Inserted: 22 oct 2019
Last Updated: 12 may 2021
Journal: Advances in Calculus of Variations
We introduce a notion of geodesic curvature 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.
The sub-Riemannian distance is not smooth on the diagonal, hence the result contains the existence of such an asymptotics. In this sense, we show a higher-order differentiability property of the sub-Riemannian distance along smooth horizontal curves. This generalizes the previously known results on the Heisenberg group.