Local rods and holonomy

On a sufficiently small angular interval choose a real lift of $\varphi$. The developing map

is a local isometry from (eq:metric) to Minkowski space with metric $-c^2\mathrm{d}{T}^2+\mathrm{d}{x}^2+\mathrm{d}{y}^2+\mathrm{d}{z}^2$. Its spatial coordinates are three local rods.