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Research on classical differential geometry from modern view points and its applications

Research Project

Project/Area Number 22540107
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKwansei Gakuin University (2011-2014)
Fukuoka University (2010)

Principal Investigator

KUROSE Takashi  関西学院大学, 理工学部, 教授 (30215107)

Co-Investigator(Renkei-kenkyūsha) SUYAMA Yoshihiko  福岡大学, 理学部, 教授 (70028223)
HAMADA Tatsuyoshi  福岡大学, 理学部, 助教 (90299537)
KAWAKUBO Satoshi  福岡大学, 理学部, 助教 (80360303)
MATSUURA Nozomu  福岡大学, 理学部, 助教 (00389339)
INOGUCHI Junichi  山形大学, 理学部, 教授 (40309886)
FURUHATA Hitoshi  北海道大学, 大学院理学研究院, 准教授 (80282036)
FUJIOKA Atsushi  関西大学, システム理工学部, 教授 (30293335)
Project Period (FY) 2010-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords古典的微分幾何 / 曲線の運動 / 可積分系方程式 / 多重ハミルトン系 / 統計多様体 / ヘッセ多様体 / ヘッセ断面曲率 / 情報幾何 / 幾何的ダイバージェンス / ガウスの補題 / 可積分系 / 変形KdV方程式 / ハミルトン系 / ミウラ変換 / アフィン微分幾何 / 接束の幾何 / 幾何的ミウラ変換 / 双ハミルトン系 / 計量的tt^*構造 / アフィンはめ込み
Outline of Final Research Achievements

In this research program, classical differential geometry, geometry of curves, surfaces and hypersurfaces in various spaces, have been studied, mainly with the method of the theory of integrable systems. Many results on classical differential geoemtry and its application have been achieved; for instance, through the observation that certain sorts of changes with time of curves yield equations dealt with in the theory of integrable systems, geometric descriptions and/or interpretations of several accomplishments of the theory have been given. Moreover, by applying geometry of hypersurfraces in affine spaces, new properties of statistical manifolds, which appear in informtion geometry, the study of mathematical statistics and information theory with differential geometric tools and methods, have been obtained and the statistical manifolds satisfying some curvature condition have been explicitely constructed and classified.

Report

(6 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (9 results)

All 2014 2013 2012 2011 2010 Other

All Journal Article (3 results) (of which Peer Reviewed: 2 results) Presentation (6 results) (of which Invited: 2 results)

  • [Journal Article] Multi-Hamiltonian structures on spaces of closed equicentroaffine plane curves associated to higher KdV flows2014

    • Author(s)
      Atsushi Fujioka and Takashi Kurose
    • Journal Title

      Symmetry, Integrability and Geometry: Methods and Applications

      Volume: 10

    • DOI

      10.3842/sigma.2014.048

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hessian manifolds of nonpositive constant Hessian sectional curvature2013

    • Author(s)
      Hitoshi Furuhata and Takashi Kurose
    • Journal Title

      Tohoku Mathematical Journal

      Volume: 65 Pages: 31-42

    • NAID

      130005147683

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 曲線の空間上の双ハミルトン系2012

    • Author(s)
      黒瀬俊
    • Journal Title

      数理解析研究所講究録

      Volume: 1775 Pages: 25-32

    • Related Report
      2011 Annual Research Report
  • [Presentation] 可積分系方程式が付随する曲線の運動2012

    • Author(s)
      黒瀬俊
    • Organizer
      日本数学会年会幾何学分科会特別講演
    • Place of Presentation
      東京理科大学神楽坂キャンパス(招待講演)
    • Year and Date
      2012-03-26
    • Related Report
      2011 Annual Research Report
  • [Presentation] 曲線の空間上の双ハミルトン系2011

    • Author(s)
      黒瀬俊
    • Organizer
      RIMS研究集会「部分多様体の微分幾何学的研究」
    • Place of Presentation
      京都大学数理解析研究所(招待講演)
    • Year and Date
      2011-06-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] 曲線の空間上のシンプレクティック構造2010

    • Author(s)
      黒瀬俊
    • Organizer
      福岡大学微分幾何研究会
    • Place of Presentation
      福岡大学セミナーハウス
    • Year and Date
      2010-10-11
    • Related Report
      2010 Annual Research Report
  • [Presentation] 3 次元ユークリッド空間曲線の空間上の変形 KdV 流

    • Author(s)
      黒瀬俊
    • Organizer
      福岡大学微分幾何研究会
    • Place of Presentation
      福岡大学セミナーハウス
    • Related Report
      2013 Annual Research Report
  • [Presentation] Equiaffine plane curves from the viewpoint of billiards

    • Author(s)
      黒瀬俊
    • Organizer
      ミニワークショップ「統計多様体の幾何学とその周辺(4)」
    • Place of Presentation
      北海道大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] ヘッセ多様体と双曲性

    • Author(s)
      黒瀬俊
    • Organizer
      「情報幾何と関連分野 小研究集会」
    • Place of Presentation
      大阪市立大学
    • Related Report
      2012 Annual Research Report
    • Invited

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Published: 2010-08-23   Modified: 2019-07-29  

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