Surfaces with one constant principal curvature in three-dimensional space forms

Author(s)

Anciaux, Henri

• Document type: Journal article
• Source: BEITRAGE ZUR ALGEBRA UND GEOMETRIE-CONTRIBUTIONS TO ALGEBRA AND GEOMETRY; v. 56, n. 2, p. 641-653, OCT 2015.
• Web of Science Citations: 0

Abstract

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of them being centered at a point of a regular curve and contained in its normal plane. In some cases, a kind of trichotomy phenomenon is observed: the curves of the foliations may be circles, hyperbolas or horocycles, depending of whether the constant principal curvature is respectively larger, smaller of equal to one (not necessarily in this order). We describe explicitly some examples showing that there do exist complete surfaces with one constant principal curvature enjoying both umbilic and non-umbilic points. (AU)

• FAPESP's process: 11/21362-2 - Group actions, submanifold theory and global analysis in Riemannian and pseudo-Riemannian geometry
• Grantee: Paolo Piccione
• Support Opportunities: Research Projects - Thematic Grants