Abstract
We study the asymptotic and characteristic curves in the neighbourhood of a parabolic cross-cap, that is, on a singular surface with a cross-cap singularity with a parabolic set having a cusp singularity at the singular point. We obtain the topological configurations of these foliations both in the domain of a parametrisation of such a surface, and on the surface itself. We construct a natural one-parameter family of surfaces with cross-cap singularities in which the parabolic cross-cap is the transition from a hyperbolic cross-cap to an elliptic cross-cap. We study the bifurcations of the asymptotic and characteristic curves in this family.
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