A Formula for Popp’s Volume in Sub-Riemannian Geometry

Davide Barilari 1 Luca Rizzi 2, 3
3 GECO - Geometric Control Design
Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR7641
Abstract : For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub-Laplacian, namely the one associated with Popp's volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp's volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp's volume is essentially the unique volume with such a property.
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https://hal.archives-ouvertes.fr/hal-00751222
Contributor : Luca Rizzi <>
Submitted on : Wednesday, March 4, 2015 - 5:52:58 PM
Last modification on : Tuesday, May 14, 2019 - 11:07:49 AM
Long-term archiving on : Friday, June 5, 2015 - 11:16:09 AM

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Davide Barilari, Luca Rizzi. A Formula for Popp’s Volume in Sub-Riemannian Geometry. Analysis and Geometry in Metric Spaces, 2013, 1, ⟨10.2478/agms-2012-0004⟩. ⟨hal-00751222⟩

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