This article is cited in 6 papers

On Integrability of Certain Rank 2 Sub-Riemannian Structures

Boris S. Kruglikov^a, Andreas Vollmer^bc, Georgios Lukes-Gerakopoulos<sup>de

a</sup> 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.

MSC: 37J30, 37J60; 70G45, 37J15, 35N10

Received: 31.01.2017
Accepted: 15.08.2017

Language: English

DOI: 10.1134/S1560354717050033

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