On integrability of certain rank 2 sub-Riemannian structures
Boris Kruglikov, Andreas Vollmer, Georgios Lukes-Gerakopoulos
备注：
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
发表信息：
Regular and Chaotic Dynamics, September 2017, Volume 22, Issue 5, pp 502-519
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.
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In contrast to SPD matrices, few tools exist to perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as the quotient of the affine-invariant metric by the congruence action of positive diagonal matrices. The space of SPD matrices had always been thought of as a Riemannian homogeneous space. In contrast, we view in this work SPD matrices as a Lie group and the affine-invariant metric as a left-invariant metric. This unexpected new viewpoint allows us to generalize the construction of the quotient-affine metric and t…
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