Chaotic Geodesics in Carnot Groups
نویسندگان
چکیده
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 geodesic flow on any Carnot group is completely integrable. We prove this conjecture is false by showing that that the subRiemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebraN− consists of 4 by 4 nilpotent triangular matrices. We use this to prove that the centralizer for the corresponding quadratic “quantum” Hamiltonian H in the universal enveloping algebra of N− is “as small as possible”.
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