Curvature of a principal connection ($\mathcal{F}=d\mathcal{A}+\frac{1}{2}[\mathcal{A}\wedge \mathcal{A}]$)

The curvature of a principal connection is the horizontal equivariant two-formFor horizontal vector fields $X,Y$, it satisfies $\mathcal{F}(X,Y)=-\mathcal{A}([X,Y])$.

A principal connection is flat when its curvature vanishes. The identity $\mathcal{F}(X,Y)=-\mathcal{A}([X,Y])$ shows that this is equivalent to integrability of its horizontal distribution.