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

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 14 Jan 2013Embargo end date: 01 Jan 2012 France, Italy, Italy Publisher:Walter de Gruyter GmbHJournal:Analysis and Geometry in Metric Spaces, volume 1, pages 42-57 (eissn: 2299-3274,

Authors: Barilari, Davide; Rizzi, Luca;

doi: 10.2478/agms-2012-0004 , 10.48550/arxiv.1211.2325

arXiv: http://arxiv.org/abs/1211.2325

handle: 11577/3401936 , 20.500.11767/128678

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

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Abstract

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.

Countries

France, Italy, Italy

Related Organizations

University of Padua
Italy
University of Paris
France
UNIVERSITE PARIS DESCARTES
France
École Polytechnique
France
French National Centre for Scientific Research
France

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Keywords

Mathematics - Differential Geometry, sub-riemannian isometries, popp’s volume, 53C17, 28D05, Popp's volume, 510, Popp's volume; Sub-laplacian; Sub-riemannian geometry; Sub-riemannian isometries, Sub-laplacian, sub-laplacian, FOS: Mathematics, [MATH.MATH-MG] Mathematics [math]/Metric Geometry [math.MG], Sub-riemannian geometry, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics - Optimization and Control, QA299.6-433, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], sub-riemannian geometry, 53c17, 28d05, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Optimization and Control (math.OC), Sub-riemannian isometries, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Analysis

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