Measures of transverse paths in sub-Riemannian geometry
type: Item
instance of: scholarly article
title: Measures of transverse paths in sub-Riemannian geometry
zbMATH Open document ID: 1073.53046
DOI: 10.1007/BF02788789
author: Elisha Falbel
author: Frédéric Jean
published in: Journal d'Analyse Mathématique
publication date: 2004-09-27T00:00:00Z
review text: Let \(M\) be a \(\mathcal{C}^{\infty}\) manifold, let \(\mathcal{D} \subset TM\) be a distribution on \(M\) and let \(g\) be a \(\mathcal{C}^{\infty}\) Riemannian metric on \(\mathcal{D}\), so that \((M, \mathcal{D}, g)\) is a sub-Riemannian manifold. The authors assume that Chow's condition is satisfied, namely if \(\mathcal{D}^{s}\) denotes the linear span of all brackets of degree \(\leq s\) of the vector fields tangent to \(\mathcal{D}^{1} = \mathcal{D}\), then at every point \(p \in M\) there exists an integer \(r = r(p)\) such that \(\mathcal{D}^{r} (p) = T_{p} M\). The horizontal paths are those paths which are tangent to \(\mathcal{D}\). Their lengths are obtained as in the Riemannian geometry by integrating the norm of their corresponding tangent vectors. The Chow condition implies that one can join any two points of the manifold by a horizontal path, and therefore a distance \(d\) can be defined. A transverse path is one whose tangent vector does not belong to the distribution. In the paper under review the authors introduce a length or a measure for a transverse path. Their definition is based on a careful study of the metric structure of the tangent space. The metric tangent space, called here tangent cone is itself a sub-Riemannian manifold and has moreover an algebraic structure, namely it is a group at a regular point. The authors propose an intrinsic definition of the tangent cone at any point. Furthermore, they introduce a filtration of the tangent cone and relate it to the natural filtration of the tangent space defined by the filtration. A class of lengths of paths is defined by integration of infinitesimal measures on the tangent cone. One of those lengths coincides with the usual length. As an application, the measures for contact and Martinet sub-Riemannian manifolds are computed.
reviewed by: Adrian Sandovici
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 2103496
zbMATH Keywords: sub-Riemannian manifold
zbMATH Keywords: distribution
zbMATH Keywords: Chow's condition
zbMATH Keywords: horizontal path
zbMATH Keywords: transverse path
zbMATH Keywords: tangent cone
zbMATH Keywords: regular point
zbMATH Keywords: filtration of the tangent cone
zbMATH Keywords: infinitesimal measure on the tangent cone
Wikidata QID: Q115391605
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2103496