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.