1997/04/25 by R. Montgomery, Richard Montgomery, Michael Shapiro +6
Mathematics · #53C22 #53Cxx #58A30 #58F07 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #dg-ga #math.DG #msc:53C22 #msc:53Cxx #msc:58A30 #msc:58F07
paper · pdf · doi:10.48550/arxiv.dg-ga/9704013
LaTeX, 10 pages
arxiv created 1997/04/25 · openalex publication_date 1997/04/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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''.