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

TL;DR

This paper derives a general formula for Popp's volume in equiregular sub-Riemannian manifolds, enabling explicit computation of the sub-Laplacian and analyzing isometry invariance, thus advancing geometric analysis in this setting.

Contribution

It provides a new explicit formula for Popp's volume using adapted frames, facilitating computations and understanding of sub-Riemannian geometric structures.

Findings

Derived a general formula for Popp's volume in terms of adapted frames.

Obtained an explicit formula for the canonical sub-Laplacian associated with Popp's volume.

Proved that sub-Riemannian isometries preserve Popp's volume and that it is essentially unique under certain conditions.

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.