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Izv. RAN. Ser. Mat., 2019, Volume 83, Issue 6, Pages 63–103 (Mi izv8863)  

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

Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards

V. V. Vedyushkina (Fokicheva), A. T. Fomenko

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The authors have recently introduced the class of topological billiards. Topological billiards are glued from elementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classical linearly and quadratically integrable geodesic flows on tori and spheres are Liouville equivalent to appropriate topological billiards. Moreover, the linear and quadratic integrals of the geodesic flows reduce to a single canonical linear integral and a single canonical quadratic integral on the billiard. These results are obtained within the framework of the Fomenko–Zieschang theory of the classification of integrable systems.

Keywords: integrable system, topological billiard, geodesic flow, Liouville equivalence, Fomenko–Zieschang invariant.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation НШ-6399.2018.1
Russian Foundation for Basic Research 19-01-00775-a
This paper was written with the support of the Russian Federation President's Programme for the support of leading scientific schools (grant no. NSh-6399.2018.1, contract no. 075-02-2018-867), and the Russian Foundation for Basic Research (grant no. 19-01-00775-a).


DOI: https://doi.org/10.4213/im8863

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English version:
Izvestiya: Mathematics, 2019, 83:6, 1137–1173

Bibliographic databases:

UDC: 517.938.5
MSC: Primary 37D50; Secondary 37J35
Received: 13.09.2018
Revised: 04.03.2019

Citation: V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. RAN. Ser. Mat., 83:6 (2019), 63–103; Izv. Math., 83:6 (2019), 1137–1173

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Vedyushkina, “Integrable billiard systems realize toric foliations on lens spaces and the 3-torus”, Sb. Math., 211:2 (2020), 201–225  mathnet  crossref  crossref  isi  elib
    2. G. V. Belozerov, “Topologicheskaya klassifikatsiya integriruemykh geodezicheskikh billiardov na kvadrikakh v trekhmernom evklidovom prostranstve”, Matem. sb., 211:11 (2020), 3–40  mathnet  crossref
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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