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Chaotic Geodesics in Carnot Groups

R. Montgomery·M. Shapiro·A. Stolin
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Abstract

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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BibTeXRIS

R. Montgomery, M. Shapiro, A. Stolin. 1997-04-25. Chaotic Geodesics in Carnot Groups. https://arxiv.org/abs/dg-ga/9704013

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