Carnot-Carathéodory distance ($d_{CC}(g,h)={\inf }_{\gamma }\int |{\dot{x}}_t|dt$)

On a graded nilpotent Lie group whose first layer generates its Lie algebra, a horizontal curve obeys ${\dot{\gamma }}_t=(L_{{\gamma }_t}{)}_{\ast }{\dot{x}}_t$ for a control in the first layer. The displayed infimum is over horizontal curves with prescribed endpoints. The distance is left invariant and homogeneous under the graded dilations. On a free nilpotent Lie group it measures the least Euclidean length of a path realizing a prescribed truncated path signature.

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.