Differential geometric study of surfaces with singularities

• Project/Area Number: 20740028
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Gifu 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: ¥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.

Research Products

• [Journal Article] Criteria for D4 singularities of fronts (2011)
  • Author(s): Kentaro Saji
  • Journal Title: Tohoku Math.J. 63 Pages: 137-147
  • Peer Reviewed
• [Journal Article] A2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space (2011)
  • Author(s): Kentaro Saji, Masaaki Umehara, Kotaro Yamada
  • Journal Title: Kodai Math.J. (to appear)
  • NAID: 130004687956
  • Peer Reviewed
• [Journal Article] Criteria for D4 singualrities of fronts (2011)
  • Author(s): Kentaro Saji
  • Journal Title: Tohoku Math.J. Volume: 63 Pages: 137-147
  • Peer Reviewed
• [Journal Article] Criteria for singularities of smooth maps from the plane into the plane and their applications (2010)
  • Author(s): Kentaro Saji
  • Journal Title: Hiroshima Math.J. 40 Pages: 229-239
  • Peer Reviewed
• [Journal Article] Singularities of Blaschke normal maps of convex surfaces (2010)
  • Author(s): Kentaro Saji, Masaaki Umehara, Kotaro Yamada
  • Journal Title: C.R.Math.Acad.Sci.Paris 348 Pages: 665-668
  • Peer Reviewed
• [Journal Article] The duality between singular points and inflection points on wave fronts (2010)
  • Author(s): Kentaro Saji, Masaaki Umehara, Kotaro Yamada
  • Journal Title: Osaka J.Math. 47 Pages: 591-607
  • NAID: 120004844268
  • Peer Reviewed
• [Journal Article] Horospherical flat surfaces in hyperbolic 3-space (2010)
  • Author(s): Shyuichi Izumiya, Kentaro Saji, Masatomo Takahashi
  • Journal Title: J.Math.Soc.Japan 62 Pages: 789-849
  • NAID: 10027870826
  • Peer Reviewed
• [Journal Article] Criteria for cuspidal Sk singularities and their applications (2010)
  • Author(s): Kentaro Saji
  • Journal Title: J.Gokova Geom.Topol. 4 Pages: 67-81
  • Peer Reviewed
• [Journal Article] The mandala of Legendrian dualities for pseudo-spheres of Lorentz-Minkowski space and "flat" spacelike surfaces (2010)
  • Author(s): Shyuichi Izumiya, Kentaro Saji
  • Journal Title: J.Singularity 2 Pages: 97-127
  • NAID: 120006459641
  • Peer Reviewed
• [Journal Article] Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts (2010)
  • Author(s): Kentaro Saji, Masaaki Umehara, Kotaro Yamada
  • Journal Title: J.Geometric Analysis. Online First (to appear)
  • Peer Reviewed
• [Journal Article] Criteria for singularities of smooth maps from the plane into the plane and their applications (2010)
  • Author(s): Kentaro Saji
  • Journal Title: Hiroshima Math.J. Volume: 40 Pages: 229-239
  • Peer Reviewed
• [Journal Article] Singularities of Blaschke normal maps of convex surfaces (2010)
  • Author(s): Kentaro Saji, Masaaki Umehara, Kotaro Yamada
  • Journal Title: C.R.Math.Acad.Sci.Paris Volume: 348 Pages: 665-668
  • Peer Reviewed
• [Journal Article] The duality between singular points and inflection points on wave fronts (2010)
  • Author(s): Kentaro Saji, Masaaki Umehara, Kotaro Yamada
  • Journal Title: Osaka J.Math. Volume: 47 Pages: 591-607
  • NAID: 120004844268
  • Peer Reviewed
• [Journal Article] Horospherical flat surfaces in hyperbolic 3-space (2010)
  • Author(s): Shyuichi Izumiya, Kentaro Saji, Masatomo Takahashi
  • Journal Title: J.Math.Soc.Japan Volume: 62 Pages: 789-849
  • NAID: 10027870826
  • Peer Reviewed
• [Journal Article] Criteria for cuspidal Sk singularities and their applications (2010)
  • Author(s): Kentaro Saji
  • Journal Title: J.Gokova Geom.Topol. Volume: 4 Pages: 67-81
  • Peer Reviewed
• [Journal Article] The mandala of Legendrian dualities for pseudo-spheres of Lorentz-Min kowski space and "flat" spacelike surfaces (2010)
  • Author(s): Shyuichi Izumiya, Kentaro Saji
  • Journal Title: J.Singularity Volume: 2 Pages: 97-127
  • NAID: 120006459641
  • Peer Reviewed
• [Journal Article] Criteria for singularities of smooth maps from the plane into the plane and their applications (2010)
  • Author(s): K.Saji
  • Journal Title: Hiroshima Math. J. 40(予定)(出版受理)
  • Peer Reviewed
• [Journal Article] Ak singularities of wave fronts (2009)
  • Author(s): Kentaro Saji, Masaaki Umehara, Kotaro Yamada
  • Journal Title: Math.Proc.Cambridge Philos.Soc. 146 Pages: 731-746
  • Peer Reviewed
• [Journal Article] Hyperbolic Schwarz map of the confluent hypergeometric differential equation (2009)
  • Author(s): Kentaro Saji, Takeshi Sasaki, Masaaki Yoshida
  • Journal Title: J.Math.Soc.Japan 61 Pages: 559-578
  • NAID: 10024905825
  • Peer Reviewed
• [Journal Article] Flat lightlike hypersurfaces in Lorentz Minkowski 4-space (2009)
  • Author(s): Shyuichi Izumiya, Marila del Carmen Romero Fuster, Kentaro Saji
  • Journal Title: J.Geom.Phys. 59 Pages: 1528-1546
  • NAID: 120001672718
  • Peer Reviewed
• [Journal Article] The geometry of fronts (2009)
  • Author(s): Kentaro Saji, Masaaki Umehara, Kotaro Yamada
  • Journal Title: Ann.of Math. 169 Pages: 491-529
  • Peer Reviewed
• [Journal Article] A_k singularities of wave fronts (2009)
  • Author(s): K.Saji, M.Umehara, K.Yamada
  • Journal Title: Math. Proc. Camb. Philos. Soc. 146 Pages: 731-746
  • Peer Reviewed
• [Journal Article] Hyperbolic Schwarz map of the confluent hypergeometric differential equation (2009)
  • Author(s): K.Saji, T.Sasaki, M.Yoshida
  • Journal Title: J.Math. Soc. Japan 61 Pages: 559-578
  • NAID: 10024905825
  • Peer Reviewed
• [Journal Article] A_k-singularities of wave fronts (2009)
  • Author(s): K. Saji, M. Umahara, K. Yamada
  • Journal Title: Math. Proc. Camb. Philos. Soc (掲載決定)
  • Peer Reviewed
• [Journal Article] Cuspidal cross caps and singularities of maximal surfaces (2008)
  • Author(s): S. Fujimori, K. Saji, M. Umehara, K. Yamada
  • Journal Title: Math. Z 259 Pages: 827-848
  • 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
  • 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)
  • 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
  • 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
  • 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
• [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
• [Presentation] Geometry of cuspidal edges (2010)
  • 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
• [Presentation] Geometry of cuspidal edges (2010)
  • 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
• [Presentation] Fronts to be graphs of continuous functions (2010)
  • 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
• [Presentation] Fronts to be graphs of continuous functions (2010)
  • 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
• [Presentation] Recognition criteria for cuspidal Mond singularities and their applications (2009)
  • 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
• [Presentation] Recognition criteria for cuspidal Mond singularities and their applications (2009)
  • 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
• [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
• [Presentation] Criteria for singualrities of fronts and its applications (2009)
  • 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
• [Presentation] Criteria for singularities of fronts and its applications (2009)
  • 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
• [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
• [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
• [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
• [Presentation] Shapes of swallowtail as discriminant sets (2008)
  • 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
• [Presentation] Criteria for singularities of the maps from the plane into the plane and their applications (2008)
  • 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
• [Remarks] ホームページ等
  • URL: http://www1.gifu-u.ac.jp/~ksaji/math/kaken.html
• [Remarks]
  • URL: http://www1.gifu-u.ac.jp/~ksaji/math/kaken.html
• [Remarks]
  • URL: http://www1.gifu-u.ac.jp/~ksaji/math/kaken.html
• [Remarks]
  • URL: http://wwwl.gifu-u.ac.jp/~ksaji/math/kaken.html