Commentarii Mathematici Helvetici


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Volume 74, Issue 2, 1999, pp. 306–321
DOI: 10.1007/s000140050091

Published online: 1999-06-30

Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid

S. Tabachnikov[1]

(1) University of Arkansas, Fayetteville, USA

We give a new proof of the complete integrability of the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose non-parameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (Klein's model).

Keywords: Riemannian and Finsler metrics, symplectic and contact structures, geodesic flow, exact transverse line fields

Tabachnikov S.: Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid. Comment. Math. Helv. 74 (1999), 306-321. doi: 10.1007/s000140050091