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Regul. Chaotic Dyn., 2017, Volume 22, Issue 5, Pages 502–519 (Mi rcd272)  

On Integrability of Certain Rank 2 Sub-Riemannian Structures

Boris S. Kruglikova, Andreas Vollmerbc, Georgios Lukes-Gerakopoulosde

a Institute of Mathematics and Statistics, University of Tromsø, Tromsø 90-37, Norway
b Mathematisches Institut, Friedrich-Schiller-Universität, 07737 Jena, Germany
c INdAM - Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
d Astronomical Institute of the Academy of Sciences of the Czech Republic, Boční II 1401/1a, CZ-141 31 Prague, Czech Republic
e Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, 121 16 Prague, Czech Republic

Abstract: We discuss rank 2 sub-Riemannian structures on low-dimensional manifolds and prove that some of these structures in dimensions 6, 7 and 8 have a maximal amount of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing vector fields and the Hamiltonian, thus indicating nonintegrability of the corresponding geodesic flows.

Keywords: Sub-Riemannian geodesic flow, Killing tensor, integral, symmetry, Tanaka prolongation, overdetermined system of PDE, prolongation

Funding Agency Grant Number
Czech Science Foundation 17-06962Y
Deutsche Forschungsgemeinschaft 1523
BK and AV were supported by the NFR and DAAD cooperation grant 2014-2015, respectively. AV is a research fellow of Istituto Nazionale di Alta Matematica, a member of GNSAGA, and thanks GRK 1523 (DFG) and the project FIR-2013 Geometria delle equazioni differenziali for financial support. GLG was supported by the UNCE-204020 and GACR-17-06962Y grants.


DOI: https://doi.org/10.1134/S1560354717050033

References: PDF file   HTML file

Bibliographic databases:

Document Type: Article
MSC: 37J30, 37J60; 70G45, 37J15, 35N10
Received: 31.01.2017
Language: English

Citation: Boris S. Kruglikov, Andreas Vollmer, Georgios Lukes-Gerakopoulos, “On Integrability of Certain Rank 2 Sub-Riemannian Structures”, Regul. Chaotic Dyn., 22:5 (2017), 502–519

Citation in format AMSBIB
\Bibitem{KruVolLuk17}
\by Boris S. Kruglikov, Andreas Vollmer, Georgios Lukes-Gerakopoulos
\paper On Integrability of Certain Rank 2 Sub-Riemannian Structures
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 5
\pages 502--519
\mathnet{http://mi.mathnet.ru/rcd272}
\crossref{https://doi.org/10.1134/S1560354717050033}
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