Integrability criterion of geodesical equivalence: hierarchies
The subject of geodesically equivalent Riemannian metrics, i.e., metrics with the same geodesic lines, has a long history, beginning with \textit{U. Dini} [Brioschi Ann. (2) III, 269-294 (1870; JFM 02.0622.02)]. Because Riemannian metrics are produced by Hamiltonians in a straightforward manner, it results that the implications of geometry of such metrics in dynamics will be of great interest. On this theme the author gives an important contribution by the main result of the paper that two metrics are geodesically equivalent if and only if some canonical associated functions are pairwise commuting constants of motion. This result is applied to obtain a criterion of geodesical equivalence of left-invariant metrics on Lie groups and also, to yield families of metrics geodesically equivalent to the metrics of the standard ellipsoid and the Poisson sphere.
scientific article; zbMATH DE number 1436945