TY - JOUR
T1 - A formula for Popp\'s volume in sub-Riemannian geometry
JF - Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57
Y1 - 2012
A1 - Luca Rizzi
A1 - Davide Barilari
KW - subriemannian, volume, Popp, control
AB - For an equiregular sub-Riemannian manifold M, Popp\'s volume is a smooth\r\nvolume which is canonically associated with the sub-Riemannian structure, and\r\nit is a natural generalization of the Riemannian one. In this paper we prove a\r\ngeneral formula for Popp\'s volume, written in terms of a frame adapted to the\r\nsub-Riemannian distribution. As a first application of this result, we prove an\r\nexplicit formula for the canonical sub-Laplacian, namely the one associated\r\nwith Popp\'s volume. Finally, we discuss sub-Riemannian isometries, and we prove\r\nthat they preserve Popp\'s volume. We also show that, under some hypotheses on\r\nthe action of the isometry group of M, Popp\'s volume is essentially the unique\r\nvolume with such a property.
PB - SISSA
UR - http://hdl.handle.net/1963/6501
N1 - 16 pages, minor revisions
U1 - 6446
U2 - Mathematics
U4 - 1
U5 - MAT/03 GEOMETRIA
ER -