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

TL;DR

This paper characterizes surfaces with one constant principal curvature in 3D space forms, revealing their foliation structure and classifying the types of curvature curves involved, with explicit examples of complete surfaces.

Contribution

It provides a detailed geometric description and classification of such surfaces, including explicit examples of complete surfaces with mixed umbilic and non-umbilic points.

Findings

Surfaces are foliated by curves of constant curvature centered at points of a regular curve.

Depending on the principal curvature, the curvature curves are circles, hyperbolas, or horocycles.

Existence of complete surfaces with both umbilic and non-umbilic points is demonstrated.

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 these curves 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.