Singularities of flat fronts in hyperbolic space

Masatoshi Kokubu, Wayne Rossman, Kentaro Saji, Masaaki Umehara and Kotaro Yamada

Vol. 221 (2005), No. 2, 303–351

DOI: 10.2140/pjm.2005.221.303

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.

Keywords

flat front, hyperbolic 3-space, cuspidal edge singularity, swallowtail singularity

Mathematical Subject Classification 2000

Primary: 53C42

Secondary: 53A99

Milestones

Received: 27 December 2003

Revised: 23 March 2004

Accepted: 20 December 2004

Published: 1 October 2005

Correction: 20 February 2018