Singularities of flat fronts in hyperbolic space

Abstract

It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure.A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle.We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail.Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails.We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.