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Applications of Clifford and Cayley algebras to Geometry

Research Project

Project/Area Number 15K04860
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionMeijo University

Principal Investigator

Hashimoto Hideya  名城大学, 理工学部, 教授 (60218419)

Research Collaborator Mashimo Katsuya  
Nakata Fuminori  
Ohashi Misa  
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
KeywordsClifford環 / Cayley 代数 / 例外型単純Lie群G2 / 非平坦な全測地的曲面 / スピノール 群 / 零因子 / 既約表現 / ケーリー代数 / 例外型単純Lie群 / 四元数ケーラー構造 / Twistor space / 複素接触構造 / 佐々木構造 / 3-佐々木構造 / Fano 多様体 / Clifford algebra / Moving frame method / 四元数ケーラー多様体 / principal fibre bundle / contact structure / Quaternionic Kahler / Legendrian submanifold
Outline of Final Research Achievements

We obtain the method of construction of the G2 principal fibre bundle of any oriented 3-dimensional Riemannian manifold, by using Clifford algebras and octonions.
We give some fibre bundle structures related to the special unitary group SU(4). The classical Lie group SU(4) is isomorphic to the spinor group Spin(6) which is a double covering group of the special orthogonal group SO(6). This isomorphism gives rise to some fibre bundle structures on some homogeneous spaces related to SU(4). By using this structure, we give the relationship between the non-flat totally geodesic surfaces in SU(4)/SO(4) and in Sp(2)/U(2) =Spin(5)/U(2).

Academic Significance and Societal Importance of the Research Achievements

本研究は例外型単純Lie 群の対称性を幾何学的に理解する事にある。古典群の対称性では得られない特殊な対称性を用いて幾何学的によい条件を満たす等質空間の構成を行う事が可能であることを示すことが学術的意義である。特に例外型単純Lie 群G2の幾何学的対称性についての研究を行っている。ある四元数ケーラー多様体(8次元)上の2種のTwistor空間の射影空間への具体的な埋め込みを表現できることを示した。

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (14 results)

All 2020 2019 2018 2017 2016 2015

All Journal Article (6 results) (of which Peer Reviewed: 6 results,  Open Access: 1 results,  Acknowledgement Compliant: 3 results) Presentation (7 results) (of which Int'l Joint Research: 1 results,  Invited: 6 results) Book (1 results)

  • [Journal Article] Non-flat totally geodesic surfaces of SU(4)/SO(4) and fibre bundle structures related to SU(4)2020

    • Author(s)
      Hideya Hashimoto, Misa Ohashi and Kazuhiro Suzuki
    • Journal Title

      -

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Relationships among non-flat totally geodesic surfaces in symmetric spaces of type A and their polynomial representations2019

    • Author(s)
      Hideya Hashimoto, Misa Ohashi and Kazuhiro Suzuki
    • Journal Title

      Kodai math. J

      Volume: 印刷中

    • NAID

      130007674788

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Fundamental relationship between Cartan imbeddings of type A and Hopf fibrations2018

    • Author(s)
      Hideya Hashimoto and Misa Ohashi
    • Journal Title

      Contemporary perspectives in differential geometry and its related fields

      Volume: 1 Pages: 79-94

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] G2 fibre bundle structure on 3-dimensional manifolds2017

    • Author(s)
      Hideya Hashimoto and Misa Ohashi
    • Journal Title

      Note di Mathematica

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] On generalized cylindrical helices and Lagrangian surfaces of R^42015

    • Author(s)
      Hideya Hashimoto and Misa Ohashi
    • Journal Title

      Journal of geometry

      Volume: 106 Issue: 2 Pages: 405-420

    • DOI

      10.1007/s00022-015-0273-3

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On fibre bundle structures of Stiefel manifolds related to the octonions2015

    • Author(s)
      Hideya Hashimoto and Misa Ohashi
    • Journal Title

      Topology and its application

      Volume: 196 Pages: 483-491

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] G2/SO(4) 上の Hyper-Kaehler manifold の実現2019

    • Author(s)
      橋本英哉
    • Organizer
      淡路島幾何学研究集会2019
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Geometrical structures and the Calabi-Bryant formula of G22018

    • Author(s)
      橋本英哉
    • Organizer
      ICDG2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] G2 に関連した等質空間上の微分形式と幾何構造2018

    • Author(s)
      橋本英哉
    • Organizer
      淡路島幾何学研究集会2018
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] G2 に関連した等質空間の幾何構造の関連2018

    • Author(s)
      橋本英哉
    • Organizer
      部分多様体論・湯沢2018
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] G2に関連した等質空間上の微分形式と幾何構造2018

    • Author(s)
      橋本英哉
    • Organizer
      淡路島幾何学研究集会2018
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] G2に関連した等質空間の幾何構造の局所座標による具体化2017

    • Author(s)
      橋本英哉
    • Organizer
      多様体上の微分方程式(金沢シリーズ第16回)
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] A1型の対称空間内の平坦でない全測地的曲面の多項式表示の計算方法2016

    • Author(s)
      鈴木和大、大橋美佐、橋本英哉
    • Organizer
      日本数学会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
  • [Book] Contemporary perspectives in differential geometry and its related fields2018

    • Author(s)
      Toshiaki Adachi, Hideya Hashimoto, Milen J Hristov
    • Total Pages
      192
    • Publisher
      World Scientific
    • ISBN
      9789813220904
    • Related Report
      2017 Research-status Report

URL: 

Published: 2015-04-16   Modified: 2020-03-30  

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