On the Hausdorff volume in sub-Riemannian geometry

By Andrei A. Agrachev, Davide Barilari, Ugo V. Boscain. 2010

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For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is $${\mathcal {C}^{3}}$$ (and $${\mathcal {C}^{4}}$$ on every smooth curve) but in general not $${\mathcal {C}^{5}}$$. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/633de097f14879cd24612a56dc6afad5a881a912

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In Applied Mathematics

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Sub-Riemannian structures on 3D lie groups (2010, background): “Three dimensional contact subRiemannian structures have been deeply studied in the last years (for example see [2, 4, 5])”
Measure contraction properties of Carnot groups (2015, background): “We extend the results of [1], where the case of a non-degenerate A is considered.”