On Pairs of Foliations of a Parabolic Cross-Cap

作者：Oliver J M^*

来源：Qualitative Theory of Dynamical Systems, 2011, 10(1): 139-166.

DOI：10.1007/s12346-011-0042-0

• Cross-cap
• Asymptotic curves
• Characteristic curves
• Singularities
• Binary differential equations

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.

• 出版日期2011-4