Existence of abnormal minimizing geodesics in sub-Riemannian geometry
type: Item
instance of: scholarly article
title: Existence of abnormal minimizing geodesics in sub-Riemannian geometry
zbMATH Open document ID: 0844.53032
author: N. N. Petrov
published in: Vestnik St. Petersburg University. Mathematics
publication date: 1996-01-03T00:00:00Z
review text: In sub-Riemannian geometry one studies manifolds equipped with a subbundle \(S\) of the tangent bundle \(TM\) and a Riemannian metric on \(S\). If \(S\) is fully nonholonomic, meaning that vector fields tangential to \(S\) together with their iterated Lie brackets generate all of \(TM\), then any two points of \(M\) can be joint by a curve tangential to \(S\), a so-called horizontal curve. This induces a distance function on \(M\), sometimes called Carnot-Caratheodory metric. It has been stated many times that shortest horizontal curves satisfy the corresponding Hamilton-Jacobi equations. In Riemannian geometry this is true: shortest curves satisfy the geodesic equation. But in sub-Riemannian geometry it turned out that there are shortest horizontal curves not satisfying the Hamilton-Jacobi equations. In the present paper a simple example for such an abnormal geodesic is constructed.
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 815611
zbMATH Keywords: sub-Riemannian geometry
zbMATH Keywords: Carnot-Caratheodory metric
zbMATH Keywords: abnormal geodesic
reviewed by: Christian Bär
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 815611