 Congruent Displacement of Length:  With respect to a choice of gauge for a neighborhood of \(p\), the transport of a length \(l_{p}\) at \(p\) to an infinitesimally neighboring point \(q\) constitutes a congruent displacement if and only if there exists a choice  of gauge for the neighborhood of \(p\) relative to which the  transported length  \(\overline{l}_{q}\) has the same value as  \(\overline{l}_{p}\); that is 