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.
& Zhang, P.M., Carroll versus Newton and Galilei: Two Dual Non-Einsteinian Concepts of Time,Class
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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.