 Local principal connection form

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.