Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid
type: Item
instance of: scholarly article
title: Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid
zbMATH Open document ID: 0958.37008
published in: Commentarii Mathematici Helvetici
publication date: 2001-04-18T00:00:00Z
review text: A number of proofs of the complete integrability of the geodesic flow on the ellipsoid is known. In this paper a new proof of this result is given. The proof is based on the construction of a metric on the ellipsoid whose non-parametrized geodesics coincide with those of the standard metric, that is, the obtained metric is projectively equivalent to the Euclidean one. To do it, the author considers the field of Euclidean normals to the ellipsoid and the condition to be exact. This construction is closely related with a result from himself [Geom. Funct. Anal. 7, No. 3, 594-608 (1997)] concerning the billiard inside the ellipsoid and the corresponding billiard map. Such metric is induced by the hyperbolic metric inside the ellipsoid considered as the Klein model of the hyperbolic space. Here, the ellipsoid plays the roel of the sphere at infinity. Moreover using the same technique, it is proved that the geodesic flow on the ellipsoid in the spherical or hyperbolic space is also completely integrable.
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37J05
zbMATH DE Number: 1333325
zbMATH Keywords: geodesic flow
zbMATH Keywords: exact transverse line fields
zbMATH Keywords: symplectic structures
zbMATH Keywords: Riemannian metric
zbMATH Keywords: Finsler metric
author: S. Tabachnikov
reviewed by: J. C. González-Dávila
MaRDI profile type: Publication
full work available at URL: s000140050091
OpenAlex ID: W2163681122
DOI: 10.1007/S000140050091
description: scientific article; zbMATH DE number 1333325