Tohoku Mathematical Journal

Contact properties of surfaces in ${\mathbb R}^3$ with corank 1 singularities

Luciana F. Martins and Juan J. Nuño-Ballesteros

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Abstract

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.

Article information

Source
Tohoku Math. J. (2), Volume 67, Number 1 (2015), 105-124.

Dates
First available in Project Euclid: 20 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1429549581

Digital Object Identifier
doi:10.2748/tmj/1429549581

Mathematical Reviews number (MathSciNet)
MR3337965

Zentralblatt MATH identifier
1320.58023

Subjects
Primary: 58K05: Critical points of functions and mappings
Secondary: 57R45: Singularities of differentiable mappings 53A05: Surfaces in Euclidean space

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

Citation

Martins, Luciana F.; Nuño-Ballesteros, Juan J. Contact properties of surfaces in ${\mathbb R}^3$ with corank 1 singularities. Tohoku Math. J. (2) 67 (2015), no. 1, 105--124. doi:10.2748/tmj/1429549581. https://projecteuclid.org/euclid.tmj/1429549581


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