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The tangent space in sub-Riemannian geometry

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Part of the Progress in Mathematics book series (PM,volume 144)

Abstract

Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian, case, they are indeed vector spaces, that is, abelian groups with dilations. Actually, the above is true only for regular points. At singular points, instead of nilpotent Lie groups one gets quotient spaces G/H of such groups G.

Keywords

  • Vector Field
  • Tangent Space
  • Heisenberg Group
  • Regular Point
  • Carnot Group

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bellaïche, A. (1996). The tangent space in sub-Riemannian geometry. In: Bellaïche, A., Risler, JJ. (eds) Sub-Riemannian Geometry. Progress in Mathematics, vol 144. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9210-0_1

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  • DOI: https://doi.org/10.1007/978-3-0348-9210-0_1

  • Publisher Name: Birkhäuser Basel

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