 Solution

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.