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A local section $s:U\to P$ is horizontal when $ds(TU)\subseteq H$, equivalently when $s^{\ast }\mathcal{A}=0$. A flat principal connection has horizontal sections locally, while its holonomy can obstruct a global horizontal section.

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.

For $f=t\cos \theta +1$ and $g=\sin \theta$,The connection is therefore flat, so the Frobenius theorem gives horizontal sections locally.

A global section has the form $z=h(\theta ,t)$ and is horizontal exactly whenThe second equation gives $h=t\sin \theta +k(\theta )$, and the first then forces $k^{\prime }(\theta )=1$. No such $k$ is periodic on $S^1$, so no global horizontal section exists. Equivalently, the horizontal lift of one positive circuit in the $\theta$ direction changes $z$ bywhich is nontrivial holonomy.