Foundations of Carrollian Geometry

TL;DR

This work develops a fully intrinsic, covariant framework for Carrollian geometry, focusing on degenerate metrics that arise on null hypersurfaces and the necessary Carrollian vector in their kernel. It shows that in Carrollian settings the Levi-Civita connection is not uniquely determined; instead, a most general intrinsic Carrollian connection exists, and a preferred connection emerges from embedding Carrollian structures as null hypersurfaces in a pseudo-Riemannian bulk. The authors then build curvature notions suited to Carrollian geometry, including the Riemann–Carroll tensor and horizontal curvature, and derive Gauss–Codazzi–Mainardi relations with novel null-hypersurface results such as the null Brown–York stress tensor and its conservation laws. Finally, the Rigging framework is used to relate intrinsic Carrollian data to ambient spacetime geometry, and the authors extend the Carrollian program to stretched (non-null) hypersurfaces, producing sCarrollian structures and conserved stress tensors that unify hypersurfaces of all causal character and reproduce Einstein equations on the surface. Collectively, these results provide a coherent, embedding-friendly, and broadly applicable geometric toolkit for null and Carrollian physics with potential applications to holography and gravitational dynamics on null boundaries.

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. A preferred connection is then identified intrinsically and later shown to coincide with the one 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.