Abstract
We prove that the Riemannian metrics g and \(\bar g\) (given in `general position") are geodesically equivalent if and only if some canonically given functions are pairwise commuting integrals of the geodesic flow of the metric g. This theorem is a multidimensional generalization of the well-known Dini theorem proved in the two-dimensional case. A hierarchy of completely integrable Riemannian metrics is assigned to any pair of geodesically equivalent Riemannian metrics. We show that the metrics of the standard ellipsoid and the Poisson sphere lie in such an hierarchy.
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