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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
Source Normalized Impact per Paper (SNIP) 2017: 0.833

Mathematical Citation Quotient (MCQ) 2017: 0.86

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

Davide Barilari / Luca Rizzi
Published Online: 2013-01-14 | DOI: https://doi.org/10.2478/agms-2012-0004


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.

Keywords: Sub-Riemannian geometry; Popp’s volume; Sub-Laplacian; Sub-Riemannian isometries

MSC: 53C17; 28D05

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About the article

Received: 2012-11-15

Accepted: 2012-12-06

Published Online: 2013-01-14

Citation Information: Analysis and Geometry in Metric Spaces, Volume 1, Pages 42–57, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2012-0004.

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©2012 Versita Sp. z o.o.. This content is open access.

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