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Totally complex submanifolds of a quaternion projective space

Research Project

Project/Area Number 15540065
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TSUKADA Kazumi  Ochanomizu University, Department of Mathematics, Professor, 理学部, 教授 (30163760)

Co-Investigator(Kenkyū-buntansha) EJIRI Norio  Meijo University, Department of Mathematics, Professor, 理学部, 教授 (80145656)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywordsa quaternion projective space / a quaternionic Kahler manifold / totally complex submanifolds / twistor space / totally real submanifolds / Einstein-Kahler submanifolds / a complex quadric / isotropic Kahler immersions / 複素二次超曲面 / 複素共形構造 / ルジャンドル部分多様体
Research Abstract

In this research, we investigate totally complex submanifolds of a quaternion projective space (more generally a quaternionic Kahler manifold). It is one aspect of the interplay of quaternionic differential geometry and complex differential geometry and makes so-called "quaternionic complex differential geometry". A submanifold M of a quaternionic Kahler manifold (M^^~, Q, g^^~) is said to be totally complex if there exists a section I^^~ of the bundle Q|_M such that (1)I^^~^2 = -id, (2)I^^~TM = TM (3)KTM ⊥ TM for any K ∈ Q|_M with <I^^~, K> = 0. Typical examples are half dimensional totally complex submanifolds of a quaternion projective space HP^n with parallel second fundamental form, which have been classified by Tsukada(head investigator of this research). The twistor space Z of (M^^~,Q,g^^~) is defined by Z = {I^^~ ∈ Q|I^^~^2 =-id}, which is an S^2-bundle over M^^~. The twistor space Z has a natural complex structure and it admits an Einstein-Kahler metric if M^^~ has positive scalar curvature.
Main results of this research are the following :
1.We show fundamental theorem on the existence and the uniqueness for half dimensional totally complex submanifolds of HP^n or the quaternion hyperbolic space HH^n. This result is an affirmative answer to the conjecture by Alekseevsky and Marchiafava.
2.For a totally complex submanifold M of M^^~, we consider a new natural lift to the twistor space Z of M^^~ and construct a totally real and minimal submanifold of Z.
3.We characterize half dimensional totally complex submanifolds of HP^n with parallel second fundamental form under some curvature condition such as Einstein-Kahler.
4.We investigate fundamental properties of isotropic Kahler submanifolds of a complex quadric, whose theory is analogous to that of totally complex submanifolds.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (27 results)

All 2005 2004 2003 Other

All Journal Article (22 results) Publications (5 results)

  • [Journal Article] Symmetric sub manifolds associated with irreducible symmetric R-spaces2005

    • Author(s)
      J.Berndt, J-H.Eschenburg, H.Naitoh,, K.Tsukada
    • Journal Title

      Math. Ann. 332

      Pages: 721-737

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Another natural lift of a Kahler sub manifold of quater nionic Kahler manifold to the twistor space2005

    • Author(s)
      N.Ejiri, K.Tsukada
    • Journal Title

      Tokyo J. Math. 28

      Pages: 71-78

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Symmetric submanifolds associated with irreducible symmetric R-spaces2005

    • Author(s)
      J.Berndt, J-H.Eschenburg, H.Naitoh, K.Tsukada
    • Journal Title

      Math.Ann. 332

      Pages: 721-737

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Another natural lift of a Kahler Submanifold of a quaternionic Kahler manifold to the twistor space2005

    • Author(s)
      N.Ejiri, K.Tsukada
    • Journal Title

      Tokyo J.Math. 28

      Pages: 71-78

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Symmetric submanifolds associated with irreducible symmetric R-spaces2005

    • Author(s)
      J.Berndt, J-H.Eschenberg, H.Naitoh, K.Tsukada
    • Journal Title

      Math.Ann. 332

      Pages: 721-737

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Another natural lift of a Kahler submanifold of a quaternionic Kahler manifold to the twistor space2005

    • Author(s)
      N.Ejiri, K.Tsukada
    • Journal Title

      Tokyo J.Math. 28

      Pages: 71-78

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Isotropic Kahler immersions into a complex quadric2005

    • Author(s)
      K.Tsukada
    • Journal Title

      Topics in Almost Hermitian Geometry and the related fields

      Pages: 248-256

    • NAID

      110006559629

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Conformably flat semi-Remarries manifolds with nilpotent Ricci operators and affine differential geometry2004

    • Author(s)
      K.Honda, k.Tsukada
    • Journal Title

      Annals of Global Analysis and Geometry 25

      Pages: 253-275

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Einstein Kahler sub manifolds in a quaternion projective space2004

    • Author(s)
      k.Tsukada
    • Journal Title

      Bull of London Math.Soc. 36

      Pages: 527-536

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Conformally flat semi-Riemannian manifolds with nilpotent Ricci operators and affine differential geometry2004

    • Author(s)
      K.Honda, K.Tsukada
    • Journal Title

      Annals of Global Analysis and Geometry 25

      Pages: 253-275

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Einstein Kahler Submanifolds in a quaternion projective space2004

    • Author(s)
      K.Tsukada
    • Journal Title

      Bull of London Math.Soc. 36

      Pages: 527-536

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Einstein-Kahler submanifolds in a quaternion projective space2004

    • Author(s)
      K.Tsukada
    • Journal Title

      Bull.London Math.Soc. 36

      Pages: 527-536

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Conformally flat semi-Riemannian Manifolds with nilpotent Ricci operators and affine differential geometry2004

    • Author(s)
      K.Honda, K.Tsukada
    • Journal Title

      Annals of Global Analysis and Geometry 25

      Pages: 253-275

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Kahler sub manifolds of a quaternion projective space2003

    • Author(s)
      塚田由美
    • Journal Title

      京都大学数理解析研究所講究録 1346

      Pages: 1-9

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 対称空間の対称部分多様体の分類2003

    • Author(s)
      塚田和美, 内藤博志
    • Journal Title

      数学 55

      Pages: 266-281

    • NAID

      10011797064

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Kahler submanifolds of a quaternion projective space2003

    • Author(s)
      K.Tsukada
    • Journal Title

      Geometry of homogeneous spaces and submanifolds, Kyoto,2003, Surikaisekikenkyusho Kokyuroku (Japanese) No.1346

      Pages: 1-9

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Classification of symmetric submanifolds in symmetric spaces.2003

    • Author(s)
      H.Naitoh, K.Tsukada
    • Journal Title

      Sugaku (Japanese) 55

      Pages: 266-281

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Isotropic Kahler immersions into a complex quadric

    • Author(s)
      k.Tsukada
    • Journal Title

      Natural Science Report of Ochanomizu University (出版予定)

    • NAID

      110006559629

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Isotropic Kahler immersions into a complex quadric

    • Author(s)
      K.Tsukada
    • Journal Title

      Natural Science Report of the Ochanomizu University (to appear)

    • NAID

      110006559629

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Symmetric submanifolds associated with irreducible symmetric R-spaces

    • Author(s)
      J.Berndt, J-H.Eschenburg, H.Naitoh, K.Tsukada
    • Journal Title

      Math.Ann. (出版予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Another natural lift of a Kahler submanifold of a quaternionic Kahler manifold to the twistor space

    • Author(s)
      N.Ejiri, K.Tsukada
    • Journal Title

      Tokyo J.Math. (出版予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Isotropic Kahler immersions into a complex quadric

    • Author(s)
      K.Tsukada
    • Journal Title

      Topics in Almost Hermitian Geometry and the Related Fields (出版予定)

    • NAID

      110006559629

    • Related Report
      2004 Annual Research Report
  • [Publications] K.Tsukada: "Einstein Kahler submanifolds on a quaternion projective space"Bull of London Math.Soc.. (出版予定)(未定).

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Berndt, J-H.Eschenburg, H.Naitoh, K.Tsukada: "Symmetric submanifolds associated with irreducible symmetric R-spaces"Math.Ann.. (出版予定)(未定).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Honda, K.Tsukada: "Conformably flat semi-Riemamian manifolds with nilpotent Ricci operators and affine differential geometry"Annals of Global Analysis and Geometry. (出版予定)(未定).

    • Related Report
      2003 Annual Research Report
  • [Publications] 塚田 和美, 内藤博夫: "対称空間の対称部分多様体の分類"数学. 55巻3号. 266-281 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Tsukada: "Kahler submanifolds of a quaternion projective space"京都大学数理解析研究所講究録. 1346. 1-9 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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