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A nonintegrable sub-Riemannian geodesic flow on a Carnot group

R. Montgomery ; M. Shapiro ; Alexander Stolin (Institutionen för matematik)
Journal of dynamical and control systems (1079-2724). Vol. 3 (1997), 4, p. 519-530.
[Artikel, refereegranskad vetenskaplig]

Graded nilpotent Lie groups, or Carnot groups, are to sub-Riemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between sub-Riemannian and Riemannian geometry is a strong one, one might conjecture that the sub-Riemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture to be false by showing that the sub-Riemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebra consists of 4 by 4 upper triangular matrices. As a corollary, we prove that the centralizer for the corresponding quadratic "quantum" Hamiltonian in the universal enveloping algebra of this Lie algebra is "as small as possible."

Nyckelord: Carnot groups, Nonholonomic distributions, Nonintegrable, Sub-Riemannian, Universal enveloping algebra

Denna post skapades 2017-07-05.
CPL Pubid: 250498


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Institutionen för matematik (1987-2001)



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