A class of Riemannian manifolds with integrable geodesic flows
type: Item
instance of: scholarly article
title: A class of Riemannian manifolds with integrable geodesic flows
zbMATH Open document ID: 0845.53029
author: Kazuo Yamato
published in: Journal of the Mathematical Society of Japan
publication date: 1996-09-11T00:00:00Z
review text: In the two-dimensional case \(n= 2\), the main theorem of this paper is easily stated: Theorem. Let \((M^2, {\mathbf g})\) be a two-dimensional Riemannian manifold. Suppose that the geodesic flow of \((M^2, {\mathbf g})\) has a first integral \(f: T(M^2)\to \mathbb{R}\) which can be expressed as \(f(X)= {\mathbf g}(\iota(X), X)\), where \(\iota\) is a symmetric tensor field of type \((1, 1)\) on \(M^2\). Assume that the determinant of \(\iota\) is positive on \(M\), and let \(\widetilde{\mathbf g}= (\text{det } \iota)^{- 1} {\mathbf g}\) be the conformal change of the metric. Then the geodesic flow of \((M^2, \widetilde{\mathbf g})\) has the first integral \(\widetilde f: T(M^2)\to \mathbb{R}\) given by \(\widetilde f(X)= \widetilde{\mathbf g}(\iota^{- 1}(X), X)\). The main theorem for general \(n\) is used for giving a differential-geometric proof of the complete integrability of geodesic flows of ellipsoids.
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 894106
zbMATH Keywords: Poisson bracket
zbMATH Keywords: simple integrability
zbMATH Keywords: geodesic flow
zbMATH Keywords: first integral
zbMATH Keywords: integrability of geodesic flows of ellipsoids
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04740719
OpenAlex ID: W2074167669
Wikidata QID: Q115225396
DOI: 10.2969/JMSJ/04740719
description: scientific article; zbMATH DE number 894106