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Journal of Dynamical and Control Systems

, Volume 15, Issue 1, pp 133–156 | Cite as

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

  • Marcos M. Diniz
  • José M. M. Veloso
Article

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.

Key words and phrases

Sub-Riemannian manifolds k-step exponential map Taylor series singularities of wave front set 

2000 Mathematics Subject Classification

53C17 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Faculdade de MatemáticaUniversidade Federal do ParáBelém, ParaBrazil

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