Singularities of flat fronts in hyperbolic 3-space

Details Singularities of flat fronts in hyperbolic 3-space Singularities of flat fronts in hyperbolic 3-space

2004-01-12

arxiv.org/abs/math/0401110v4

Pacific J. Math. 221(2) (2005), 303-352

Mathematics

Differential Geometry

35 pages, 13 figures

Scientific paper

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 (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we shall prove that generically flat fronts in the hyperbolic 3-space admit only cuspidal edges and swallowtails. Moreover, we will show that every complete flat front (which is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.

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