Geodesic
Geodesic equivalence of metrics as a particular case of integrability of geodesic flows
Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 285-293 
V. S. Matveev; P. J. Topalov. Geodesic equivalence of metrics as a particular case of integrability of geodesic flows. Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 285-293. http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a8/
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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 $\Delta$ for each metric admits sufficiently many linear differential operators commuting with $\Delta$. 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.

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