Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 115 / 4 / c

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