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
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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.
@article{TMF_2000_123_2_a8,
author = {V. S. Matveev and P. J. Topalov},
title = {Geodesic equivalence of metrics as a particular case of integrability of geodesic flows},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {285--293},
publisher = {mathdoc},
volume = {123},
number = {2},
year = {2000},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a8/}
}
TY - JOUR AU - V. S. Matveev AU - P. J. Topalov TI - Geodesic equivalence of metrics as a particular case of integrability of geodesic flows JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2000 SP - 285 EP - 293 VL - 123 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a8/ LA - ru ID - TMF_2000_123_2_a8 ER -
%0 Journal Article %A V. S. Matveev %A P. J. Topalov %T Geodesic equivalence of metrics as a particular case of integrability of geodesic flows %J Teoretičeskaâ i matematičeskaâ fizika %D 2000 %P 285-293 %V 123 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a8/ %G ru %F TMF_2000_123_2_a8
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/