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).
Cite this article
S. Tabachnikov, Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid. Comment. Math. Helv. 74 (1999), no. 2, pp. 306–321DOI 10.1007/S000140050091