Regularity and Continuity properties of the sub-Riemannian exponential map

Abstract

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.