Wave Front Sets Singularities of Homogeneous Sub-Riemannian Three Dimensional Manifolds

V´Ä±ctor Ayala; Marcos M. Diniz; Jos´e C.P. Lima; Jos´e M.M. Veloso; Ivan Tribuzy

Authors

V´Ä±ctor Ayala Universidad Cat´olica del Norte – Departamento de Matem´aticas Avenida Angamos 0610, Casilla 1280 Antofagasta – Chile.
Marcos M. Diniz Universidade Federal do Par´a – CCEN – Departamento de Matem´atica, Rua Augusto Corrˆea, 01, CEP: 66075-110, Bel´em – Par´a, Brasil.
Jos´e C.P. Lima Universidade Federal do Par´a – CCEN – Departamento de Matem´atica, Rua Augusto Corrˆea, 01, CEP: 66075-110, Bel´em – Par´a, Brasil.
Jos´e M.M. Veloso Universidade Federal do Par´a – CCEN – Departamento de Matem´atica, Rua Augusto Corrˆea, 01, CEP: 66075-110, Bel´em – Par´a, Brasil.
Ivan Tribuzy Universidade Federal do Amazonas – Departamento de Matem´atica Japiim, Manaus – Amazonas, 69077-000, Brasil.

Keywords:

sub-Riemannian geometry, exponential map, wave front sets, singularities, three dimensional manifolds

Abstract

A graphic study of wave front sets of exponential sub-Riemannian maps is performed for homogeneous three dimensional sub-Riemannian manifolds. We verify that depending on dimension of the sub-Riemannian isometry group of the manifold, the first singularities of wave front sets are of two types. If the group is four dimensional, the singularity is a conjugate point. If the group is three dimensional, there are two conjugate points and the wave front set intersects along a segment which connects both points.