The Chow–Rashevskii theorem is a fundamental result in differential geometry and the theory of control systems. It pertains to the accessibility of points in a control system defined by smooth vector fields.
Articles by others on the same topic
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.