Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid
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.
scientific article; zbMATH DE number 1333325