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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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

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

First available in Project Euclid: 20 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Singular surfaces umbilic curvature curvature parabola Whitney umbrella cross-cap


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.

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