2013/07/25 by Henri Anciaux, Anciaux, Henri
Mathematics · Engineering · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Elasticity and Material Modeling #Advanced Differential Geometry Research
paper · pdf · doi:10.48550/arxiv.1307.6735
We study surfaces with one constant principal curvature in Riemannian and\nLorentzian three-dimensional space forms. Away from umbilic points they are\ncharacterized as one-parameter foliations by curves of constant curvature, each\nof these curves being centered at a point of a regular curve and contained in\nits normal plane. In some cases, a kind of trichotomy phenomenon is observed:\nthe curves of the foliations may be circles, hyperbolas or horocycles,\ndepending of whether the constant principal curvature is respectively larger,\nsmaller of equal to one (not necessarily in this order). We describe explicitly\nsome examples showing that there do exist complete surfaces with one constant\nprincipal curvature enjoying both umbilic and non-umbilic points.\n