= Carnot-Carathéodory distance
{c}
{title2=$d_{CC}(g,h)=\inf_\gamma\int|\dot x_t|\,dt$}
On a graded nilpotent <Lie group> whose first layer generates its <Lie algebra>, a horizontal curve obeys $\dot\gamma_t=(L_{\gamma_t})_*\dot x_t$ 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>.
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