 Parallel-frame compatibility criterion

For $D_i={\partial }_i+A_i$, compatibility for arbitrary initial vectors is equivalent to zero gauge curvature. An invertible parallel bundle frame $S$ obeys $D_{i}S=0$, so $[D_i,D_j]S=F_{ij}S=0$ forces $F_{ij}=0$. Conversely, solve the two ordinary transport equations along one coordinate axis and then along parallel lines. The unused transport residual solves a homogeneous transport equation with zero initial value when the gauge curvature is zero; it therefore vanishes. Thus $A_i=-({\partial }_{i}S)S^{-1}=g^{-1}{\partial }_{i}g$ with $g=S^{-1}$. One particular parallel vector requires only $F_{ij}v=0$, a weaker condition.