Local principal connection form ($A=s^{\ast }\mathcal{A},A^{\prime }=g^{-1}Ag+g^{-1}dg$)

Pulling a principal connection back along a local section $s$ gives a Lie-algebra-valued one-form on the base. If the section changes to $sg$, the form has the displayed inhomogeneous transformation. Its gauge curvature $F=dA+A\wedge A$ transforms homogeneously as $F^{\prime }=g^{-1}Fg$. The collection of such local forms encodes a global principal connection even when no single section exists globally.

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 $s^{\prime }=su$ 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 $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$.