Sub-Riemannian homogeneous spaces in dimensions 3 and 4

Let (M, , g) be a sub-Riemannian manifold (i.e. M is a smooth manifold, is a smooth distribution on M and g is a smooth metric defined on ) such that the dimension of M is either 3 or 4 and is a contact or odd-contact distribution, respectively. We construct an adapted connection on M and use it to study the equivalence problem. Furthermore, we classify the 3-dimensional sub-Riemannian manifolds which are sub-homogeneous and show the relation to Cartan's list of homogeneous CR manifolds. Finally, we classify the 4-dimensional sub-Riemannian manifolds which are sub-symmetric.