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Study on the geometry of symmetric spaces and their totally geodesic submanifolds

Research Project

Project/Area Number 13640077
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

NAITOH Hiroo  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10127772)

Co-Investigator(Kenkyū-buntansha) SHIMA Hirohiko  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (70028182)
INOUE Toru  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (00034728)
ANDO Yoshifumi  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (80001840)
NAKAUCHI Nobumitsu  Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (50180237)
MAKINO Tetsu  Yamaguchi University, Faculty of Engineering, Professor, 工学部, 教授 (00131376)
久田見 守  山口大学, 理学部, 教授 (80034734)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordssymmetric space / symmetric submanifold / totally geodesic submanifold / Grassmann geometry / Lie group / Lie algebra / Jordan algebra / symmetric R-space / 部分多様体 / 左不変計量 / ジョルダン三項対 / コンパクトLie群
Research Abstract

This investigation is on totally geodesic submanifolds of Riemannian symmetric spaces and the Grassmann geometry of submanifolds associated with them. Such typical submanifolds are symmetric submanifolds.
1.Fundamental results on symmetric submanifolds
(1)We clarified the relationship between the construction of symmetric submanifolds and the theory of Jordan triple system and the associated symmetric R-space, and obtained a summary on the history and transition on these research fields.
(2)We next clarified the details of symmetric submanifolds in the higher-rank irreducible Riemannian symmetric spaces of noncompact type. This is a collaboration with Berndt, Eschenburg, and Tsukada.
(3)Summing up these results, we published a paper on the classification of symmetric submanifolds of general Riemannian symmetric spaces in Japanese. This is a collaboration with Tsukada. This result was announced in a JSPS-DFG seminar held at Kyoto University. A translation of this paper will be also issued in the journal "Sugaku Expositions" of the American Mathematical Society.
2.Development into another Grassmann geometry
As a development of this research, we understood the study the Grassmann geometries on Lie groups with left invariant metric. So we studied the Grassmann geometries on the 3-dimensional nilpotent Lie group called Heisenberg group and two 3-dimensional unimodular Lie groups, and obtained the classification of their Grassmann geometries and the details about the associated surface theories. As a result, we found that the Grassmann geometry is closely related to the structure of Lie group. The study for Heisenberg case is a collaboration with Inoguchi and Kuwabara.
3.A view for future study
A problem remaining in this research is the complete classification of general totally geodesic submanifolds of Riemannian symmetric spaces. Aso, the Grassmann geometry on Lie groups should be developed much.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (14 results)

All 2005 2003 2002 Other

All Journal Article (10 results) Publications (4 results)

  • [Journal Article] Symmetric submanifolds associated with the irreducible symmetric R-spaces2005

    • Author(s)
      J.Berndt, J.H.Eschenburg, Hiroo Naitoh, Kazumi Tsukada
    • Journal Title

      Mathematische Annalen (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Grassmann geometry on the 3-dimensional Heisenberg group2005

    • Author(s)
      Jun-ichi Inoguchi, Kenji Kuwabara, Hiroo Naitoh
    • Journal Title

      Hokkaido Mathematical Journal (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 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, 内藤博夫
    • Journal Title

      雑誌「数学」(日本数学会編集) 55-3

      Pages: 266-281

    • NAID

      10011797064

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classification of symmetric submanifolds of symmetric spaces (in Japanese)2003

    • Author(s)
      Kazumi Tsukada, Hiroo Naitoh
    • Journal Title

      Sugaku (ed.Math.Soc.of Japan) 55-3

      Pages: 266-281

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Symmetric submanifolds and Jordan triple systems2002

    • Author(s)
      Hiroo Naitoh
    • Journal Title

      Sophia Kokyuroku in Mathematics (Sophia University) 45

      Pages: 21-38

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] 対称部分多様体と対称R-空間2002

    • Author(s)
      内藤 博夫
    • Journal Title

      Lecture Notes Series in Mathematics (Osaka University) 7

      Pages: 195-219

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Symmetric submanifolds and symmetric R-spaces (in Japanese)2002

    • Author(s)
      Hiroo Naitoh
    • Journal Title

      Lecture Notes Series in Mathematics (osaka University) 7

      Pages: 195-219

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classifcation of symmetric submanifolds of symmetric spaces

    • Author(s)
      Kazumi Tsukada, Hiroo Naitoh
    • Journal Title

      Sugaku Expositions (translation, ed. : American Mathematical Society) (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classification of symmetric submanifolds of symmetric spaces (translation of the paper 4)

    • Author(s)
      Kazumi Tsukada, Hiroo Naitoh
    • Journal Title

      Sugaku Expositions (ed.Amer.Math.Soc.) (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classification of symmetric submanifolds of symmetric spaces

    • Author(s)
      Kazumi Tsukada
    • Journal Title

      SUGAKU EXPOSITIONS (American Mathematical Society 編集) (掲載予定)

    • Related Report
      2004 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(日本数学会編集). 55・3. 266-281 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Jun-ichi Inoguchi, Kenji Kuwabara, Hiroo Naito: "Grassmann geometry on the 3-dimensional Heisenberg group"Hokkaido Mathematical Journal. 掲載決定(発表予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] 内藤 博夫: "対称部分多様体と対称R-空間(竹内勝先生メモリアル研究会,小林亮一編)"Lecture Note Series in Mathematics, Osaka University. 7. 195-219 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hiroo Naitoh: "Symmetric submanifolds and Jordan triple systems"Sophia Kokyuroku in Mathematics, Sophia University. 45. 21-38 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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