Singularity theoretic study of surface singularities

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03301
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kobe 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
• Keywords: 特異点 / 波面 / フロンタル / 特異点の判定法 / 特異点の曲率

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.

Free Research Field

幾何学

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。