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

TL;DR

This paper introduces a novel geometric framework for Carrollian manifolds using principal a0a0-bundles with degenerate metrics, enabling analysis of connections and geodesics despite metric degeneracy.

Contribution

It develops a new formalism for Carrollian geometry using principal a0a0-bundles, allowing for non-trivial bundles and a canonical non-degenerate metric after choosing a connection.

Findings

Established a method to analyze Levi-Civita connection in Carrollian geometry.

Derived properties of null geodesics within the new formalism.

Provided tools to handle non-trivial Carrollian bundles.

Abstract

We propose an approach to Carrollian geometry using principal $\mathbb{R}^\times$-bundles ($\mathbb{R}^\times := \matthbb{R} \setminus \{0\}$) 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.