Geodesic equivalence of metrics as a particular case of integrability of geodesic flows

We consider the recently found connection between geodesically equivalent metrics and integrable geodesic flows. If two different metrics on a manifold have the same geodesics, then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution, and vice versa. The quantum version of this result is also true: if two metrics on one manifold have the same geodesics, then the Beltrami Laplace operator Δ for each metric admits sufficiently many linear differential operators communiting with Δ. This implies that the topology of a manifold with two different metrics with the same geodesics must be sufficiently simple. We also have that the nonproportionality of the metrics at a point implies the nonproportionality of the metrics at almost all points. In memory of Mikhail Vladimirovich Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 285–293, May, 2000.