  NIIGATA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (20160145)
HASHIMOTO Hideya  Meijo University, School of Science and Technology, Professor, 理工学部, 教授 (60218419)
Project Period (FY) 2002 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2002: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsAlmost complex manifold / Integrability / (Almost) Kahler manifold / Einstein metric / Goldberg conjecture / Nearly Kahler 6-dimensional sphere / J-holomorphic curve / CR-submanifold / 概超複素多様体 / 概ケーラー多様体 / Goldherg予想 / Walker計量 / 可解多様体 / 概複素構造 / 概エルミート構造 / 四元数構造 / 6次元球面 / リッチ固有値
Research Abstract

A smooth manifold M admitting (1,1) tensor field J satisfying J^2=-I is called an almost complex manifold. The concept of almost complex manifold is a natural generalization of complex manifold. There are known many examples of almost complex manifolds which are not complex manifolds. It is well-known that 6-dimensional sphere S^6 admits an almost complex structure which is not complex one. Almost complex manifold(M,J) is said to be integrable if M admits a complex structure and the associated almost complex structure coincides with the almost complex structure J.In the research project, we considered mainly the following topics in Almost Complex Geometry :
(1)The Goldberg conjecture
(2)Submanifolds in nearly Kahler 6-dimensional sphere S^6
(3)Related topics to the above (1) and (2)
Concerning (1),we obtained some affirmative answers to the conjecture under some additional curvature conditions in 4-dimensional case. Recently, Y.Matsushita found a counter example to the indefinite-version of the conjecture in dimension 4.Concerning (2),we obtained some topological conditions for a given 4-dimensional submanifold of S^6 to be CR-submanifold and constructed several examples of 4-dimensional CR-submanifolds of S^6. On one hand, H.Hashimoto(and et al.) constructed J-holomorphic flat tori of type(III)in S^6 and classified them from the integral system view point. Concerning (3),we examined local structure of Kahler surfaces with distinct constant Ricci eigenvalues and determind all homogeous Kahler surfaces. We note that the existence of compact KShler surface with distinct constant negative scalar curvatures is deeply concerned with the possibility of existence of counter example of the Goldberg conjecture. Further, K.Hasegawa proved that a 4-dimensional compact solvmanifold admitting Kahler structure is holomorphically equivalent to a torus bundle over a torus with a certain finite abelian structure group.

Report

(3 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • Research Products

    (19 results)

All Other

All Publications (19 results)

  • [Publications] L.Richard: "Notes on four-dimensional almost Kahler manifolds with negative scalar curvature"International Journal of Mathematics & Mathematical Sciences. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nihonyanagi: "Notes on Kahler surfaces with dictinct constant Ricci eigenvalues"Journal of the Korean Mathematical Society. 5. 1015-1029 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Oguro: "On some four-dimensional almost Kahler Einstein manifolds with Hermitian Weyl tensor"International Journal of Differential Equations and Applications. 4. 285-301 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Matsushita: "Four-dimensional Walker metrics and symplectic structures"Journal of Geometry and Physics. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Hashimoto: "Constructions of almost complex 2-tori of type (III) in the nearly Kahler 6-sphere"Differential Geometry and Its Applications. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Hashimoto: "On 4-dimensional CR-submanifolds of a 6-dimensional sphere"Adv.Stud.Pure Math.. 34. 143-154 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] L.Richard, T.Oguro, K.Sekigawa: "Notes on four-dimensional almost Kahler manifolds with negative scalar curvature"International Journal of Mathematics & Mathematical Sciences. (in printing).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nihonyanagi, T.Oguro, K.Sekigawa: "Notes on Kahler surfaces with distinct constant Ricci curvature"Journal of the Korean Math.Soc.. 5. 1015-1029 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Oguro, K.Sekigawa: "On some four-dimensional almost Kahler Einstein manifolds With Hermitian Weyl tensor"International J.Differential Equations and Applications. 4. 285-301 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Matsushita: "Four-dimensional Walker metrics and symplectic structures"Journal of Geometry and Physics. (in printing).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Hashimoto, T.Taniguchi, S.Udagawa: "Constructions of almost complex 2-tori of type(III) in the nearly Kahler 6-sphere"Differential Geometry and Us Aplications. (in printing).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Hashimoto, K.Mashimo, K.Sekigawa: "On 4-dimensional CR-submanifolds of a 6-dimensional sphere"Adv.Stud.Pure Math.. 34. 143-154 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] L.Richard: "Notes on four-dimensional almost Kahler manifolds with negative scalar curvature"International Journal of Mathematics & Mathematical Sciences. 印刷中.

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nihonyanagi: "Notes on Kahler surfaces with distinct constant Ricci eigenvalues"Journal of the Korean Mathematical Society. 5. 1015-1029 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Matsushita: "Four-dimensional Walker metrics and symplectic structures"Journal of Geometry and Physics. 印刷中.

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Hashimoto: "Constructions of almost complex 2-tori of type (III) in the nearly Kahler 6-sphere"Differential Geometry and Its Applications. 印刷中.

    • Related Report
      2003 Annual Research Report
  • [Publications] 橋本英哉: "On 4-dimendional Cr-submanifolds of a 6-dimensional sphere"Advance Studies in Pure Mathematics. 34. 43-54 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 橋本英哉: "6次元球面のグラスマン幾何学"数学. 55・1. 37-54 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 印南信宏: "Geometry of geodesics for convex billiards and circular billiards"Nihonkai Mathematical Journal. 13・1. 73-120 (2002)

    • Related Report
      2002 Annual Research Report

URL: 

Published: 2002-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi