螺旋CR结构与次黎曼测地线
Helical CR Structures and Sub-Riemannian Geodesics
摘要：
螺旋CR结构是将实欧几里得空间分解为偶数维水平子空间及其正交垂直补空间，以及水平空间上的殆复结构和垂直空间中的标记向量。我们证明了这样的结构与具有特殊的正规测地线的第二步卡诺群之间的等价性，以及这样的结构与导数具有常欧氏范数的光滑实曲线之间的等价性.因此，我们将第二步卡诺群配备亚黎曼测地线与这个家庭的曲线。对某些平面齐次多项式映射的单位圆的限制给出了一类有指导意义的例子。我们详细描述这些例子。
英文摘要：
A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step two Carnot groups equipped with a distinguished normal geodesic, and also between such structures and smooth real curves whose derivatives have constant Euclidean norm. As a consequence, we relate step two Carnot groups equipped with sub-Riemannian geodesics with this family of curves. The restriction to the unit circle of certain planar homogeneous polynomial mappings gives an instructive class of examples. We describe these examples in detail.
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