On the geometry of folded cuspidal edges

Abstract

We study the geometry of cuspidal $S_k$ singularities in ${\mathbb{R}}^3$ obtained by folding generically a cuspidal edge. In particular we study the geometry of the cuspidal cross-cap $M$, i.e. the cuspidal $S_0$ singularity. We study geometrical invariants associated to $M$ and show that they determine it up to order 5. We then study the flat geometry (contact with planes) of a generic cuspidal cross-cap by classifying submersions which preserve it and relate the singularities of the resulting height functions with the geometric invariants.

Second version generalizes some results for all cuspidal $S_k$ singularities, not only the cuspidal cross-cap

References in corpus (2)

• Geometry of cuspidal edges with boundary
• Geometric invariants of cuspidal edges