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Classical differential geometry from the modern viewpoint and its application

Research Project

Project/Area Number 18540103
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KUROSE Takashi  Fukuoka University, 理学部, 教授 (30215107)

Co-Investigator(Kenkyū-buntansha) SUAYMA 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)
YAMADA Kotaro  東京工業大学, 大学院・理工学研究科, 教授 (10221657)
INOGUCHI Junichi  山形大学, 理学部, 准教授 (40309886)
FURUHATA Hitoshi  北海道大学, 大学院・理学研究院, 准教授 (80282036)
Project Period (FY) 2006 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥4,080,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥780,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Keywords古典的微分幾何 / 可積分系 / 曲線の運動 / ハミルトン系 / 共形平坦超曲面 / 実超曲面 / ヘッセ幾何 / 情報幾何 / アフィン微分幾何 / ヘッセ多様体 / 差分幾何 / 離散化 / キルヒホッフ弾性棒 / 特異点 / 極大面 / 表現公式 / 統計多様体 / 平坦フロント / 零はめ込み / 等質空間 / 積分表示公式 / 差分KdV方程式
Research Abstract

In this research, we studied classical differential geometry from modern viewpoints, such as of the theory of integral systems and of the theory of singularities ; we obtained results on various fields of classical differential geometry and their applications, in particular, the motions of curves associated with integrable systems, explicit construction and the classification of conformally flat hypersurfaces of four-dimensional space forms, real hypersurfaces of complex space forms, surfaces of three-dimensional spaces, affine differential geometry and its applications to Hessian geometry and information geometry, and so on.

Report

(6 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report   Self-evaluation Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (50 results)

All 2010 2009 2008 2007 2006 Other

All Journal Article (32 results) (of which Peer Reviewed: 26 results) Presentation (16 results) Book (2 results)

  • [Journal Article] Hamiltonian formalism for the higher KdV flows on the space of closed complex equicentroaffine curves2010

    • Author(s)
      A. Fujioka, T. Kurose
    • Journal Title

      International Journal of Geometric Methods in Modern Physics 7

      Pages: 165-175

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Ruled real hypersurfaces of complex space forms2010

    • Author(s)
      T. Hamada, J. Inoguchi
    • Journal Title

      Kodai Mathematical Journal 33

      Pages: 123-134

    • NAID

      130004687926

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Hamiltonian formalism for the higher KdV flows on the space of closed complex equicentroaffine curves2010

    • Author(s)
      A.Fujioka, T.Kurose
    • Journal Title

      International Journal of Geometric Methods in Modern Physics 7

      Pages: 165-175

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ruled real hypersurfaces of complex space forms2010

    • Author(s)
      T.Hamada, J.Inoguchi
    • Journal Title

      Kodai Mathematical Journal 33

      Pages: 123-134

    • NAID

      130004687926

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Geometry of the space of closed curves in the complex hyperbola2009

    • Author(s)
      A. Fujioka, T. Kurose
    • Journal Title

      Kyushu Journal of Mathematics 63

      Pages: 161-165

    • NAID

      130000135277

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Complete real hypersurfaces in compact rank one symmetric spaces2009

    • Author(s)
      T. Hamada, K. Shiohama
    • Journal Title

      Proceedings of the American Mathematical Society 137

      Pages: 3905-3910

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] The geometry of fronts2009

    • Author(s)
      K. Saji, M. Umehara, K. Yamada
    • Journal Title

      Annals of Mathematics 169

      Pages: 491-529

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of flat urface in hyperbolic 3-space2009

    • Author(s)
      M. Kokubu, W. Rossman, M. Umehara, K. Yamada
    • Journal Title

      Journal of the Mathematical Society of Japan 61

      Pages: 799-852

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Lightlike surfaces in Minkowski 3-space2009

    • Author(s)
      J. Inoguchi, S. Lee
    • Journal Title

      International Journal of Geometric Methods in Modern Physics 6

      Pages: 267-283

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Hypersurfaces in statistical manifolds2009

    • Author(s)
      Hitoshi Furuhata
    • Journal Title

      Differential Geometry and its Applications 27

      Pages: 420-429

    • NAID

      120006459554

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Geometry of the space of closed curves in the complex hyperbola2009

    • Author(s)
      A.Fujioka and T.Kurose
    • Journal Title

      Kyushu Journal of Mathematics 63

      Pages: 161-165

    • NAID

      130000135277

    • Related Report
      2008 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Geometry of the space of closed curves in the complex hyperbola2009

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

      Kyushu Journal of Mathematics 63

      Pages: 161-165

    • NAID

      130000135277

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Motions of curves in the complex hyperbola and the Burgers hierarchy2008

    • Author(s)
      A. Fujioka, T. Kurose
    • Journal Title

      Kyushu Journal of Mathematics 45

      Pages: 1057-1065

    • NAID

      120004840816

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Kirchhoff elastic rods in three-dimensional space forms2008

    • Author(s)
      Satoshi Kawakubo
    • Journal Title

      Journal of the Mathematical Society of Japan 60

      Pages: 551-582

    • NAID

      10024331776

    • Related Report
      2009 Final Research Report 2008 Annual Research Report 2008 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] A Weierstrass representation for minimal surfaces in Sol2008

    • Author(s)
      J. Inoguchi, S. Lee
    • Journal Title

      Proceedings of the American Mathematical Society 136

      Pages: 2209-2216

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Motions of curves in the complex hyperbola and the Burgers hierarchy2008

    • Author(s)
      A.Fujioka and T.Kurose
    • Journal Title

      Osaka Journal of Mathematics 45

      Pages: 1057-1065

    • NAID

      120004840816

    • Related Report
      2008 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Motions of curves in the complex hyperbola and the Burgers hierarchy2008

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

      Osaka Journal of Mathematics 45

      Pages: 1057-1065

    • NAID

      120004840816

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Kirchihoff elastic rods in three-dimensional space forms2008

    • Author(s)
      Satoshi Kawakubo
    • Journal Title

      Journal of the Mathematical Society of Japan(掲載確定) 60

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Weierstrass representation for minimal surfaces in Sol2008

    • Author(s)
      J.Inoguchi and S.Lee
    • Journal Title

      Proceedings of the American Mathematical Society(掲載確定) 136

      Pages: 2209-2216

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Conformally flat hypersurfaces with cyclic Guichard net2007

    • Author(s)
      U. Hertrich-Jeromin, Y. Suyama
    • Journal Title

      International Journal of Mathematics 17

      Pages: 301-329

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Conformally flat hypersurfaces withcyclic Guichard net2007

    • Author(s)
      U.Hertrich-Jeromin and Y.Suyama
    • Journal Title

      International Journal of Mathematics 18

      Pages: 301-329

    • Related Report
      2008 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Flat fronts in hyperbolic 3-space and their caustics2007

    • Author(s)
      M.Kokubu, W.Rossman, M.Umehara, K.Yamada
    • Journal Title

      Journal of the Mathematical Society of Japan 59

      Pages: 265-299

    • NAID

      10019540696

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The center map of an affine immersion2006

    • Author(s)
      H.Furuhata, L.Vrancken
    • Journal Title

      Results in Mathematics 49

      Pages: 201-217

    • NAID

      120006459376

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Three dimensional real hypersurfaces in complex space forms

    • Author(s)
      J.-T. Cho, T. Hamada, J. Inoguchi
    • Journal Title

      Tokyo Journal of Mathematics (to appear)

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Complete real hypersurfaces in compact rank one symmetric spaces

    • Author(s)
      T. Hamada and K Shiohama
    • Journal Title

      Proceedings of the American Mathematical Society (掲載決定)

    • Related Report
      2008 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Complete real hypersurfaces in compact rank one symmetric spaces

    • Author(s)
      T. Hamada, K. Shiohama
    • Journal Title

      Proceedings of the American Mathematical Society (to appear確認済)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On three dimensional real hypersurface in complex space form

    • Author(s)
      J. T. Cho, T. Hamada, J. Inoguch
    • Journal Title

      Tokyo Journal of Mathematics (to appear確認済)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Singularities of maximal surfaces

    • Author(s)
      Shoichi Fujimori, Kentaro Saji, Masaaki Umehara and Kotaro Yamada
    • Journal Title

      Mathematische Zeitschrift(掲載確定)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Conformally flat hypersurfaces with cyclic Guichard net

    • Author(s)
      U.Hertlich-Jeromin, Y.Suyama
    • Journal Title

      International Journal of Mathematics To appear

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The geometry of fronts

    • Author(s)
      K.Saji, M.Umehara, K.Yamada
    • Journal Title

      Annals of Mathematics To appear

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Biminimal submanifolds in 3-dimensional contact manifolds

    • Author(s)
      Jun-ichi Inoguchi
    • Journal Title

      Balkan Journal of Geometry and its Applications To appear

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Helicoids and axially symmetric minimal surfaces in 3-dimensional homogeneous spaces

    • Author(s)
      M.Bekkar, F.Bouziani, Y.Boukhatema, J.Inoguchi
    • Journal Title

      Differential Geometry-Dynamical Systems To appear

    • Related Report
      2006 Annual Research Report
  • [Presentation] Lie hypersurfaces in the complex hyperbolic space2009

    • Author(s)
      濱田龍義
    • Organizer
      微分幾何学ワークショップ「部分多様体の幾何学とリー群論」
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2009-10-30
    • Related Report
      2009 Final Research Report
  • [Presentation] 3次元空間形内の局所誘導階層の進行波解2009

    • Author(s)
      川久保哲
    • Organizer
      第56回幾何学シンポジウム
    • Place of Presentation
      佐賀大学
    • Year and Date
      2009-08-31
    • Related Report
      2009 Annual Research Report 2009 Final Research Report
  • [Presentation] 3次元双曲空間の平均曲率一定曲面2009

    • Author(s)
      井ノ口順一
    • Organizer
      第56回幾何学シンポジウム
    • Place of Presentation
      佐賀大学
    • Year and Date
      2009-08-30
    • Related Report
      2009 Final Research Report
  • [Presentation] 複素空間型内の実超曲面に関する考察2008

    • Author(s)
      濱田龍義
    • Organizer
      第55回幾何学シンポジウム
    • Place of Presentation
      弘前大学
    • Year and Date
      2008-08-22
    • Related Report
      2009 Final Research Report
  • [Presentation] 定曲率ヘッセ多様体の分類2008

    • Author(s)
      黒瀬俊
    • Organizer
      RIMS研究集会「部分多様体の微分幾何学およびその周辺領域の研究」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-06-23
    • Related Report
      2009 Final Research Report
  • [Presentation] 部分多様体の微分幾何学およびその周辺領域2008

    • Author(s)
      黒瀬俊
    • Organizer
      定曲率ヘッセ多様体の分類、RIMS研究集会
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-06-23
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] KNOPPIX/ MATH/ 2008について2008

    • Author(s)
      濱田龍義
    • Organizer
      第17回日本数式処理学会
    • Place of Presentation
      城西大学東京紀尾井町キャンパス
    • Year and Date
      2008-06-06
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] KNOPPIX/Math/2008について2008

    • Author(s)
      演田龍義
    • Organizer
      第17回日本数式処理学会
    • Place of Presentation
      城西大学東京紀尾井町キャンパス
    • Year and Date
      2008-06-06
    • Related Report
      2008 Annual Research Report
  • [Presentation] Kirchhoff弾性棒のエネルギー汎関数のコンパクト性2008

    • Author(s)
      川久保哲
    • Organizer
      日本数学会2008年度年会
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-24
    • Related Report
      2009 Final Research Report 2008 Self-evaluation Report
  • [Presentation] Kirchhoff弾性棒のエネルギー汎関数のコンパクト性2008

    • Author(s)
      川久保 哲
    • Organizer
      日本数学会
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-24
    • Related Report
      2007 Annual Research Report
  • [Presentation] *-Einstein real hypersurfaces of complex space forms2008

    • Author(s)
      濱田龍義
    • Organizer
      The 12th International Workshop on Differential Geometry and Related Topics
    • Place of Presentation
      Kyungpook National University (Korea)
    • Related Report
      2009 Final Research Report
  • [Presentation] Biharmonic maps and bi-Yang Mills fields2007

    • Author(s)
      一山稔之・井ノ口順一・浦川肇
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-21
    • Related Report
      2009 Final Research Report
  • [Presentation] Discrete time evolution of planner discrete curves2006

    • Author(s)
      松浦望
    • Organizer
      DMHF2007: COE Conference on the Development of Dynamic Mathematics with High Functionality
    • Place of Presentation
      福岡リーセントホテル
    • Year and Date
      2006-10-02
    • Related Report
      2009 Final Research Report
  • [Presentation] Affine spheres of finite type2006

    • Author(s)
      井ノ口順一
    • Organizer
      DMHF2007: COE Conference on the Development of Dynamic Mathematics with High Functionality
    • Place of Presentation
      福岡リーセントホテル
    • Year and Date
      2006-10-02
    • Related Report
      2009 Final Research Report
  • [Presentation] COE Conference on the Development of Dynamic Mathematics with High Functionality2006

    • Author(s)
      松浦望
    • Organizer
      Discrete time evolution of planar discrete curves, DMHF2007
    • Place of Presentation
      福岡リーセントホテル
    • Year and Date
      2006-10-02
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Biharmonic maps and bi-Yang Millsfields2006

    • Author(s)
      一山稔之・井ノロ順一・浦川肇
    • Organizer
      日本数学会秋季総合分科会
    • Year and Date
      2006-09-21
    • Related Report
      2008 Self-evaluation Report
  • [Book] 曲線とソリトン2010

    • Author(s)
      井ノ口順一
    • Total Pages
      177
    • Publisher
      朝倉書店
    • Related Report
      2009 Final Research Report
  • [Book] 幾何学いろいろ2007

    • Author(s)
      井ノ口順一
    • Total Pages
      209
    • Publisher
      日本評論社
    • Related Report
      2009 Final Research Report

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Published: 2006-04-01   Modified: 2016-04-21  

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