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 TMF, 2000, Volume 123, Number 2, Pages 285–293 (Mi tmf602)

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

V. S. Matveeva, P. J. Topalovb

a Chelyabinsk State University
b Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/tmf602

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English version:
Theoretical and Mathematical Physics, 2000, 123:2, 651–658

Bibliographic databases:

Citation: V. S. Matveev, P. J. Topalov, “Geodesic equivalence of metrics as a particular case of integrability of geodesic flows”, TMF, 123:2 (2000), 285–293; Theoret. and Math. Phys., 123:2 (2000), 651–658

Citation in format AMSBIB
\Bibitem{MatTop00} \by V.~S.~Matveev, P.~J.~Topalov \paper Geodesic equivalence of metrics as a particular case of integrability of geodesic flows \jour TMF \yr 2000 \vol 123 \issue 2 \pages 285--293 \mathnet{http://mi.mathnet.ru/tmf602} \crossref{https://doi.org/10.4213/tmf602} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1794160} \zmath{https://zbmath.org/?q=an:0996.53054} \elib{http://elibrary.ru/item.asp?id=13340359} \transl \jour Theoret. and Math. Phys. \yr 2000 \vol 123 \issue 2 \pages 651--658 \crossref{https://doi.org/10.1007/BF02551397} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000165897000008} 

• http://mi.mathnet.ru/eng/tmf602
• https://doi.org/10.4213/tmf602
• http://mi.mathnet.ru/eng/tmf/v123/i2/p285

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This publication is cited in the following articles:
1. S. L. Tabachnikov, “Ellipsoids, complete integrability and hyperbolic geometry”, Mosc. Math. J., 2:1 (2002), 183–196
2. Mikes J., Stepanova E., Vanzurova A., “Differential Geometry of Special Mappings”, Differential Geometry of Special Mappings, Palacky Univ, 2015, 1–566
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