Sub-Riemannian geometry
type: Item
instance of: scholarly article
title: Sub-Riemannian geometry
zbMATH Open document ID: 0609.53021
author: Robert S. Strichartz
published in: Journal of Differential Geometry
publication date: 1986-01-01T00:00:00Z
review text: A sub-Riemannian or singular Riemannian geometry is given by a smoothly varying positive definite quadratic form defined only on a subbundle \(S\) of the tangent bundle \(TM\) of a differentiable manifold, \(S\) being bracket-generating, that is sections of \(S\) together with their Lie brackets generate the \(C^{\infty}(M)\)-module \(V(M)\) of vector fields. Such a metric is also called aCarnot-Carathéodory metric. The paper gives a fairly self-contained description of the known results concerning length minimizing curves and geodesics, the exponential map, completeness and isometries, the latter allowing the investigation of sub-Riemannian symmetric spaces, which, in the case of three dimensions, are completely classified up to local isometry. For piecewise smooth curves tangent to the subbundle \(S\) there is a natural notion of length. If \(M\) is connected, by a theorem of Chow (1939) it is possible to connect any two points of \(M\) by such a curve, giving \(M\) the structure of a metric space. The sub-Riemannian metric gives a natural positive semidefinite quadratic form on covectors and therefore a quadratic function on the cotangent bundle, the Hamiltonian vector field of which replaces the geodesic spray of a Riemannian manifold. The projections of the integrals of this vectorfield are the geodesics of the sub-Riemannian structure. The main results concerning geodesics are: Every length-minimizing curve is a geodesic and locally every geodesic is length-minimizing, the latter result as some others only being derived under the so-called strong bracket generating hypothesis, that is the sections of S and the brackets of such sections with any non-zero section of \(S\) generate \(V(M)\). The proof requires a careful study of the exponential map which can be defined in a natural way as a map from \(T^*M\) to \(M\). In contrast to the Riemannian situation the exponential map never is a diffeomorphism at the origin. Another basic result is the analogue of the Ho…
Mathematics Subject Classification ID: 53C99
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 93B99
zbMATH DE Number: 3986104
zbMATH Keywords: singular Riemannian geometry
zbMATH Keywords: Carnot-Carathéodory metric
zbMATH Keywords: sub- Riemannian symmetric spaces
zbMATH Keywords: sub-Riemannian structure
zbMATH Keywords: geodesics
zbMATH Keywords: length- minimizing curve
zbMATH Keywords: exponential map
zbMATH Keywords: Hopf-Rinow theorem
reviewed by: Klaus Horneffer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214440436
OpenAlex ID: W1512416545
Wikidata QID: Q115185143
DOI: 10.4310/JDG/1214440436
description: scientific article; zbMATH DE number 3986104