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.
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Diniz, M.M., Veloso, J.M.M. Regions Where the Exponential Map at Regular Points of Sub-Riemannian Manifolds is a Local Diffeomorphism. J Dyn Control Syst 15, 133–156 (2009). https://doi.org/10.1007/s10883-008-9054-8
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DOI: https://doi.org/10.1007/s10883-008-9054-8