Foundations of Carrollian Geometry

TL;DR

This review systematically develops the geometry of Carrollian manifolds, including connections and curvature, and introduces novel results linking null hypersurfaces, ambient space embeddings, and Einstein equations.

Contribution

It provides a comprehensive framework for Carrollian geometry, including new intrinsic connections, curvature tensors, and a unified description of hypersurfaces of all causal types.

Findings

Constructed the most general intrinsic Carrollian connection.

Derived the induced geometry from ambient spacetime using rigging techniques.

Showed Einstein equations as conservation laws for null Brown-York stress tensor.

Abstract

Carrollian physics provides the natural framework for describing null hypersurfaces. This review explores the geometry of Carrollian manifolds -- spaces endowed with a degenerate metric. We begin with an algebraic overview of the Carroll group, its conformal extension, and its relation to the BMS group. Then, in the core of the review, we follow the standard pseudo-Riemannian narrative: metric $\to$ connection $\to$ curvature. We first introduce the modern, general definition of a Carrollian structure, the analogue of the metric on such manifolds, reviewing the historical developments, symmetries, and link with the algebraic groups. The second part concerns connections. We show the breakdown of the Levi-Civita theorem in the Carrollian setting and construct the most general intrinsic Carrollian connection. The standard connection is then identified intrinsically and later shown to coincide with the rigged connection induced by embedding a null hypersurface in an ambient spacetime. The third part develops the associated curvature tensors. We include novel results presented here for the first time. Two advanced topics highlight the broader scope of this framework. The first treats null hypersurfaces via the rigging technique, deriving the induced geometry from the ambient space. This provides a unified language for spacelike, timelike, and null hypersurfaces, and shows that the induced rigged connection exactly reproduces the intrinsic Carrollian one. From this, the Gauss and Codazzi-Mainardi equations follow, and the Einstein equations emerge as conservation laws for the null Brown-York stress tensor. The second topic extends the Carrollian setup to generic, non-null hypersurfaces, enabling a smooth null limit and completing the unified geometric description of hypersurfaces of all causal characters.