Geodesic equivalence via integrability
type: Item
instance of: scholarly article
title: Geodesic equivalence via integrability
zbMATH Open document ID: 1017.37029
DOI: 10.1023/A:1022166218282
published in: Geometriae Dedicata
publication date: 2003-04-27T00:00:00Z
review text: Given two symplectic manifolds \((M,\omega)\) and \((M', \omega')\) with Hamiltonian functions \(H\) and \(H'\), let \(\Phi: M\to M'\) be a diffeomorphism mapping an isoenergy surface \(Q\) in \(M\) to an isoenergy surface \(Q'\) in \(M'\). The diffeomorphism is said to be orbital if its restriction to \(Q\) takes the orbits of the Hamiltonian system \((M,\omega, H)\) to the orbits of \((M',\omega',H')\). In the paper under review, the authors shows how to construct a one-paramter family of integrals from a given orbital diffeomorphism between two Hamiltonian systems. As a significant application of this result, whenever two geodesically equivalent Riemannian metrics \(g\) and \(g'\) are given on the same manifold \(M\), a set of \(n=\text{dim} M\) integrals for the geodesic flow is obtained. Such integrals Poisson commute as a consequence of a classical result proved by Levi-Civita in 1896. Let \(G\) be the endomorphism of the tangent bundle defined by raising one index of \(g'\) by means of \(g\). If the eigenvalues of \(G\) are all distinct at least at one point and \(M\) is connected, then they are all distinct at almost every point, and the integrals are proved to be functionally independent. Therefore, the geodesics flows of the two metrics are completely integrable in the sense of Arnol'd-Liouville. As a corollary of this result, the authors find out a topological condition preventing a closed connected real-analytic manifold from admitting a pair of geodesically equivalent metrics such that the eigenvalues of \(G\) are all distinct at least at one point.
reviewed by: Claudio Bartocci
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 53A20
zbMATH DE Number: 1901489
zbMATH Keywords: geodesically equivalent metrics
zbMATH Keywords: integrable systems
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: orbital diffeomorphism
author: Peter Topalov
author: Vladimir S. Matveev
MaRDI profile type: Publication
full work available at URL: a:1022166218282
OpenAlex ID: W1505781718
description: scientific article; zbMATH DE number 1901489