A Nonintegrable Sub-Riemannian Geodesic Flow on a Carnot Group
A Nonintegrable Sub-Riemannian Geodesic Flow on a Carnot Group
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卡诺群上不可积的亚黎曼测地流
DOI：
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发表时间：
1997
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通讯作者：
A. Stolin
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作者：
R. Montgomery;M. Shapiro;A. Stolin
分次幂零李群或卡诺群是次黎曼几何，正如欧几里德空间是黎曼几何。它们是该几何体的公制切圆锥体。希望次黎曼几何和黎曼几何之间的类比很强，人们可能会猜想，任何卡诺群上的次黎曼测地线流是完全可积的。在李代数由4×4上三角矩阵组成的群中，次黎曼测地线流不是代数完全可积的，从而证明了这一猜想是错误的。作为推论，我们证明了这个李代数的万能包络代数中对应的二次量子哈密顿量的中心化子是“尽可能小的”。
Graded nilpotent Lie groups, or Carnot groups, are to sub-Riemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between sub-Riemannian and Riemannian geometry is a strong one, one might conjecture that the sub-Riemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture to be false by showing that the sub-Riemannian geodesic flow 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.”