Geodesic equivalence of metrics as a particular case of integrability of geodesic flows
type: Item
instance of: scholarly article
title: Geodesic equivalence of metrics as a particular case of integrability of geodesic flows
zbMATH Open document ID: 0996.53054
DOI: 10.1007/BF02551397
published in: Theoretical and Mathematical Physics
publication date: 2002-10-29T00:00:00Z
review text: Let \(M\) be smooth n-manifold with two metrics \(g\) and \(\overline{g}\). These are geodesically equivalent if they have the same geodesics, considered as unparametrized curves. The authors solve the following problem, on closed manifolds: If the metrics \(g\) and \(\overline{g}\) are geodesically equivalent, then the geodesic flows of the metrics are completely integrable [see also \textit{S. Yu. Dobrokhotov} and \textit{A. I. Shafarevich}, Funct. Anal. Appl. 34, No. 2, 133-134 (2000; Zbl 1019.58009], by using the \(n\) functions: \[ I_k (\xi) = \left(\frac{\det (g)}{\det (\overline{g}}\right) g (S_k \xi,\xi), \] where \(S_k\) is the linear operator given by \[ S_k \underset{\text{def}} = \sum_{i=0}^k c_i G^{k-i+1} \] and \(g(v,\xi)\) denotes the dot product of the tangent vectors \(v\) and \(\xi\) in the metric \(g\). The authors point out that: ``\dots{} the integral \(I_0\) was considered by Painlevé [\textit{T. Levi-Civita}, Ann. Mat. (2), 24, 255-300 (1896; JFM 27.0603.04)]''. They give various nontrivial examples on the \(n\)-sphere, on the \(n\)-torus and on the Poisson sphere, and analyse on real analytic manifolds admissibility of non proportional geodesically equivalent metrics. They also give an approach to the question: ``How many metrics are geodesically equivalent to a given one?'' by calculating the dimension of the manifold of the metrics that are geodesically equivalent to a given one. The second author has worked out more details of one of the examples [see \textit{P. Topalov}, Families of metrics geodesically equivalent to the analogs of the Poisson sphere, J. Math. Phys. 41, No. 11, 7510-7520 (2000; Zbl 1001.53005]. This is a very beautiful and readable article.
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 1551588
zbMATH Keywords: integrable geodesic flows
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: orbitally equivalent metrics
zbMATH Keywords: geodesically equivalent metrics
zbMATH Keywords: JFM 27.0603.04
author: Vladimir S. Matveev
author: Peter Topalov
reviewed by: L. Del Riego
MaRDI profile type: Publication
cites work: Conjugate points of hyperbolic geodesics of square integrable geodesic flows on closed surfaces
cites work: TOPOLOGICAL OBSTRUCTIONS TO INTEGRABILITY OF GEODESIC FLOWS ON NON-SIMPLY-CONNECTED MANIFOLDS
cites work: Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid
cites work: CONSTRUCTION OF COMPLETELY INTEGRABLE GEODESIC FLOWS ON COMPACT SYMMETRIC SPACES
cites work: Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry
Recommended article: Hierarchy of integrable geodesic flows
Recommended article: Trajectory equivalence and corresponding integrals
Recommended article: Integrable geodesic flows on Riemannian manifolds
Recommended article: Dynamical and topological methods in the theory of geodesically equivalent metrics
Recommended article: Q5457350
Recommended article: Integrable geodesic flows on \(n\)-step nilmanifolds
Recommended article: Rational integrals of 2-dimensional geodesic flows: new examples
Recommended article: Q5430911
Recommended article: A class of Riemannian manifolds with integrable geodesic flows
Recommended article: Integrability in the theory of geodesically equivalent metrics
description: scientific article; zbMATH DE number 1551588