Calculus of Variations and Geometric Measure Theory

D. Barilari - L. Rizzi

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

created by barilari on 10 Nov 2012
modified by rizzi1 on 15 May 2017


Published Paper

Inserted: 10 nov 2012
Last Updated: 15 may 2017

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

ArXiv: 1211.2325v2 PDF
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 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.