On sub-Riemannian geodesic curvature in dimension three

Abstract

We introduce a notion of geodesic curvature $k_{\zeta }$ for a smooth horizontal curve $\zeta$ 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 $$d_{SR}^2(\zeta (t),\zeta (t+\epsilon ))={\epsilon }^2-\frac{k_{\zeta }^2(t)}{720}{\epsilon }^6+o({\epsilon }^6).$$ 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.

26 pages, to appear on Advances in Calculus of Variations

References in corpus (1)

• Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$

Cited by in corpus (1)

• Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds