On sub-Riemannian geodesic curvature in dimension three

We introduce a notion of geodesic curvature kζ for a smooth horizontal curve in a three-dimensional contact sub-Riemannian manifold, measuring how much a horizontal curve is far from being a geodesic. We show that the geodesic curvature appears as the first corrective term in the Taylor expansion of the sub-Riemannian distance between two points on a unit speed horizontal curve dSR2(ζ(t),ζ(t+ϵ))=ϵ2-kζ2(t)720ϵ6+o(ϵ6). The sub-Riemannian distance is not smooth on the diagonal; hence the result contains the existence of such an asymptotics. This can be seen as a higher-order differentiability property of the sub-Riemannian distance along smooth horizontal curves. It generalizes the previously known results on the Heisenberg group.