= Parallel-frame compatibility criterion
{title2=$[D_i,D_j]=F_{ij}$}
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 $D_iS=0$, 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 $A_i=-(\partial_iS)S^{-1}=g^{-1}\partial_i g$ with $g=S^{-1}$. One particular parallel vector requires only $F_{ij}v=0$, a weaker condition.
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