Q1917799
type: Item
instance of: scholarly article
title: Application of classification theory for integrable Hamiltonian systems to geodesic flows on 2-sphere and 2-torus and to the description of the topological structure of momentum mapping near singular points
zbMATH Open document ID: 0855.58046
DOI: 10.1007/BF02363855
author: A. T. Fomenko
author: A. V. Bolsinov
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-02-16T00:00:00Z
review text: The authors give a survey of recent results on integrable geodesic flows for Riemannian metrics on the 2-sphere \(S^2\) and the 2-torus \(T^2\). The known examples for such flows on \(T^2\) (resp. \(S^2\)) have a second integral whose restriction to the fibres of the fibration \(TT^2\to T^2\) (resp. \(TS^2\to S^2\)) is polynomial of degree 1 or 2 (resp. 1, 2, 3, 4). To the class of such a flow can be associated a numerical invariant with respect to equivalence under topological conjugacy, the so called complexity. A main conjecture discussed in the paper states that the complexity of any such system coincides with the complexity of a system with a fibrewise linear or quadratic integral. Some evidence supporting the conjecture is discussed. The paper consists mainly of a succession of definitions and theorems and does not contain any proof. It is not very informative for non-specialists.
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 903430
zbMATH Keywords: topological invariants
zbMATH Keywords: classification
zbMATH Keywords: survey
zbMATH Keywords: integrable geodesic flows
zbMATH Keywords: Riemannian metrics
zbMATH Keywords: 2-sphere
zbMATH Keywords: 2-torus
zbMATH Keywords: numerical invariant
zbMATH Keywords: complexity
MaRDI profile type: Publication
cites work: Q3994727
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full work available at URL: https://doi.org/10.1007/bf02363855
OpenAlex ID: W1995270306
label: Application of classification theory for integrable Hamiltonian systems to geodesic flows on 2-sphere and 2-torus and to the description of the topological structure of momentum mapping near singular points
description: scientific article; zbMATH DE number 903430