We prove a version of Warner's regularity and continuity properties for the sub-Riemannian exponential map. The regularity property is established by considering sub-Riemannian Jacobi fields while the continuity property follows from studying the Maslov index of Jacobi curves. We finally show how this implies that the exponential map of the three dimensional Heisenberg group is not injective in any neighbourhood of a conjugate vector.

We prove that the sub-Riemannian exponential map is not injective in any neighbourhood of certain critical points. Namely that it does not behave like the injective map of reals given by f (x) = x 3 near its critical point x = 0. As a consequence, we characterise conjugate points in ideal sub-Riemannian manifolds in terms of the metric structure of the space. The proof uses the Hilbert invariant integral of the associated variational problem.

We prove a version of Warner’s regularity and continuity properties for the sub-Riemannian exponential map. The regularity property is established by considering sub-Riemannian Jacobi fields while the continuity property follows from studying the Maslov index of Jacobi curves. We finally show how this implies that the exponential map of the three-dimensional Heisenberg group is not injective in any neighbourhood of a conjugate vector.