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

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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.

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Authors and Affiliations

  1. Universidade de São Paulo, Instituto de Matemática e Estatística, Rua do Matão, 1010, São Paulo, 05508-090, Brazil

    Henri Anciaux

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  1. Henri Anciaux
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Correspondence to Henri Anciaux.

Additional information

H. Anciaux was supported by CNPq (PQ 306154/2011-0) and Fapesp (2011/21362-2).

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Anciaux, H. Surfaces with one constant principal curvature in three-dimensional space forms. Beitr Algebra Geom 56, 641–653 (2015). https://doi.org/10.1007/s13366-014-0215-5

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  • DOI: https://doi.org/10.1007/s13366-014-0215-5

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