A Non-integrable Subriemannian Geodesic Flow on a Carnot Group

AND Abstract. Graded nilpotent Lie groups, or Carnot Groups are to subRiemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between subRiemannian and Riemannian geometry is a strong one, one might conjecture that the subRiemannian geo-desic ow on any Carnot group is completely integrable. We prove this conjecture to be false by showing that the subRiemannian geo-desic ow 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". 1. Introduction Geometry would be in a poor state if Euclidean geodesic ow was not completely integrable { in other words, if we did not have an explicit algebraic description of straight lines in Euclidean space. Riemannian geometry, being innnitesimally Euclidean, makes frequent use of this explicit description. For example the exponential map takes Euclidean lines through the origin to geodesics. SubRiemannian geometries, also called Carnot-Caratheodory geometries, are not innnitesimally Euclidean. Rather they are, at typical points, in-nitesimally modelled by Carnot groups. We will review these geometries and groups, and the relation between them momentarily. The point of this note is to show that the Carnot geodesic ows need not be integrable. We do this by giving an example of a Carnot group of dimension 6 whose geodesic ow cannot be integrated by rational functions. A subRiemannian geometry consists of a nonintegrable subbundle (distribution) V of the tangent bundle T of a manifold, together with a ber-inner product on this bundle. These geometries arise, among other places, as the