Almost-Riemannian geometry on Lie groups

A simple almost-Riemannian structure (ARS) on a Lie group G is defined by a linear vector field (that is, an infinitesimal automorphism) and dim(G) - 1 left-invariant ones. We state results about the singular locus, the abnormal extremals, and the desingularization of such ARSs, and these results are illustrated by examples on the two-dimensional affine and the Heisenberg groups. These ARSs are extended in two ways to homogeneous spaces, and a necessary and sufficient condition for an ARS on a manifold to be equivalent to a general ARS on a homogeneous space is stated.