Open Access
2015 Contact properties of surfaces in ${\mathbb R}^3$ with corank 1 singularities
Luciana F. Martins, Juan J. Nuño-Ballesteros
Tohoku Math. J. (2) 67(1): 105-124 (2015). DOI: 10.2748/tmj/1429549581


We study the geometry of surfaces in $\mathbb{R}^3$ with corank 1 singularities. At a singular point we define the curvature parabola using the first and second fundamental forms of the surface, which contains all the local second order geometrical information about the surface. The curvature parabola is used to introduce the concepts of asymptotic directions and umbilic curvature, which are related to contact properties of the surface with planes and spheres.


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Luciana F. Martins. Juan J. Nuño-Ballesteros. "Contact properties of surfaces in ${\mathbb R}^3$ with corank 1 singularities." Tohoku Math. J. (2) 67 (1) 105 - 124, 2015.


Published: 2015
First available in Project Euclid: 20 April 2015

zbMATH: 1320.58023
MathSciNet: MR3337965
Digital Object Identifier: 10.2748/tmj/1429549581

Primary: 58K05
Secondary: 53A05 , 57R45

Keywords: cross-cap , curvature parabola , Singular surfaces , umbilic curvature , Whitney umbrella

Rights: Copyright © 2015 Tohoku University

Vol.67 • No. 1 • 2015
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