Singularities of flat fronts in hyperbolic 3-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 (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.

35 pages, 13 figures

Cited by in corpus (44)

• Parallel and dual surfaces of cuspidal edges
• A_k singularities of wave fronts
• Criteria for cuspidal S_k singularities and their applications
• A characterization of Weingarten surfaces in hyperbolic 3-space
• Pseudospherical surfaces with singularities
• The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications
• Principal curvatures and parallel surfaces of wave fronts
• Focal surfaces of wave fronts in the Euclidean 3-space
• Spherical Surfaces
• Loop Group Methods for Constant Mean Curvature Surfaces
• Timelike Constant Mean Curvature Surfaces with Singularities
• Affine connections and singularities of tangent surfaces to space curves
• Surfaces of revolution of frontals in the Euclidean space
• Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space
• Discrete cyclic systems and circle congruences
• Surfaces of constant curvature in R^3 with isolated singularities
• Holomorphic endomorphisms of P^3(C) related to a Lie algebra of type A_3 and catastrophe theory
• The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces
• The fundamental theorem for singular surfaces with limiting tangent planes
• Discrete constant mean curvature surfaces via conserved quantities
• Geometric invariants of cuspidal edges
• Geometric properties near singular points of surfaces given by certain representation formulae
• Singularities of flat extensions from generic surfaces with boundaries
• Duality on generalized cuspidal edges preserving singular set images and first fundamental forms
• Singularities of a surface given by Kenmotsu-type formula in Euclidean three-space
• The flat geometry of the $I_1$ singularity: $(x,y)\mapsto (x,xy,y^2,y^3)$
• Wave front evolution and pedal evolution
• Bloch-Ros principle and its application to surface theory
• Singularities of spacelike mean curvature one surfaces in de Sitter space
• Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifolds
• Generic bifurcations of framed curves in a space form and their envelopes
• On pairs of geometric foliations on a cuspidal edge
• Normal form of swallowtail and its applications
• Characterizing singularities of a surface in Lie sphere geometry
• On the geometry of folded cuspidal edges
• Geometry of lightlike locus on mixed type surfaces in Lorentz-Minkowski 3-space from a contact viewpoint
• Circular evolutes and involutes of framed curves in the Euclidean space
• Cuspidal crosscaps and folded singularities on a maxface and a minface
• Maximal surfaces in the Lorentzian Heisenberg group
• On quasicomplete $k$-surfaces in $3$-dimensional space-forms
• Local normal forms of em-wavefronts in affine flat coordinates
• On Gaussian curvatures and singularities of Gauss maps of cuspidal edges
• Singularities of the asymptotic completion of developable Möbius strips
• Jacobian-squared function-germs