On Carrollian R^×-bundles, a chosen connection turns the degenerate metric into a Lorentzian metric, defining Hodge star, codifferential, and Hodge-de Rham Laplacian, with a Schwarzschild horizon example and a Carrollian Maxwell theory.
Carrollian $\mathbb{R}^\times$-bundles: Connections and Beyond
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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.
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Carrollian $\mathbb{R}^\times$-bundles III: The Hodge Star and Hodge--de Rham Laplacians
On Carrollian R^×-bundles, a chosen connection turns the degenerate metric into a Lorentzian metric, defining Hodge star, codifferential, and Hodge-de Rham Laplacian, with a Schwarzschild horizon example and a Carrollian Maxwell theory.