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Singularity theoretic study of surface singularities

Research Project

Project/Area Number 18K03301
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionKobe University

Principal Investigator

Saji Kentaro  神戸大学, 理学研究科, 教授 (70451432)

Project Period (FY) 2018-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords特異点 / 波面 / フロンタル / 特異点の判定法 / 特異点の曲率 / 接触構造 / 束準同型 / 判定法 / 漸近線 / 特性線 / 特異点判定法 / 平均曲率 / コースティック / 型変化 / 特異点の微分幾何 / 型変化曲面 / リボクール変換 / 曲率線 / カスプ辺 / ツバメの尾 / 特異点論 / 曲面論 / 特異点の認識問題 / 輪郭線
Outline of Final Research Achievements

A useful criteria for sharksfin and deltoid singularities which is the rank zero singularities that appear most frequently from the plane into the plane is given. Furthermore, the geometric meaning of this condition is given. A developable surface along the singular curve of a swallowtail is constructed, and the properties of this surface, in particular singularity, conditions for cylinders and cones are given. A new fundamental invariant, the axial curvature, is given for non-frontal singularities. A method to construct for a surface of revolution with given mean curvature with diverging points is given. A cylindrical direction for singular surfaces, cuspidal edges and Whitney umbrella, which measures the contact of a surface with a cylinder, is given and its properties are investigated. For caustics, which are surfaces without parametrization is given by using the blow-up method, and the configuration of curves of vanishing Gaussian curvature is clarified.

Academic Significance and Societal Importance of the Research Achievements

平面間の写像の余階数2の特異点で一番よく現れる特異点に対して使いやすい判定法を得た。さらにこの条件の幾何学的意味も与えた。この基本的な特異点に対して使いやすい判定法が与えられたことは今後これらの特異点に対応する特異性が調べやすくなったという意味であり、非常に大きな学術的意味を持っている。ツバメの尾の特異点曲線に沿う可展面を構成、フロンタルでない特異点に対して軸曲率なる新しい基本的な不変量を与えたこと、特異点を持つ曲面の円柱との接触による円柱方向の定義、与えられた発散することを許す平均曲率を持つものの構成は今後これらの特異点の研究が進むことを意味し、意義深い。

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (58 results)

All 2024 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (13 results) Journal Article (23 results) (of which Int'l Joint Research: 8 results,  Peer Reviewed: 23 results,  Open Access: 5 results) Presentation (16 results) (of which Int'l Joint Research: 9 results,  Invited: 7 results) Remarks (1 results) Funded Workshop (5 results)

  • [Int'l Joint Research] Valencia University(スペイン)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Warszawa University of Technology(ポーランド)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Sao Paulo State University(ブラジル)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] サンパウロ州立大学/サンパウロ大学(ブラジル)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] バレンシア大学(スペイン)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] ICMC-USP/IBLICE-UNESP(ブラジル)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] バレンシア大(スペイン)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] バレンシア大学(スペイン)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] サンパウロ大学/サンパウロ州立大学(ブラジル)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] サンパウロ州立大学/サンパウロ大学(ブラジル)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] バレンシア大学(スペイン)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] バレンシア大学(スペイン)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] サンパウロ州立大学(ブラジル)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Symmetries of cross caps2023

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

      Tohoku Mathematical Journal

      Volume: 75 Issue: 1 Pages: 131-141

    • DOI

      10.2748/tmj.20211203

    • Related Report
      2023 Annual Research Report 2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Legendrian dual surfaces of a spacelike curve in the 3-dimensional lightcone2022

    • Author(s)
      Saji Kentaro、Yildirim Handan
    • Journal Title

      Journal of Geometry and Physics

      Volume: 179 Pages: 104593-104593

    • DOI

      10.1016/j.geomphys.2022.104593

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A generalization of Zakalyukin's lemma, and symmetries of surface singularities2022

    • Author(s)
      Honda Atsufumi、Naokawa Kosuke、Saji Kentaro、Umehara Masaaki、Yamada Kotaro
    • Journal Title

      Journal of Singularities

      Volume: 25 Pages: 299-324

    • DOI

      10.5427/jsing.2022.25m

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The axial curvature for corank 1 singular surfaces2022

    • Author(s)
      Oset Sinha Raul、Saji Kentaro
    • Journal Title

      Tohoku Mathematical Journal

      Volume: 74 Issue: 3 Pages: 365-388

    • DOI

      10.2748/tmj.20210322

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Flat surfaces along swallowtails2022

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

      Kobe Journal of Mathematics

      Volume: 39 Pages: 63-80

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Geometry of bifurcation sets of generic unfoldings of corank two functions2021

    • Author(s)
      Saji Kentaro、dos Santos Samuel Paulino
    • Journal Title

      Monatshefte fur Mathematik

      Volume: 196 Issue: 3 Pages: 553-575

    • DOI

      10.1007/s00605-021-01607-8

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Behavior of principal curvatures of frontals near non-front singular points and their applications2021

    • Author(s)
      Saji Kentaro、Teramoto Keisuke
    • Journal Title

      Journal of Geometry

      Volume: 112 Issue: 3

    • DOI

      10.1007/s00022-021-00605-3

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Geometry of lightlike locus on mixed type surfaces in Lorentz-Minkowski 3-space from a contact viewpoint2021

    • Author(s)
      Honda Atsufumi、Izumiya Shyuichi、Saji Kentaro、Teramoto Keisuke
    • Journal Title

      Tsukuba Journal of Mathematics

      Volume: 45 Issue: 1 Pages: 51-68

    • DOI

      10.21099/tkbjm/20214501051

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Capturing information on curves and surfaces from their projected images2020

    • Author(s)
      Hasegawa Masaru、Kabata Yutaro、Saji Kentaro
    • Journal Title

      International Journal of Mathematics for Industry

      Volume: 12 Issue: 01 Pages: 2050004-2050004

    • DOI

      10.1142/s2661335220500045

    • NAID

      120006993629

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Duality on generalized cuspidal edges preserving singular set images and first fundamental forms2020

    • Author(s)
      Honda Atsufumi、Naokawa Kosuke、Saji Kentaro、Umehara Masaaki、Yamada Kotaro
    • Journal Title

      Journal of Singularities

      Volume: 22 Pages: 59-91

    • DOI

      10.5427/jsing.2020.22e

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Horo-flat surfaces along cuspidal edges in the hyperbolic space2020

    • Author(s)
      Izumiya Shyuichi、Romero-Fuster Maria Carmen、Saji Kentaro、Takahashi Masatomo
    • Journal Title

      Journal of Singularities

      Volume: 22 Pages: 40-58

    • DOI

      10.5427/jsing.2020.22d

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Cuspidal edges with the same first fundamental forms along a knot2020

    • Author(s)
      Honda Atsufumi、Naokawa Kosuke、Saji Kentaro、Umehara Masaaki、Yamada Kotaro
    • Journal Title

      Journal of Knot Theory and Its Ramifications

      Volume: 29 Issue: 07 Pages: 2050047-2050047

    • DOI

      10.1142/s0218216520500479

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Curved foldings with common creases and crease patterns2020

    • Author(s)
      Honda Atsufumi、Naokawa Kosuke、Saji Kentaro、Umehara Masaaki、Yamada Kotaro
    • Journal Title

      Advances in Applied Mathematics

      Volume: 121 Pages: 102083-102083

    • DOI

      10.1016/j.aam.2020.102083

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifolds2020

    • Author(s)
      Honda Atsufumi、Saji Kentaro、Teramoto Keisuke
    • Journal Title

      Advances in Mathematics

      Volume: 365 Pages: 107036-107036

    • DOI

      10.1016/j.aim.2020.107036

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Dualities of Differential Geometric Invariants on Cuspidal Edges on Flat Fronts in the Hyperbolic Space and the de Sitter Space2020

    • Author(s)
      Saji, Kentaro; Teramoto, Keisuke
    • Journal Title

      Mediterr. J. Math.

      Volume: 17 Issue: 2

    • DOI

      10.1007/s00009-020-1474-z

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Geometric invariants of 5/2-cuspidal edges2019

    • Author(s)
      Honda, Atsufumi; Saji, Kentaro
    • Journal Title

      Kodai Mathematical Journal

      Volume: 42 Issue: 3 Pages: 496-525

    • DOI

      10.2996/kmj/1572487230

    • NAID

      130007742214

    • ISSN
      0386-5991, 1881-5472
    • Year and Date
      2019-10-31
    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Singular surfaces of revolution with prescribed unbounded mean curvature2019

    • Author(s)
      Martins, Luciana F.; Saji, Kentaro; Santos, Samuel P. Dos; Teramoto, Keisuke
    • Journal Title

      An. Acad. Brasil. Cienc.

      Volume: 91 Issue: 3

    • DOI

      10.1590/0001-3765201920170865

    • NAID

      120007017122

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] MR4025586 Pending Martins, Luciana F.; Saji, Kentaro; Teramoto, Keisuke Singularities of a surface given by Kenmotsu-type formula in Euclidean three-space2019

    • Author(s)
      Martins, Luciana F.; Saji, Kentaro; Teramoto, Keisuke
    • Journal Title

      Sao Paulo J. Math. Sci.

      Volume: 13 Issue: 2 Pages: 663-677

    • DOI

      10.1007/s40863-019-00138-3

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank2019

    • Author(s)
      K. Saji and A. Tsuchida
    • Journal Title

      Rocky Mountain J. Math.

      Volume: 49 Issue: 1 Pages: 335-354

    • DOI

      10.1216/rmj-2019-49-1-335

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Geometry of cuspidal edges with boundary2018

    • Author(s)
      Martins Luciana F.、Saji Kentaro
    • Journal Title

      Topology and its Applications

      Volume: 234 Pages: 209-219

    • DOI

      10.1016/j.topol.2017.11.024

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space2018

    • Author(s)
      A. Honda, M. Koiso and K. Saji
    • Journal Title

      Hokkaido Math. J.

      Volume: 47 Pages: 245-267

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Normal form of the swallowtail and its applications2018

    • Author(s)
      K. Saji
    • Journal Title

      Internat. J. Math.

      Volume: 29 Issue: 07 Pages: 1850046-1850046

    • DOI

      10.1142/s0129167x18500465

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the geometry of folded cuspidal edges2018

    • Author(s)
      Oset Sinha Raul、Saji Kentaro
    • Journal Title

      Revista Matematica Complutense

      Volume: - Issue: 3 Pages: 627-650

    • DOI

      10.1007/s13163-018-0257-6

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 波面のD4特異点の幾何2024

    • Author(s)
      佐治 健太郎
    • Organizer
      特異点論及びその周辺分野の深化と異分野への応用
    • Related Report
      2023 Annual Research Report
  • [Presentation] Geometry of singular surfaces and curves 12024

    • Author(s)
      Kentaro Saji
    • Organizer
      WORKSHOP on Algebraic and Analytic Singularities
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of singular surfaces and curves 22024

    • Author(s)
      Kentaro Saji
    • Organizer
      WORKSHOP on Algebraic and Analytic Singularities
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of singular surfaces and curves 32024

    • Author(s)
      Kentaro Saji
    • Organizer
      WORKSHOP on Algebraic and Analytic Singularities
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of D4 singularities of fronts2024

    • Author(s)
      Kentaro Saji
    • Organizer
      WORKSHOP on Algebraic and Analytic Singularities
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Geometry of central singular point of D4-plus bifurcation of fronts2024

    • Author(s)
      Kentaro Saji
    • Organizer
      International Workshop on Singularities in Geometry and applications
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Criteria for sharksfin and deltoid singularities from the plane into the plane and their applications2022

    • Author(s)
      佐治健太郎
    • Organizer
      位相幾何・微分幾何及びその周辺分野への特異点論の応用
    • Related Report
      2022 Research-status Report
  • [Presentation] Geometry of D4 singularities of fronts2022

    • Author(s)
      佐治健太郎
    • Organizer
      Singularities of Differentiable Maps and Its applications
    • Related Report
      2022 Research-status Report
  • [Presentation] D_4 関数の分岐集合の幾何2021

    • Author(s)
      佐治健太郎
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2021 Research-status Report
  • [Presentation] D_4 関数の分岐集合の幾何2021

    • Author(s)
      佐治健太郎
    • Organizer
      幾何や自然科学に現れる特異点
    • Related Report
      2020 Research-status Report
  • [Presentation] Geometric invariants of lightlike points of surfaces in 3-dimensional Lorentzian manifolds2019

    • Author(s)
      Kentaro Saji
    • Organizer
      6th International Workshop on Singularities in Generic Geometry and its Applications
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Contact-like relation of vector fields and functions and its applications2019

    • Author(s)
      Kentaro Saji
    • Organizer
      Hyperplane arrangements and Japanese Australian workshop on Real and Complex Singularities
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] 特異点をもつ曲面とカスプ辺との接触2019

    • Author(s)
      佐治健太郎
    • Organizer
      日本数学会トポロジー分科会
    • Related Report
      2018 Research-status Report
  • [Presentation] 与えられた発散する平均曲率をもつ特異点付き曲面2018

    • Author(s)
      佐治健太郎
    • Organizer
      微分幾何学・微分式系・特異点論の応用
    • Related Report
      2018 Research-status Report
  • [Presentation] SO(3) normal forms and differential geometric invariants of singular surfaces2018

    • Author(s)
      Kentaro Saji
    • Organizer
      5th edition of the International Workshop on Real and Complex Singularities
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Singularities of wave fronts and curvatures2018

    • Author(s)
      佐治健太郎
    • Organizer
      幾何学的視点からの形状形成
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] 研究費と得られた主な成果

    • URL

      http://www.math.kobe-u.ac.jp/HOME/saji/math/kaken.html

    • Related Report
      2023 Annual Research Report 2022 Research-status Report 2021 Research-status Report 2020 Research-status Report 2019 Research-status Report 2018 Research-status Report
  • [Funded Workshop] 特異点論特別月間-1 可微分写像の特異点論とその応用2022

    • Related Report
      2022 Research-status Report
  • [Funded Workshop] 特異点論特別月間-2 特異点論と幾何的トポロジー2022

    • Related Report
      2022 Research-status Report
  • [Funded Workshop] 特異点論特別月間-3 特異点論と微分幾何学2022

    • Related Report
      2022 Research-status Report
  • [Funded Workshop] 特異点論特別月間-4 特異点論における分類問題とその応用2022

    • Related Report
      2022 Research-status Report
  • [Funded Workshop] MSJ-SI 2022 応用特異点論の深化と展開2022

    • Related Report
      2022 Research-status Report

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Published: 2018-04-23   Modified: 2025-01-30  

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