Sub-Riemannian homogeneous spaces in dimensions 3 and 4

Abstract

Let (M, \(\mathcal{D}\), g) be a sub-Riemannian manifold (i.e. M is a smooth manifold, \(\mathcal{D}\) is a smooth distribution on M and g is a smooth metric defined on \(\mathcal{D}\)) such that the dimension of M is either 3 or 4 and \(\mathcal{D}\) 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.