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2006 Fiscal Year Final Research Report Summary

Study of topis related to almost complex structures

Research Project

Project/Area Number 16540057
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

SEKGAWA Kouei  Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (60018661)

Co-Investigator(Kenkyū-buntansha) INNAMI Nobuhiro  Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (20160145)
HASEGAW Keizo  Niigata University, Institute of Humanities, Social Science and Education, Associate Professor, 人文社会・教育科学系, 助教授 (00208480)
MATSUSHITA Yasuo  Shiga Prefectural University, Haculty of Technology, Professor, 工学部, 教授 (90144336)
HASHIMOTO Hideya  Mijyo University, Scol of Science and Technology, Professor, 理工学部, 教授 (60218419)
Project Period (FY) 2004 – 2006
KeywordsAlmost complex structure / Kaehler manifold / 6-dimensional sphere / Einstein manifold / Goldberg conjecture / J-holomorphic curve / Solvmanifold / Walker metric
Research Abstract

A smooth manifold M admitting a (1,1)-tensor field J satisfying J=・I is called an almost complex manifold and the tensor field J is called the almost complex structure. The concept of almost complex manifold is a generalization of complex manifold.. Almost complex manifold (M, J) is said to be integrable if M admits a complex structure and the derived almost complex structure coincides with the almost complex structure J. Any 2-dimensional almost complex manifold is always integrable. However, this is not true for higher dimensional cases in general. An almost complex manifold (M, J) equipped with a compatible (pseudo) Riemannian metric g is called an almost Hermitian manifold. A Kaehler manifold is the most typical one. In this research project, we study mainly the following topics related to the almost complex structures :
(1) Integrability of almost complex structure
(2) Submanifolds in almost complex manifolds
(3) Intermediate and Related topics to (1),(2)
Concerning (1), we study the integrability of almost Kaehler manifolds, for example Goldberg conjecture. Y.Matsushita et al. constructed an 8-dimensional counter example with a neutral Walker metric to the conjecture in the pseuo-Riemannian case. However, the conjecture itself is still remaining open in the case where the scalar curvature is negative. Concerning (2), H.Hashimoto studied several topics related to J-holomorphic curves in the nearly Kaehler 6-sphere S6 from the viewpoint of the Grassmann geometry and obtained interesting results on the deformations of super-minimal J-holomorphic curves and on some tubes around J-holomorphic curves in S6. Recently, the head investigator and H.Hashimoto et al. began to study 6-dimensional oriented submanifolds in the Octonions.and succeed to classify all extrinsic homogeneous almost hermitian 6-manifolds. Concerning (3), for example, K.Hasegawa gave an affirmative answer to the generalized Benson-Gordon conjecture.

  • Research Products

    (12 results)

All 2007 2006 2005 Other

All Journal Article (11 results) Book (1 results)

  • [Journal Article] Almost Kaehler-Einstein structure on 8-dimensional Walker manifolds2007

    • Author(s)
      Y.Matsushita
    • Journal Title

      Monatshefte fur Mathematik 150

      Pages: 41-48

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Almost Kaehler-Einstein structures on 8-dimensional Walker manifolds2007

    • Author(s)
      Y.Matsushita, S.Haze, P.Law
    • Journal Title

      Monatshefte fur Mathematik 150

      Pages: 41-48

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A note on the interability of a class of almost quaternionic manifolds2006

    • Author(s)
      K.Seigawa
    • Journal Title

      Indian J. Math. 48

      Pages: 239-248

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A note on compact solmanifolds with Kaehler structures2006

    • Author(s)
      K.Hasegawa
    • Journal Title

      Osaka J. Math. 43

      Pages: 131-135

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Grassmann geometry of the 6-dimensional sphere2006

    • Author(s)
      H.Hashimoto
    • Journal Title

      Sugaku Expositions 19

      Pages: 1-18

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A note on compact solvmanifolds with Kaehler structures2006

    • Author(s)
      K.Hasegawa
    • Journal Title

      Osaka Journal of Mathematics 43

      Pages: 131-135

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A remark on an example of a 6-dimensional Einstein almost Kaehler manifold

    • Author(s)
      K.Hirobe
    • Journal Title

      J. Geometry (To appear in)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Compression theorems for surfaces and their applications

    • Author(s)
      N.Innami
    • Journal Title

      J. Math. Soc. Japan (To appear in)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A remark on an example of A 6-dimensional Einstein almost Kaehler manifold

    • Author(s)
      K.Hirobe, T.Oguro, K.Sekigawa
    • Journal Title

      Journal of Geometry (in printing)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A note on the integrability of a class of almost quaternionic manifolds

    • Author(s)
      K.Sekigawa, A.Yamada
    • Journal Title

      Indian Journal of mathemayics 48(2006)

      Pages: 239-248

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Compression theorems for surfaces and their applications

    • Author(s)
      N.Innami
    • Journal Title

      Journal of Mathematical Society of Japan (in printing)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 別冊 数理科学「相対論の歩み」2005

    • Author(s)
      松下泰雄
    • Total Pages
      108-115
    • Publisher
      サイエンス社
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2008-05-27  

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