Integrability criterion of geodesical equivalence: hierarchies
type: Item
instance of: scholarly article
title: Integrability criterion of geodesical equivalence: hierarchies
zbMATH Open document ID: 0972.53048
DOI: 10.1023/A:1006369525091
published in: Acta Applicandae Mathematicae
publication date: 2001-11-05T00:00:00Z
review text: The subject of geodesically equivalent Riemannian metrics, i.e., metrics with the same geodesic lines, has a long history, beginning with \textit{U. Dini} [Brioschi Ann. (2) III, 269-294 (1870; JFM 02.0622.02)]. Because Riemannian metrics are produced by Hamiltonians in a straightforward manner, it results that the implications of geometry of such metrics in dynamics will be of great interest. On this theme the author gives an important contribution by the main result of the paper that two metrics are geodesically equivalent if and only if some canonical associated functions are pairwise commuting constants of motion. This result is applied to obtain a criterion of geodesical equivalence of left-invariant metrics on Lie groups and also, to yield families of metrics geodesically equivalent to the metrics of the standard ellipsoid and the Poisson sphere.
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 58J70
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 1436945
zbMATH Keywords: integrable geodesic flow
zbMATH Keywords: geodesical equivalence
zbMATH Keywords: projective mapping
zbMATH Keywords: ellipsoid
zbMATH Keywords: Poisson sphere
author: Peter Topalov
reviewed by: Mircea Crasmareanu
MaRDI profile type: Publication
full work available at URL: a:1006369525091
OpenAlex ID: W145301763
description: scientific article; zbMATH DE number 1436945