= Reconstruction of a principal connection from local gauge potentials
{title2=$\omega=\gamma^{-1}A\gamma+\gamma^{-1}d\gamma$}
A <local principal connection form> $A$ and a fiber coordinate $\gamma$ reconstruct the connection on a <principal bundle> in a local section. Changing that section by $s'=su$ changes $\gamma$ to $u^{-1}\gamma$ and $A$ to $u^{-1}Au+u^{-1}du$. Substitution cancels the two terms involving $du$, so the reconstructed forms agree on overlaps. Here $A$ is Lie-algebra-valued and matrix conjugation denotes the adjoint action. Agreement across trivializations differs from the right-action equivariance $R_h^*\omega=\operatorname{Ad}_{h^{-1}}\omega$.
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