Carrollian manifolds can be described as principal R^x-bundles with a degenerate metric, and a chosen connection yields a canonical non-degenerate metric and geodesic dynamics.
& Pons, J.M., Dynamics of Carroll strings,J
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Carrollian $\mathbb{R}^\times$-bundles: Connections and Beyond
Carrollian manifolds can be described as principal R^x-bundles with a degenerate metric, and a chosen connection yields a canonical non-degenerate metric and geodesic dynamics.