Chow-Rashevskii theorem
= Chow-Rashevskii theorem
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= Chow theorem
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On a connected manifold, a smooth bracket-generating family of vector fields connects every pair of points by a piecewise smooth horizontal curve. For a <free step-N nilpotent Lie group>, this says that every group element is the <path signature> of a sufficiently regular path. In the planar step-two case, a closed rectangle prescribes the signed area and a following straight segment prescribes the displacement.