Carrollian Manifolds: Geometry and Applications

• Carrollian manifolds are smooth (d+1)-dimensional spaces equipped with a degenerate metric whose kernel is generated by a nowhere-vanishing vector field, defining an ultra‐relativistic causal structure.
• They emerge from an Inönü–Wigner contraction of the Poincaré group and are characterized by unique symmetry groups and duality with Galilean (Newton–Cartan) limits.
• These structures underpin innovative field theories and holographic models, with applications to null hypersurfaces, gravitational memory, and black hole boundary analysis.

A Carrollian manifold is a geometric object consisting of a smooth manifold equipped with a degenerate metric whose kernel is generated by a nowhere-vanishing vector field, encapsulating an “ultra-relativistic” causal structure in which the lightcone collapses and spatial propagation is frozen in a preferred “Carrollian time” direction. This geometry is inherently linked to the Carroll group, which arises from an Inönü–Wigner contraction of the Poincaré group as the speed-of-light parameter $c \to 0$, and is characterized by the preservation of the degenerate metric and its kernel. Carrollian manifolds give rise to a wide spectrum of applications in mathematical physics, from the geometry of null hypersurfaces in Lorentzian spacetimes to the structure of null boundaries in gravitational holography, and support a unique class of field theories and symmetry algebras.

1. Geometric Structure and Defining Data

A Carrollian manifold is most generally defined as a tuple $(C, g, \xi, \nabla)$, with:

• $C$ a smooth $(d+1)$-dimensional manifold,
• $g$ a symmetric, rank-$d$ covariant tensor field (degenerate metric) whose kernel is a line bundle generated by a complete, nowhere-vanishing vector field $\xi$ (so $g(\xi, \cdot) = 0$),
• $\nabla$ an affine symmetric (torsion-free) connection, compatible with both the degenerate metric and the distinguished vector field ($\nabla g = 0$, $\nabla \xi = 0$) (Duval et al., 2014).

In adapted coordinates (“Carroll coordinates”) on the flat model, one writes

$$C^{d+1} = \mathbb{R} \times \mathbb{R}^d,\quad g = \delta_{AB}\,dx^A \otimes dx^B,\quad \xi = \partial_s,$$

where $s$ is the “Carrollian time” and $(x^A)$ label the spatial directions (Duval et al., 2014). The degenerate metric does not specify temporal distances, and all tangent vectors proportional to $\xi$ are null with respect to $g$. There is no invertible “metric,” hence no isomorphism between tangent and cotangent bundles.

The automorphism group of a Carrollian manifold—the Carroll group $\text{Carr}(d+1)$—consists of diffeomorphisms preserving both $g$ and the vector field $\xi$. In the flat case, these include spatial rotations, translations, Carroll boosts (mixing time and space in a way distinct from Galilean boosts), and time translations. The infinitesimal Carroll boost acts as $s' = s - b\cdot x$, $x' = R x + c$ with $R \in O(d)$, $c \in \mathbb{R}^d$, and $b \in \mathbb{R}^d$ (Duval et al., 2014).

2. Duality with the Galilean Limit and Bargmann Structures

Carrollian manifolds are intimately related to Newton–Cartan (Galilean) geometries, as both originate from non-Einsteinian limits of the Poincaré group. In the Galilean limit, with absolute time $t = x^0/c$ and $c\to\infty$, the contravariant metric’s kernel is generated by $dt$. In the Carrollian limit, a new scaling $s = C x^0$ is introduced, and $C\to\infty$ (or $C\to0$ under rescaling), which leads to a degenerate covariant metric with kernel generated by $\xi = \partial_s$ (Duval et al., 2014).

In the unified Bargmann framework (an ambient higher-dimensional spacetime), both Newton–Cartan and Carrollian manifolds naturally arise: Galilean manifolds as quotients by null translations, Carrollian manifolds as null hypersurfaces. The duality is mathematically realized as an exchange $t \leftrightarrow s$ between the canonical time coordinates. Consequently, while spatial hypersurfaces are propagating in the Galilean regime, propagation is “frozen” in Carrollian spacetime (“no spatial propagation”), underscoring fundamentally different causal structures (Duval et al., 2014, Morand, 2018).

Within ambient (d+2)-dimensional spacetimes possessing a distinguished null vector field $\hat{\xi}$ (Bargmann structure), lightlike foliations induce degenerate metrics with radical spanned by a Carrollian vector field on each leaf. For classes of gravitational waves known as Bargmann–Eisenhart or Dodgson waves, construction of Carrollian structures is possible by pulling back the ambient metric, along with a projected connection. Notably, Dodgson waves allow for embedding a much wider class of (torsion-free) Carrollian manifolds, including pseudo-invariant connections relevant for (A)dS Carroll spaces (Morand, 2018).

3. Carrollian Connections and Curvature

Unlike in pseudo-Riemannian geometry, where the Levi–Civita connection is unique, the degenerate nature of the Carrollian metric $g$ prevents the existence of a unique torsion-free, metric-compatible connection. The compatibility requirement ($\nabla g = 0$, $\nabla \xi = 0$) is usually imposed, but additional torsionless connections can be constructed, with the freedom parametrized by arbitrary tensors with values in the line bundle generated by $\xi$ (Ciambelli et al., 24 Oct 2025).

If a Carrollian manifold is realized as a null hypersurface in an ambient Lorentzian spacetime, an intrinsic connection derives naturally by pulling back the Levi–Civita connection and applying a “rigging” technique to define the corresponding horizontal and vertical projectors. The null hypersurface inherits a degenerate metric $q$ and a ruling $\ell$, with the connection $D_a$ characterized by

$$D_a q_{bc} = -k_b \theta_{ac} - k_c \theta_{ab},\qquad D_a \ell^b = \theta_a^b + \omega_a \ell^b,$$

where $\theta_{ab}$ is the null expansion tensor and $k_b$ is the Ehresmann (clock) 1-form (Ciambelli et al., 24 Oct 2025). The induced curvature (Riemann–Carroll tensor) encodes both intrinsic and extrinsic properties and is the natural replacement for the Riemann curvature in the degenerate setting.

4. Carrollian Manifolds as Geometry of Null Hypersurfaces and Boundaries

Carrollian geometry is the intrinsic geometry of null hypersurfaces embedded in Lorentzian spacetimes. For a null boundary (event or Killing horizon, null infinity $\mathscr{I}$), the induced metric is necessarily degenerate, and the kernel corresponds to the null generator tangent to the hypersurface (Ciambelli et al., 2019, Blitz et al., 29 Sep 2024, Herfray, 2021).

This intrinsic geometry forms the mathematical underpinning for the analysis of physical phenomena at null boundaries. For example, at null infinity in asymptotically flat spacetime, the Carrollian conformal structure emerges as the natural geometric structure, and, under suitable gauge choices (Bondi gauge), the redundant conformal factors and connection choices can be fixed, yielding a strict Carrollian geometry (Herfray, 2021). Cartan geometry provides a unifying language, as both “strong” Carrollian and conformal Carrollian geometries arise as Cartan geometries modeled on appropriate homogeneous spaces (Herfray, 2021). The presence of gravitational radiation is reflected in the curvature of the Cartan connection and serves as an obstruction to the reduction of the full BMS group of asymptotic symmetries to the Poincaré subgroup.

Similar structures appear when considering time-/space-like infinity in projectively compact Ricci-flat Einstein manifolds, where Carrollian geometries arise on the extended boundary through holonomy reduction and projective tractor calculus (Borthwick et al., 3 Jun 2024).

The connection between Carrollian geometry and null boundaries has direct implications for the encoding of gravitational data (such as memory effects, soft hair, and asymptotic charges) and the emergent symmetry algebras (e.g., infinite-dimensional conformal Carroll/BMS algebras).

5. Carrollian Field Theories, Electromagnetism, and Symmetries

The degenerate causal structure of Carrollian manifolds leads to distinctive field theory models, notably Carrollian versions of electromagnetism and scalar electrodynamics. In the Carrollian contraction (ultra-relativistic limit), one obtains two distinct sectors of Carrollian electromagnetism—electric-like and magnetic-like—mirroring the dual Galilean (Le Bellac–Lévy-Leblond) sectors (Duval et al., 2014). In the electric sector, Maxwell’s equations reduce to

$$\nabla \times \vec{E}^{\,e} + \partial_s \vec{B}^{\,e} = 0,\quad \nabla \cdot \vec{B}^{\,e} = 0,\quad \mathrm{div}\,\vec{E}^{\,e} = 0,\quad \nabla_s \vec{E}^{\,e} = 0.$$

Owing to the degeneracy of $g$, there is no isomorphism between vectors and one-forms, and careful distinction between covariant and contravariant fields is necessary. Carrollian versions of electromagnetism and field theories typically split into decoupled spatial and time (“Carrollian time”) components; propagation is "frozen" in the spatial directions.

Field theories defined on Carrollian manifolds, such as Carrollian scalar electrodynamics, possess infinite-dimensional symmetry algebras extending the conformal Carroll group, highly relevant for flat space holography (Bagchi et al., 2019). Noether charges associated with these symmetries close under Poisson brackets into the infinite conformal Carrollian algebra.

Flat holography connects field theories on Carrollian manifolds to gravitational dynamics in asymptotically flat spacetimes and their memory/soft theorem structures (Saha, 2023, Stieberger et al., 21 Feb 2024, Alday et al., 27 Jun 2024). Enhanced symmetry structures—such as the BMS group and its versions—are naturally interpreted as extensions of the Carroll group, and conserved charges organize into infinite towers (electric/magnetic, leading/subleading), with duality symmetries (e.g., Möbius symmetry acting on Carrollian Cotton and energy multiplets) (Mittal et al., 2022). The connection with higher symmetry algebras extends to the structure of Ward identities and operator product expansions in conformal Carrollian field theory.

6. Extensions, Generalizations, and Recent Developments

Carrollian geometry has seen significant methodological and conceptual broadening:

• Principal $\mathbb{R}^\times$-bundles: Carrollian manifolds can be constructed as principal $\mathbb{R}^\times$-bundles $P$ over a base $M$, with degenerate metric $g$ whose kernel consists of vertical vector fields. Choosing a principal connection permits the definition of a canonical affine (typically non-metric-compatible) connection, closely connected to Kaluza–Klein geometry. These constructions enable Hodge theory on the total space and facilitate a bridge to pseudo-Riemannian geometry, including applications such as Carrollian electromagnetism and studying black hole horizons (Bruce, 27 May 2025, Bruce, 29 Jul 2025).
• Super-Carrollian Geometry: A super-Carrollian manifold is a supermanifold of dimension $n|1$ equipped with an even degenerate metric whose kernel is generated by a non-singular odd supersymmetry generator $Q$, with $[Q,Q]=2P$ yielding a supertranslation algebra. Compatible affine connections always exist and inherently possess torsion, a signature of the super-geometric degeneracy of the kernel (Bruce, 19 Aug 2025).
• Carrollian Lie Algebroids and Singular Carrollian Structures: To accommodate situations with singular Carroll vector fields (where the kernel of the degenerate metric images under the anchor map can drop rank), Carrollian Lie algebroids generalize the concept to a Lie algebroid $A$ with degenerate metric $g$ and a subbundle $L$ such that $\mathrm{ker}(g) = \mathrm{Sec}(L)$. The image of $L$ under the anchor, the Carroll distribution, can be a singular Stefan–Sussmann distribution fluctuating in rank, enabling the study of mixed null–spacelike hypersurfaces and structures arising from reductions on principal bundles (Bruce, 4 Oct 2025). Compatible affine connections exist even in singular cases.
• Noncommutative Carrollian Geometry: Carrollian structures are extended to almost commutative (ρ–commutative) geometry via ρ–Lie–Rinehart pairs. A degenerate metric with cyclic kernel can be implemented in graded-algebraic settings such as the quantum plane or noncommutative torus, with the Carroll structure encoded via cyclic submodules. Compatible “Carrollian” connections and Killing sections are defined, providing a framework relevant for quantum Carrollian models (Bruce, 22 Oct 2025).

7. Impact and Applications

Carrollian manifolds provide the geometric arena for a variety of phenomena:

• Null boundaries in general relativity: The intrinsic geometry of event horizons, null infinity, and generic null hypersurfaces is Carrollian (Blitz et al., 29 Sep 2024, Ciambelli et al., 24 Oct 2025), with direct implications for gravitational dynamics, black hole uniqueness, and the structure of gravitational memory/soft charges.
• Holography and Flat Space Limits: The boundary theories in flat holography, Carrollian correlators (amplitudes), and the emergence of BMS symmetries are naturally described in Carrollian geometric language (Saha, 2023, Stieberger et al., 21 Feb 2024, Alday et al., 27 Jun 2024).
• Ultra-relativistic field theories and fluids: Carrollian field theories generalize electrodynamics, hydrodynamics, and effective models for tensionless strings and high energy/saturation limits (Bagchi et al., 2019, Athanasiou et al., 8 Jul 2024).
• Cartan geometry and symmetry reduction: Intrinsic and induced Cartan connections unify the treatment of (conformal) Carrollian geometry and enable rigorous derivation of Gauss–Codazzi–Mainardi equations, as well as the construction of Brown–York-type stress tensors for null boundaries (Ciambelli et al., 24 Oct 2025).

These developments situate Carrollian manifolds at the crossroads of differential geometry, mathematical physics, and holographic dualities, with ongoing research encompassing projective compactifications, singular distribution theory, noncommutative geometry, and supersymmetric models. The synthesis of degenerate geometry, connection theory, symmetry algebras, and applications in null boundary physics underscores the foundational role of Carrollian manifolds in understanding ultra-relativistic limits and null spacetime structures.