 Carnot-Carathéodory distance

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.