You are here

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

TitleA formula for Popp\'s volume in sub-Riemannian geometry
Publication TypeJournal Article
Year of Publication2012
AuthorsRizzi, L, Barilari, D
JournalAnalysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57
Keywordssubriemannian, volume, Popp, control

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.


Sign in