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Harmonic maps into symmetric spaces and applications of the theory of integrable systems

Research Project

Project/Area Number 06640174
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

UDAGAWA Seiichi  School of Medicine, Nihon University, Lecturer, 医学部, 講師 (70193878)

Co-Investigator(Kenkyū-buntansha) IKAWA Toshihiko  School of Medicine, Nihon University, associate Professor, 医学部, 助教授 (30151252)
Project Period (FY) 1994 – 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1996: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1995: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1994: ¥600,000 (Direct Cost: ¥600,000)
KeywordsHarmonic map / Symmetric space / Torus / Finite type / Conformal map / Complex Grassmannion / Quaternicnic projective space / Quaternionic projective space / harmonic map / finite type / primitive map / complex Grassmann manifold / quaternionic projective space / two-torus
Research Abstract

The construction of harmonic two-tori in symmetric spaces are known in two ways. One way is a method using twistor fibration and the resulting harmonic map is said to be superminimal. Another way is a method using the theory of integrable system and it is constructed from two-dimensional linear flows. The latter harmonic map is said to be finite type. For example, any non-conformal harmonic two-tori in compact symmetric spaces of rank one is of finite type. Burstall proved any weakly conformal non-superminimal harmonic two-tori in a sphere or a complex projective space is covered by a primitive map of finite type. Then the following problems naturally arises :
(1) Is weakly conformal non-superminimal hatmonic two-tori in quaternionic projective space covered by a primitive map of finite type?
(2) How about the case where the target is a compact symmetric space of rank greater than one?
Our result for the problem (1) is : Any weakly conformal non-superminimal harmonic two-tori in HP^3 is covered by a primitive map of finite type or constructed by using twistor fibration. For the problem (2), weakly conformal non-superminimal harmonic two-tori in G_2(C^4) is covered by a primitive map of finite type or constructed by using twistor fibration. Under the additional condition, the same type theorem holds for G_2(C^<2n>).

Report

(4 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • 1995 Annual Research Report
  • 1994 Annual Research Report
  • Research Products

    (13 results)

All Other

All Publications (13 results)

  • [Publications] S.Udagawa: "Harmonic maps from a two-torus into a complex Grassmann manifold" International J. Math.6. 447-459 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] S.Udagawa: "Harmonic tori in complex Grassmann manifolds and quaternionic projective spaces" 日本大学医学部一般教育研究紀要. 22. 6-15 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] S.Udagawa: "Harmonic tori in quaternionic projective 3-spaces" Proc. Amer. Math. Soc.125. 275-285 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] S.Udagawa: "Harmonic maps from a two-torus into a complex Grassmann manifold" International J.Math.6. 447-459 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] S.Udagawa: "Harmonic tori in complex Grassmann manifolds and quaternionic projective spaces" Bulktin of Liberal arts and sciences, Nihon University, School of Medicine. 22. 6-15 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] S.Udagawa: "Harmonic tori in quaternionic projective 3-spaces" Proc.Amer.Math.Soc.125. 275-285 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] S.Udagawa: "Harmonic maps from a two-torus into a complex Grassmann manifold" International J.Math.6. 447-459 (1995)

    • Related Report
      1996 Annual Research Report
  • [Publications] S.Udagawa: "Harmonic tori in complex Grassmann manifolds and quaternionic projective spaces" 日本大学医学部一般教育研究紀要. 22. 6-15 (1994)

    • Related Report
      1996 Annual Research Report
  • [Publications] S.Udagawa: "Harmonic tori in quaternionic projective 3-spaces" proc.Amer.Math.Soc.125. 275-285 (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] Seiichi Udagawa: "Harmonic maps from a two-torus into a complex Grassmann manifold" International Journal of Mathematics. 6. 447-459 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] Seiichi Udagawa: "Harmonic tori in quaternionic projective 3-spaces" Proceedings of the American Mathematical Society. (to appear).

    • Related Report
      1995 Annual Research Report
  • [Publications] 宇田川,誠一: "Harmonic tori in complex Grassmann manifolds and quaternionic projective spaces" 日本大学医学部一般教育研究紀要. 22. 7-15 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 宇田川,誠一(Seiichi Udagawa): "Harmonic maps from a two-torus into a complex Grassmann manifold" International Journal of Mathematics.

    • Related Report
      1994 Annual Research Report

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

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