Research on surfaces and related differential equations from the view point of singularity theory

Research Project

Project/Area Number	15K04867
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Research Field	Geometry
Research Institution	Saitama University
Principal Investigator	FUKUI Toshizumi 埼玉大学, 理工学研究科, 教授 (90218892)
Project Period (FY)	2015-04-01 – 2018-03-31
Project Status	Completed (Fiscal Year 2017)
Budget Amount *help	¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000) Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000) Fiscal Year 2016: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000) Fiscal Year 2015: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Keywords	特異点論 / 特異点理論 / 特異点 / 曲面 / ホイットニーの傘
Outline of Final Research Achievements	The analysis of the contact between a nonsingular surface and the cylinders in the three　dimensional Euclidean space was completed satisfactorily. We give the necessary and sufficient conditions for the existence of a cylinder which has A1, A2, A3, A4, A5, D4, D5 contact with a given nonsingular surface. Cylindrical directions is defined as a generatrix of the cylinder which has contact with A >=3. A classification of their singular point is also given. Koenderink's formula show that Gauss curvature equals to the product of curvature of contour and normal curvature for a give direction. We also show Koenderink-type formulas for singular surface with Whitney umbrella.