Chaotic Geodesics in Carnot Groups

Details Chaotic Geodesics in Carnot Groups Chaotic Geodesics in Carnot Groups

1997-04-25

arxiv.org/abs/dg-ga/9704013v1

Mathematics

Differential Geometry

LaTeX, 10 pages

Scientific paper

The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first example of a Carnot group (graded nilpotent Lie group with an invariant subRiemannian structure supported on the generating subspace) with a non-integrable geodesic flow. We apply this result to prove that the centralizer for the corresponding quadratic ``quantum'' Hamiltonian in the universal enveloping algebra for this group is ``as small as possible''.

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