Regions Where the Exponential Map at Regular Points of Sub-Riemannian Manifolds is a Local Diffeomorphism

Abstract

For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, we describe a region near the point where the exponential map is a local diffeomorphism. This is proved by taking the Taylor series of the exponential map and calculating the first nonzero term, which has order $ 2{\left( {{{\mathcal{D}}}_{{{\mathcal{H}}}} - n} \right)} $ , where n is the topological dimension and $ {{\mathcal{D}}}_{{{\mathcal{H}}}} $ is the Hausdorff dimension of the metric space associated to the sub-Riemannian manifold.