# A Formula for Popp’s Volume in Sub-Riemannian Geometry

• Volume: 1, page 42-57
• ISSN: 2299-3274

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## Abstract

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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.

## How to cite

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Davide Barilari, and Luca Rizzi. "A Formula for Popp’s Volume in Sub-Riemannian Geometry." Analysis and Geometry in Metric Spaces 1 (2013): 42-57. <http://eudml.org/doc/267366>.

@article{DavideBarilari2013,
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.},
author = {Davide Barilari, Luca Rizzi},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Sub-Riemannian geometry; Popp’s volume; Sub-Laplacian; Sub-Riemannian isometries; sub-Riemannian geometry; Popp's volume; sub-Laplacian; sub-Riemannian isometries},
language = {eng},
pages = {42-57},
title = {A Formula for Popp’s Volume in Sub-Riemannian Geometry},
url = {http://eudml.org/doc/267366},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Davide Barilari
AU - Luca Rizzi
TI - A Formula for Popp’s Volume in Sub-Riemannian Geometry
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 42
EP - 57
AB - 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.
LA - eng
KW - Sub-Riemannian geometry; Popp’s volume; Sub-Laplacian; Sub-Riemannian isometries; sub-Riemannian geometry; Popp's volume; sub-Laplacian; sub-Riemannian isometries
UR - http://eudml.org/doc/267366
ER -

## References

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