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

We propose an approach to Carrollian geometry using principal ${\mathbb{R}}^{\times }$-bundles (${\mathbb{R}}^{\times }:=\text{matthbb}R\setminus \{0\}$) 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. A key result in this is that once a principal connection has been selected, there is a canonical non-degenerate metric that can be leveraged to circumvent the difficulties associated with a degenerate metric. Within this framework, we examine the Levi-Civita connection and null geodesics.