Wave Front Sets Singularities of Homogeneous Sub-Riemannian 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.