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Applications to differential geometry and singularity theory of differential equations

Research Project

Project/Area Number 23740041
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionMuroran Institute of Technology

Principal Investigator

TAKAHASHI MASATOMO  室蘭工業大学, 工学(系)研究科(研究院), 准教授 (80431302)

Project Period (FY) 2011 – 2013
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2011: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords特異点 / 微分方程式 / 微分幾何学 / 完全解 / ルジャンドル曲線 / 双対性 / 縮閉線 / 伸開線 / 特異点論 / フロンタル / 幾何構造 / フロント / implicitな常微分方程式 / ルジャンドル特異点論
Research Abstract

As applications of singularity theory, we have studied a qualitative theory for implicit ordinary differential equations. We gave a generic classification of semi-local first order ordinary differential equations of Clairaut type. Moreover, we defined complete solutions for singular first order ordinary differential equations and gave its an existence condition. We also gave types of completely integrable implicit second order ordinary differential equations.
As applications to differential geometry, we have studied a theory of Legendre curves. We gave the existence and the uniqueness for Legendre curves via Legendre curvatures. By using the Legendre curvature, we investigated the evolutes and the involutes. Moreover, we gave a generic classifications of tangent surfaces associated with G2 and D4 geometries.

Report

(4 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Research-status Report
  • 2011 Research-status Report
  • Research Products

    (28 results)

All 2014 2013 2012 2011 Other

All Journal Article (6 results) (of which Peer Reviewed: 6 results) Presentation (18 results) (of which Invited: 3 results) Remarks (4 results)

  • [Journal Article] Existence and uniqueness for Legendre curves2013

    • Author(s)
      T. Fukunaga, M. Takahashi
    • Journal Title

      Journal of Geometry

      Volume: Vol. 104 Issue: 2 Pages: 297-307

    • DOI

      10.1007/s00022-013-0162-6

    • NAID

      120005463956

    • Related Report
      2013 Annual Research Report 2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Completely integrable implicit ordinary differential equations2012

    • Author(s)
      Masatomo Takahashi
    • Journal Title

      Yokohama Mathematical Journal

      Volume: Vol. 58 Pages: 17-29

    • NAID

      120006459658

    • Related Report
      2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Pedal foliations and Gauss maps of hypersurfaces in Euclidean space2012

    • Author(s)
      Shyuichi Izumiya, Masatomo Takahashi
    • Journal Title

      Journal of Singularities

      Volume: 6 Pages: 84-97

    • DOI

      10.5427/jsing.2012.6g

    • Related Report
      2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] On caustics of submanifolds and canal hypersurfaces in Euclidean space2012

    • Author(s)
      Shyuichi Iziumiya and Masatomo Takahash
    • Journal Title

      Topology and its Applications

      Volume: 159(2) Issue: 2 Pages: 509-518

    • DOI

      10.1016/j.topol.2011.09.025

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] On caustics of submanifolds and canal hypersurfaces in Euclidean space2012

    • Author(s)
      S. Izumiya
    • Journal Title

      Topology and its Applications

      Volume: 159 Pages: 501-508

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Asymmetry in singularities of tangent surfaces in contact-cone Legendre-null duality2011

    • Author(s)
      G. Ishikawa, Y. Machida, M. Takahashi
    • Journal Title

      Journal of Singularities

      Volume: Vol. 3 Pages: 126-143

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Presentation] フロンタルの縮閉線と伸開線について2014

    • Author(s)
      高橋雅朋
    • Organizer
      第21回沼津研究集会
    • Place of Presentation
      沼津高専
    • Year and Date
      2014-03-08
    • Related Report
      2013 Final Research Report
  • [Presentation] Evolutes and involutes of fronts in the Euclidean plane, Geometric Singularity Theory2013

    • Author(s)
      Masatomo Takahashi
    • Organizer
      Polish-Japanese Singularity Theory Working Days
    • Place of Presentation
      Banach Center, Warsaw (Polish)
    • Year and Date
      2013-08-29
    • Related Report
      2013 Final Research Report
  • [Presentation] 縮閉線と伸開線について2013

    • Author(s)
      高橋雅朋
    • Organizer
      第21回沼津研究集会
    • Place of Presentation
      沼津高専
    • Year and Date
      2013-03-07
    • Related Report
      2013 Final Research Report
  • [Presentation] On Legendre curves2012

    • Author(s)
      高橋雅朋
    • Organizer
      可微分写像の特異点論とその応用
    • Place of Presentation
      日本大学文理学部
    • Year and Date
      2012-11-30
    • Related Report
      2013 Final Research Report
  • [Presentation] Evolutes of fronts in the Euclidean plane2012

    • Author(s)
      高橋雅朋
    • Organizer
      数理研研究集会「特異点論の真髄の探求」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2012-11-30
    • Related Report
      2013 Final Research Report
  • [Presentation] Completely integrable first order ordinary differential equations2012

    • Author(s)
      Masatomo Takahashi
    • Organizer
      An international workshop in Singularity Theory, its Applications and Future Prospects
    • Place of Presentation
      University of Liverpool
    • Year and Date
      2012-06-21
    • Related Report
      2013 Final Research Report
  • [Presentation] Singularities of smooth mappings with patterns2012

    • Author(s)
      Masatomo Takahashi
    • Organizer
      2012 Spring Western Section Meeting
    • Place of Presentation
      University of Hawaii
    • Year and Date
      2012-03-03
    • Related Report
      2013 Final Research Report 2011 Research-status Report
  • [Presentation] 特異点論と常微分方程式の定性理論(1)(2)2011

    • Author(s)
      高橋雅朋
    • Organizer
      広島幾何学研究集会2011
    • Place of Presentation
      広島大学
    • Year and Date
      2011-10-06
    • Related Report
      2013 Final Research Report
  • [Presentation] 常微分方程式の定性理論の構築2011

    • Author(s)
      高橋雅朋
    • Organizer
      室蘭夏の学校
    • Place of Presentation
      室蘭工業大学
    • Year and Date
      2011-08-29
    • Related Report
      2013 Final Research Report 2011 Research-status Report
  • [Presentation] Generic classifications of first order multi-Clairaut type2011

    • Author(s)
      Masatomo Takahashi
    • Organizer
      Workshop on Singularities in Geometry and Applications
    • Place of Presentation
      Bedlewo (Polish)
    • Year and Date
      2011-05-20
    • Related Report
      2013 Final Research Report
  • [Presentation] Generic classifications of first order multi-Clairaut type2011

    • Author(s)
      Masatomo Takahashi
    • Organizer
      Workshop on Singularities in Geometry and Applications
    • Place of Presentation
      Poland (Bedlewo)
    • Related Report
      2011 Research-status Report
  • [Presentation] 特異点論と常微分方程式の定性理論(1)(2)2011

    • Author(s)
      高橋 雅朋
    • Organizer
      広島幾何学研究集会2011(招待講演)
    • Place of Presentation
      広島大学
    • Related Report
      2011 Research-status Report
  • [Presentation] フロンタルの縮閉線と伸開線について

    • Author(s)
      高橋 雅朋
    • Organizer
      第21回沼津研究集会
    • Place of Presentation
      沼津高專(静岡県、沼津市)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Evolutes and involutes of fronts in the Euclidean plane

    • Author(s)
      Masatomo Takahashi
    • Organizer
      Geometric Singularity Theory. Polish-Japanese Singularity Theory Working Days
    • Place of Presentation
      Banach Center, Warsaw, Polish
    • Related Report
      2013 Annual Research Report
  • [Presentation] 縮閉線と伸開線について

    • Author(s)
      高橋 雅朋
    • Organizer
      第20回沼津研究集会
    • Place of Presentation
      沼津高専
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] On Legendre curves

    • Author(s)
      高橋 雅朋
    • Organizer
      可微分写像の特異点論とその応用
    • Place of Presentation
      日本大学文理学部
    • Related Report
      2012 Research-status Report
  • [Presentation] Evolutes of fronts in the Euclidean plane

    • Author(s)
      高橋 雅朋
    • Organizer
      数理研研究集会「特異点論の真髄の探求」
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Completely integrable first order ordinary differential equations

    • Author(s)
      Masatomo Takahashi
    • Organizer
      An international workshop in Singularity Theory, its Applications and Future Prospects
    • Place of Presentation
      University of Liverpool, United Kingdom
    • Related Report
      2012 Research-status Report
  • [Remarks]

    • URL

      http://www.mmm.muroran-it.ac.jp/~masatomo/Mathematics.index.html

    • Related Report
      2013 Final Research Report
  • [Remarks] Mathematics

    • URL

      http://www.mmm.muroran-it.ac.jp/~masatomo/Mathematics.index.html

    • Related Report
      2013 Annual Research Report
  • [Remarks] HP

    • URL

      http://www.mmm.muroran-it.ac.jp/~masatomo/Mathematics.index.html

    • Related Report
      2012 Research-status Report
  • [Remarks]

    • URL

      http://www.mmm.muroran-it.ac.jp/~masatomo/Mathematics.index.html

    • Related Report
      2011 Research-status Report

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Published: 2011-08-05   Modified: 2019-07-29  

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