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Differential geometric study of surfaces with singularities

Research Project

Project/Area Number 20740028
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionGifu University

Principal Investigator

SAJI Kentaro  Gifu University, 教育学部, 准教授 (70451432)

Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Keywords特異点 / ルジャンドル特異点 / 波面 / 曲率 / 線織面 / カスプ辺 / 判定法 / 特異曲率 / 写像の図式 / 擬球面 / ウルフ図形 / 認識問題 / 平行曲面 / ミンコフスキ空間 / 臍点 / ジェネリック微分幾何学
Research Abstract

I gave criteria for cuspidal lips, cuspidal beaks, cuspidal butterfly, umbilics of singularities fronts. Furthermore, I gave criteria for Mond singularity and higher dimensional singularities of fronts.
As applications of them, I clarified relationships between singularities and differential geometric properties of surfaces. Furthermore, I gave an intrinsic formulation of fronts.
Using this, I gave formulas about numbers of singularities.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (49 results)

All 2011 2010 2009 2008 Other

All Journal Article (29 results) (of which Peer Reviewed: 29 results) Presentation (16 results) Remarks (4 results)

  • [Journal Article] Criteria for D4 singularities of fronts2011

    • Author(s)
      Kentaro Saji
    • Journal Title

      Tohoku Math.J. 63

      Pages: 137-147

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] A2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space2011

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

      Kodai Math.J. (to appear)

    • NAID

      130004687956

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Criteria for D4 singualrities of fronts2011

    • Author(s)
      Kentaro Saji
    • Journal Title

      Tohoku Math.J.

      Volume: 63 Pages: 137-147

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Criteria for singularities of smooth maps from the plane into the plane and their applications2010

    • Author(s)
      Kentaro Saji
    • Journal Title

      Hiroshima Math.J. 40

      Pages: 229-239

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Singularities of Blaschke normal maps of convex surfaces2010

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

      C.R.Math.Acad.Sci.Paris 348

      Pages: 665-668

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] The duality between singular points and inflection points on wave fronts2010

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

      Osaka J.Math. 47

      Pages: 591-607

    • NAID

      120004844268

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Horospherical flat surfaces in hyperbolic 3-space2010

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

      J.Math.Soc.Japan 62

      Pages: 789-849

    • NAID

      10027870826

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Criteria for cuspidal Sk singularities and their applications2010

    • Author(s)
      Kentaro Saji
    • Journal Title

      J.Gokova Geom.Topol. 4

      Pages: 67-81

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] The mandala of Legendrian dualities for pseudo-spheres of Lorentz-Minkowski space and "flat" spacelike surfaces2010

    • Author(s)
      Shyuichi Izumiya, Kentaro Saji
    • Journal Title

      J.Singularity 2

      Pages: 97-127

    • NAID

      120006459641

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts2010

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

      J.Geometric Analysis. Online First (to appear)

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Criteria for singularities of smooth maps from the plane into the plane and their applications2010

    • Author(s)
      Kentaro Saji
    • Journal Title

      Hiroshima Math.J.

      Volume: 40 Pages: 229-239

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Singularities of Blaschke normal maps of convex surfaces2010

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

      C.R.Math.Acad.Sci.Paris

      Volume: 348 Pages: 665-668

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The duality between singular points and inflection points on wave fronts2010

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

      Osaka J.Math.

      Volume: 47 Pages: 591-607

    • NAID

      120004844268

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Horospherical flat surfaces in hyperbolic 3-space2010

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

      J.Math.Soc.Japan

      Volume: 62 Pages: 789-849

    • NAID

      10027870826

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Criteria for cuspidal Sk singularities and their applications2010

    • Author(s)
      Kentaro Saji
    • Journal Title

      J.Gokova Geom.Topol.

      Volume: 4 Pages: 67-81

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The mandala of Legendrian dualities for pseudo-spheres of Lorentz-Min kowski space and "flat" spacelike surfaces2010

    • Author(s)
      Shyuichi Izumiya, Kentaro Saji
    • Journal Title

      J.Singularity

      Volume: 2 Pages: 97-127

    • NAID

      120006459641

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Criteria for singularities of smooth maps from the plane into the plane and their applications2010

    • Author(s)
      K.Saji
    • Journal Title

      Hiroshima Math. J. 40(予定)(出版受理)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ak singularities of wave fronts2009

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

      Math.Proc.Cambridge Philos.Soc. 146

      Pages: 731-746

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Hyperbolic Schwarz map of the confluent hypergeometric differential equation2009

    • Author(s)
      Kentaro Saji, Takeshi Sasaki, Masaaki Yoshida
    • Journal Title

      J.Math.Soc.Japan 61

      Pages: 559-578

    • NAID

      10024905825

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Flat lightlike hypersurfaces in Lorentz Minkowski 4-space2009

    • Author(s)
      Shyuichi Izumiya, Marila del Carmen Romero Fuster, Kentaro Saji
    • Journal Title

      J.Geom.Phys. 59

      Pages: 1528-1546

    • NAID

      120001672718

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] The geometry of fronts2009

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

      Ann.of Math. 169

      Pages: 491-529

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] A_k singularities of wave fronts2009

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

      Math. Proc. Camb. Philos. Soc. 146

      Pages: 731-746

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hyperbolic Schwarz map of the confluent hypergeometric differential equation2009

    • Author(s)
      K.Saji, T.Sasaki, M.Yoshida
    • Journal Title

      J.Math. Soc. Japan 61

      Pages: 559-578

    • NAID

      10024905825

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A_k-singularities of wave fronts2009

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

      Math. Proc. Camb. Philos. Soc (掲載決定)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Cuspidal cross caps and singularities of maximal surfaces2008

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

      Math. Z 259

      Pages: 827-848

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Great circular surfaces in the three-sphere

    • Author(s)
      Shyuichi Izumiya, Takayuki Nagai, Kentaro Saji
    • Journal Title

      Differential Geom.appl. (to appear)

    • NAID

      120006459655

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts

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

      J.Geometric Analysis

      Volume: (掲載決定 未定)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Great circular surfaces in the three-sphere

    • Author(s)
      Shyuichi Izumiya, Takayuki Nagai, Kentaro Saji
    • Journal Title

      Differential Geom.appl.

      Volume: (掲載決定 未定)

    • NAID

      120006459655

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space

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

      Kodai Math.J.

      Volume: (掲載決定 未定)

    • NAID

      130004687956

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] Applications of criteria for singularities -Singularities of ruling maps in the plane-2011

    • Author(s)
      佐治健太郎
    • Organizer
      研究集会「札幌・長春特異点論セミナー」
    • Place of Presentation
      北海道大学
    • Year and Date
      2011-02-18
    • Related Report
      2010 Final Research Report
  • [Presentation] Applications of criteria for singularities-Singularities of ruling maps in the plane-2011

    • Author(s)
      佐治健太郎
    • Organizer
      札幌・長春特異点論セミナー
    • Place of Presentation
      北海道大学
    • Year and Date
      2011-02-18
    • Related Report
      2010 Annual Research Report
  • [Presentation] Geometry of cuspidal edges2010

    • Author(s)
      佐治健太郎
    • Organizer
      研究集会「The 11th Workshop on Singularities」
    • Place of Presentation
      ブラジル・サンカルロス
    • Year and Date
      2010-07-26
    • Related Report
      2010 Final Research Report
  • [Presentation] Geometry of cuspidal edges2010

    • Author(s)
      佐治健太郎
    • Organizer
      The 11th Workshop on Singularities
    • Place of Presentation
      ブラジル・サンカルロス
    • Year and Date
      2010-07-26
    • Related Report
      2010 Annual Research Report
  • [Presentation] Fronts to be graphs of continuous functions2010

    • Author(s)
      佐治健太郎
    • Organizer
      研究集会「空間認識のための特異点論」
    • Place of Presentation
      伊勢市観光文化会館
    • Year and Date
      2010-06-02
    • Related Report
      2010 Final Research Report
  • [Presentation] Fronts to be graphs of continuous functions2010

    • Author(s)
      佐治健太郎
    • Organizer
      空間認識のための特異点論
    • Place of Presentation
      伊勢市観光文化会館
    • Year and Date
      2010-06-02
    • Related Report
      2010 Annual Research Report
  • [Presentation] Recognition criteria for cuspidal Mond singularities and their applications2009

    • Author(s)
      佐治健太郎
    • Organizer
      研究集会「Third Japanese/Australian Workshop on Real and Complex Singularities」
    • Place of Presentation
      オーストラリア・シドニー
    • Year and Date
      2009-09-16
    • Related Report
      2010 Final Research Report
  • [Presentation] Recognition criteria for cuspidal Mond singularities and their applications2009

    • Author(s)
      佐治健太郎
    • Organizer
      Third Japanese Australian workshop on real and complex singularities
    • Place of Presentation
      オーストラリア(シドニー)
    • Year and Date
      2009-09-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] 波面のD型特異点とその判定法2009

    • Author(s)
      佐治健太郎
    • Organizer
      幾何セミナー
    • Place of Presentation
      大阪大学理学部
    • Year and Date
      2009-07-27
    • Related Report
      2009 Annual Research Report
  • [Presentation] Criteria for singualrities of fronts and its applications2009

    • Author(s)
      佐治健太郎
    • Organizer
      研究集会「International Conference on Mathematical Control Theory and Mechanics」
    • Place of Presentation
      ロシア・スズダリ
    • Year and Date
      2009-07-04
    • Related Report
      2010 Final Research Report
  • [Presentation] Criteria for singularities of fronts and its applications2009

    • Author(s)
      佐治健太郎
    • Organizer
      Mathematical control theory and mechanics
    • Place of Presentation
      ロシア(スーズダリ)
    • Year and Date
      2009-07-04
    • Related Report
      2009 Annual Research Report
  • [Presentation] モンド特異点の判定とその応用2009

    • Author(s)
      佐治健太郎
    • Organizer
      研究集会「トポロジーと写像の特異点」
    • Place of Presentation
      信州大学
    • Year and Date
      2009-06-05
    • Related Report
      2010 Final Research Report
  • [Presentation] モンド特異点の判定とその応用2009

    • Author(s)
      佐治健太郎
    • Organizer
      トポロジーと写像の特異点
    • Place of Presentation
      信州大学理学部
    • Year and Date
      2009-06-05
    • Related Report
      2009 Annual Research Report
  • [Presentation] 可微分写像にあらわれる特異点の認識について2009

    • Author(s)
      佐治健太郎
    • Organizer
      応用特異点論の現状と展望
    • Place of Presentation
      北海道大学数学連携研究センター
    • Year and Date
      2009-03-16
    • Related Report
      2008 Annual Research Report
  • [Presentation] Shapes of swallowtail as discriminant sets2008

    • Author(s)
      佐治健太郎
    • Organizer
      International workshop on Real and Complex singularities
    • Place of Presentation
      サンカルロス(ブラジル)
    • Year and Date
      2008-07-31
    • Related Report
      2008 Annual Research Report
  • [Presentation] Criteria for singularities of the maps from the plane into the plane and their applications2008

    • Author(s)
      佐治健太郎
    • Organizer
      幾何学的視点からの特異点論
    • Place of Presentation
      九州大学理学部
    • Year and Date
      2008-06-04
    • Related Report
      2008 Annual Research Report
  • [Remarks] ホームページ等

    • URL

      http://www1.gifu-u.ac.jp/~ksaji/math/kaken.html

    • Related Report
      2010 Final Research Report
  • [Remarks]

    • URL

      http://www1.gifu-u.ac.jp/~ksaji/math/kaken.html

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://www1.gifu-u.ac.jp/~ksaji/math/kaken.html

    • Related Report
      2009 Annual Research Report
  • [Remarks]

    • URL

      http://wwwl.gifu-u.ac.jp/~ksaji/math/kaken.html

    • Related Report
      2008 Annual Research Report

URL: 

Published: 2008-04-01   Modified: 2016-04-21  

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