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On Pairs of Foliations of a Parabolic Cross-Cap

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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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Authors and Affiliations

  1. Department of Mathematics, Durham University Science Laboratories, Durham, DH1 3LE, UK

    J. M. Oliver

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  1. J. M. Oliver
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Correspondence to J. M. Oliver.

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Joey Oliver is supported by an EPSRC studentship.

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Oliver, J.M. On Pairs of Foliations of a Parabolic Cross-Cap. Qual. Theory Dyn. Syst. 10, 139–166 (2011). https://doi.org/10.1007/s12346-011-0042-0

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