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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.

For a principal connection with connection form $\mathcal{A}$, the horizontal distribution of a principal connection is $H=\ker \mathcal{A}$, the complement of the tangent spaces to the $G$-orbits. Its curvature of a principal connection isIf $X,Y$ are horizontal vector fields, then $\mathcal{A}(X)=\mathcal{A}(Y)=0$, and henceThe Frobenius theorem says that $H$ is integrable exactly when $[X,Y]$ is horizontal for all horizontal $X,Y$. The displayed identity makes this equivalent to the vanishing of the horizontal two-form $\mathcal{F}$, hence to $\mathcal{F}=0$. Thus the horizontal distribution is integrable exactly for a flat principal connection.

The vertical tangent space of the trivial principal $\mathbb{R}$-bundle is spanned by ${\partial }_z$. Sinceprojects isomorphically onto the tangent space of $S^1\times \mathbb{R}$, it is always complementary to the vertical direction. It is the horizontal distribution of a principal connection precisely when it is invariant under the principal translations $z\mapsto z+a$. The horizontal lifts of ${\partial }_{\theta }$ and ${\partial }_t$ are unique, so this invariance is equivalent toThe functions must also be smooth and periodic in $\theta$, as is already required for them to be functions on the cylinder.

Under these conditions the connection form isIt sends ${\partial }_z$ to $1$, is translation-invariant, and has kernel $D$, proving sufficiency as well. Since the structure group $\mathbb{R}$ is abelian, the bracket term vanishes and