Lemma: local curvature transforms by conjugation

For an arbitrary Lie group,

Here $g^{-1}dg$ denotes the pullback of the left Maurer–Cartan form. For matrix Lie groups these become

Write $A=s^*\omega$ and $A^g=(s\cdot g)^*\omega$. The principal curvature $\Omega$ is horizontal and equivariant. In differentiating $s\cdot g$, terms from differentiating $g$ are vertical and therefore vanish when inserted into $\Omega$. Equivariance gives

which is the claimed formula.

On overlaps with transition function $g_{ij}$, the local curvature forms satisfy

Consequently, invariant polynomials applied to the local curvatures agree on overlaps and define global Chern–Weil forms.

1. Abelian groups (electromagnetism). If $G$ is abelian (for example $U(1)$), then $g^{-1}Fg=F$, so the curvature 2-form is gauge invariant. In particular, the transformation reduces to $A^g=A+g^{-1}dg$ while $F^g=dA^g=dA=F$.

1. Pure gauge connections have zero curvature. On a trivial bundle, if $A=g^{-1}dg$ is pure gauge, then $F=dA+\tfrac12[A\wedge A]=0$. The lemma then gives $F^h=h^{-1}0\,h=0$ for any further gauge transformation $h$.

1. Associated vector bundles (matrix conjugation). If $G$ acts on a vector space via a representation (see representation), the induced curvature on the associated vector bundle is a matrix-valued 2-form, and this lemma becomes the familiar rule “curvature matrices conjugate under change of frame.”

Adam Marsh, Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results, §5.4, especially equations (5.17)–(5.19), on horizontal equivariant curvature and its local forms. Author paper.