  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.

Differential Geometry;Optimization and Control