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Mat. Sb., 1998, Volume 189, Number 10, Pages 5–32 (Mi msb346)  

This article is cited in 40 scientific papers (total in 40 papers)

Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry

A. V. Bolsinova, V. S. Matveeva, A. T. Fomenko

a M. V. Lomonosov Moscow State University

Abstract: Classical and new results on integrable geodesic flows on two-dimensional surfaces are reviewed. The central question is the classification of such flows up to various equivalences, of which the following four kinds are the most interesting ones: 1) isometry; 2) geodesic equivalence; 3) orbital equivalence; 4) Liouville equivalence.

DOI: https://doi.org/10.4213/sm346

Full text: PDF file (406 kB)
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English version:
Sbornik: Mathematics, 1998, 189:10, 1441–1466

Bibliographic databases:

UDC: 513.83
MSC: Primary 58F17, 58F07; Secondary 58F05
Received: 04.06.1998

Citation: A. V. Bolsinov, V. S. Matveev, A. T. Fomenko, “Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry”, Mat. Sb., 189:10 (1998), 5–32; Sb. Math., 189:10 (1998), 1441–1466

Citation in format AMSBIB
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    This publication is cited in the following articles:
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    4. Dullin, HR, “A new integrable system on the sphere”, Mathematical Research Letters, 11:5–6 (2004), 715  crossref  mathscinet  zmath  isi  elib
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    7. Dullin H.R., Matveev V.S., “A new natural Hamiltonian system on T*S-2 admitting an integral of degree 3 in momenta”, Global Analysis and Applied Mathematics, Aip Conference Proceedings, 729, 2004, 141–146  crossref  mathscinet  zmath  adsnasa  isi
    8. V. S. Matveev, “The eigenvalues of the Sinyukov mapping for geodesically equivalent metrics are globally ordered”, Math. Notes, 77:3 (2005), 380–390  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Matveev, VS, “Lichnerowicz-Obata conjecture in dimension two”, Commentarii Mathematici Helvetici, 80:3 (2005), 541  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    10. Sinclair, R, “Jacobi's last geometric statement extends to a wider class of Liouville surfaces”, Mathematics of Computation, 75:256 (2006), 1779  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    11. Kruglikov, BS, “Strictly non-proportional geodesically equivalent metrics have h(top)(g)=0”, Ergodic Theory and Dynamical Systems, 26 (2006), 247  crossref  mathscinet  zmath  isi  elib  scopus  scopus
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    13. Kruglikov, B, “Invariant characterization of Liouville metrics and polynomial integrals”, Journal of Geometry and Physics, 58:8 (2008), 979  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    14. Volodymyr Kiosak, Vladimir S. Matveev, “Complete Einstein Metrics are Geodesically Rigid”, Comm Math Phys, 2009  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    15. Bolsinov, AV, “Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta”, Journal of Geometry and Physics, 59:7 (2009), 1048  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    16. Matveev V.S., Shevchishin V.V., “Differential invariants for cubic integrals of geodesic flows on surfaces”, Journal of Geometry and Physics, 60:6–8 (2010), 833–856  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
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    19. Matveev V.S., “Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics”, J Math Soc Japan, 64:1 (2012), 107–152  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    20. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Kachenie bez vercheniya shara po ploskosti: otsutstvie invariantnoi mery v sisteme s polnym naborom integralov”, Nelineinaya dinam., 8:3 (2012), 605–616  mathnet
    21. Bolsinov A.V., Borisov A.V., Mamaev I.S., “Rolling of a Ball Without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals”, Regul. Chaotic Dyn., 17:6 (2012), 571–579  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    22. Labrousse C., “Polynomial Growth of the Volume of Balls for Zero-Entropy Geodesic Systems”, Nonlinearity, 25:11 (2012), 3049–3069  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
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