Calculus of Variations and Geometric Measure Theory
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D. Barilari - L. Rizzi

A formula for Popp's volume in sub-Riemannian geometry

created by barilari on 10 Nov 2012
modified on 15 Oct 2014


Published Paper

Inserted: 10 nov 2012
Last Updated: 15 oct 2014

Journal: Analysis and Geometry in Metric Spaces
Volume: 1
Pages: 42-57
Year: 2013
Doi: 10.2478/agms-2012-0004
Links: link to the paper


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 gen- eralization 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 isome- tries, 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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