  We compare different notions of curvature on contact sub-Riemannian
manifolds. In particular we introduce canonical curvatures as the coefficients
of the sub-Riemannian Jacobi equation. The main result is that all these
coefficients are encoded in the asymptotic expansion of the horizontal
derivatives of the sub-Riemannian distance. We explicitly compute their
expressions in terms of the standard tensors of contact geometry. As an
application of these results, we obtain a sub-Riemannian version of the
Bonnet-Myers theorem that applies to any contact manifold.
