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

This article is cited in 2 scientific papers (total in 2 papers)

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
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\by V.~S.~Matveev, P.~J.~Topalov
\paper Geodesic equivalence of metrics as a particular case of integrability of geodesic flows
\jour TMF
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\issue 2
\pages 285--293
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\transl
\jour Theoret. and Math. Phys.
\yr 2000
\vol 123
\issue 2
\pages 651--658
\crossref{https://doi.org/10.1007/BF02551397}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  mathscinet  zmath  elib
    2. Mikes J., Stepanova E., Vanzurova A., “Differential Geometry of Special Mappings”, Differential Geometry of Special Mappings, Palacky Univ, 2015, 1–566  mathscinet  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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