On integrability of certain rank 2 sub-Riemannian structures
Differential Geometry
2017-10-10 v2 Dynamical Systems
Optimization and Control
Chaotic Dynamics
Abstract
We discuss the integrability of rank 2 sub-Riemannian structures on low-dimensional manifolds, and then prove that some structures of that type in dimension 6, 7 and 8 have a lot of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing fields and the Hamiltonian, thus indicating non-integrability of the corresponding geodesic flows.
Keywords
Cite
@article{arxiv.1507.03082,
title = {On integrability of certain rank 2 sub-Riemannian structures},
author = {Boris Kruglikov and Andreas Vollmer and Georgios Lukes-Gerakopoulos},
journal= {arXiv preprint arXiv:1507.03082},
year = {2017}
}
Comments
In the second version we restructured the material, improved non-existence result in dimension 7 (to degree 6 using the modular approach), and updated references. We also refined the algorithm, and we attach the corresponding commented Maple file (together with PDF outputs of its work for several cases) as the supplement