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.

31 pages, 2 figures; v2: the Bonnet-Myers theorem 1.7 now holds for any contact structure; v3: final version (with expanded introduction) to appear on Journal of Geometric Analysis; v4: fixed typos