On a step \(2(k+1)\) sub-Riemannian manifold
type: Item
instance of: scholarly article
title: On a step \(2(k+1)\) sub-Riemannian manifold
zbMATH Open document ID: 1056.53020
DOI: 10.1007/BF02921863
author: Ovidiu Calin
author: Der-Chen Chang
author: Peter C. Greiner
published in: The Journal of Geometric Analysis
publication date: 2004-06-08T00:00:00Z
review text: The article deals with the sub-Riemannian structure on \(\mathbb{R}^3= \mathbb{R}^2_x\times \mathbb{R}_t\) \((x= (x_1,x_2))\) determined by the vector fields \[ X_1= \frac{\partial}{\partial x_1}+ 2x_2| x|^{2k}\frac{\partial}{\partial t},\;X_2= \frac{\partial}{\partial x_2}- 2x_1| x|^{2k} {\partial\over\partial t}\quad (k= 0,1,\dots). \] The vector fields \(X_1\), \(X_2\) and their iterated Lie brackets span the tangent space of \(\mathbb{R}^3\) and the number of needful brackets plus 1, called the step of the space, is equal to \(2(k+1)\). By using polar coordinates, the authors solve the Euler-Lagrange system \[ \dot x= r\dot\varphi(\dot\varphi+ 4(k+1)\vartheta r^{2k}),\quad r^2(\dot\varphi+ 2\vartheta r^{2k})= C,\quad \vartheta= D \] for the geodesics, compute the sub-Riemannian distance explicitly and determine the number of geodesics between the origin (then \(C=0\)) and any other point.
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 35H20
zbMATH DE Number: 2070477
zbMATH Keywords: sub-Riemannian structure
zbMATH Keywords: sub-Riemannian geodesic
Wikidata QID: Q115391167
reviewed by: Jan Chrastina
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921863
OpenAlex ID: W2062613496
description: scientific article; zbMATH DE number 2070477