On sub-Riemannian geodesic curvature in dimension three

Abstract

Abstract We introduce a notion of geodesic curvature k ζ k_{\zeta} for a smooth horizontal curve 𝜁 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 ⁢ ( ζ ⁢ ( t ) , ζ ⁢ ( t + ε ) ) = ε 2 - k ζ 2 ⁢ ( t ) 720 ⁢ ε 6 + o ⁢ ( ε 6 ) . d_{\mathrm{SR}}^{2}(\zeta(t),\zeta(t+\varepsilon))=\varepsilon^{2}-\frac{k_{\zeta}^{2}(t)}{720}\varepsilon^{6}+o(\varepsilon^{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.