A nonintegrable sub-Riemannian geodesic flow on a Carnot group
Journal article, 1997

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."

Universal enveloping algebra

Carnot groups


Nonholonomic distributions



R. Montgomery

M. Shapiro

Alexander Stolin

Department of Mathematics

University of Gothenburg

Journal of Dynamical and Control Systems

1079-2724 (ISSN)

Vol. 3 4 519-530

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