Carrollian $\mathbb{R}^\times$-bundles: Connections and Beyond

We propose an approach to Carrollian geometry using principal $\mathbb{R}^\times$-bundles ($\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$) equipped with a degenerate metric whose kernel is the module of vertical vector fields. The constructions allow for non-trivial bundles and a large class of Carrollian manifolds can be analysed in this formalism. Alongside other results, we show that once an $\mathbb{R}^\times$-connection has been chosen, there is a canonical affine connection that is torsionless, but, in general, not compatible with the degenerate metric. The construction of an affine connection is intimately tied to a Kaluza-Klein geometry.