Tohoku Mathematical Journal
2015
March
SECOND SERIES VOL. 67, NO. 1

Tohoku Math. J.
67 (2015), 105-124

Title CONTACT PROPERTIES OF SURFACES IN $\mathbb{R}^3$ WITH CORANK 1 SINGULARITIES

Author Luciana F. Martins and Juan J. Nuño-Ballesteros

(Received September 2, 2013, revised January 29, 2014)
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.

Mathematics Subject Classification. Primary 58K05; Secondary 57R45, 53A05.
Key words and phrases. Singular surfaces, umbilic curvature, curvature parabola, Whitney umbrella, cross-cap.

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