The geodesic flow on nilpotent Lie groups of steps two and three
type: Item
instance of: scholarly article
title: The geodesic flow on nilpotent Lie groups of steps two and three
zbMATH Open document ID: 1487.53109
published in: Discrete and Continuous Dynamical Systems
publication date: 2022-01-03T00:00:00Z
review text: This work investigates the geodesic flow on \(k\)-step nilpotent Lie groups for \(k=2,3\). For such Lie groups in dimension at most five, the author obtains a left-invariant metric for which the geodesic flow is Liouville integrable. The main tools used to obtain Liouville integrability are Killing vector fields and symmetric Killing 2-tensors associated to quadratic invariant first integrals of the geodesic flow. The author obtains explicit conditions for a symmetric map to induce a first integral as well as conditions for invariant linear or quadratic polynomials to become first integrals. Several explicit involution formulas for first integrals are obtained. Moreover, they also investigate six-dimensional \(k\)-step nilpotent Lie groups for \(k=2,3\), and in most cases (all but two) show that these groups admit a left-invariant metric for which the geodesic flow is completely integrable.
reviewed by: Julie Rowlett
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 70G65
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 7451270
zbMATH Keywords: geodesic flow
zbMATH Keywords: first integrals
zbMATH Keywords: Liouville integrability
zbMATH Keywords: nilpotent Lie groups
zbMATH Keywords: Killing tensor fields
author: Gabriela Paola Ovando
MaRDI profile type: Publication
full work available at URL: dcds.2021119
OpenAlex ID: W3197053716
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DOI: 10.3934/DCDS.2021119
description: scientific article; zbMATH DE number 7451270