Intrinsic Carrollian Connection

Updated 28 October 2025

Intrinsic Carrollian Connection is a geometric structure on degenerate manifolds, defined by a degenerate metric and a preferred time direction.
It bypasses the uniqueness of the Levi-Civita connection by using minimal non-metricity and criteria tied to expansion and shear while maintaining torsion-free conditions.
This framework underpins flat holography, null hypersurface dynamics, and the emergence of Carrollian symmetry algebras in non-Lorentzian gravity.

An intrinsic Carrollian connection is a geometric structure defined on degenerate metric manifolds—Carrollian manifolds—where the kernel of the metric singles out a preferred direction interpreted as “absolute time.” Unlike pseudo-Riemannian geometry, where the Levi-Civita theorem guarantees a unique metric-compatible, torsion-free connection, the degenerate nature of the Carrollian metric precludes this uniqueness. Instead, the intrinsic Carrollian connection is constructed through compatibility requirements that respect the degenerate geometry and its causal, symplectic, and symmetry structure. This concept is foundational in flat holography, null hypersurface theory, Cartan and bundle approaches to Carrollian geometry, non-Lorentzian gravity, and emergent symmetry algebras on null boundaries.

1. Algebraic and Geometric Foundations

A Carrollian manifold is a smooth manifold $N$ equipped with a symmetric, positive semidefinite tensor $q$ of corank-1 and a global nowhere-vanishing vector field $\ell$ spanning the kernel of $q$ , i.e., $q(\ell, -) = 0$ . The pair $(q, \ell)$ constitutes the intrinsic Carrollian structure. In the modern formalism, particularly following the review (Bruce, 22 Oct 2025), one also introduces structural data such as a one-form $k$ (an Ehresmann connection) satisfying $k(\ell) = 1$ .

The degeneracy of $q$ means it cannot be inverted; there is no natural way to project tensor indices “upwards,” and the usual Levi-Civita connection construction fails.
The bundle structure $N \rightarrow \mathcal{S}$ (with $\mathcal{S}$ a spatial base and the fibre the “absolute time” direction) is canonically equipped with an Ehresmann connection, splitting the tangent bundle $TN = V \oplus H$ into vertical (along $\ell$ ) and horizontal distributions.

On a Carrollian spacetime or null hypersurface, the degenerate metric and vertical direction define ruled (“foliated by lines”) geometry that underlies all intrinsic connection constructions (Ciambelli et al., 2019, Herfray, 2021).

2. The Breakdown of the Levi-Civita Theorem

In contrast to the unique Levi-Civita connection of Riemannian geometry, on a Carrollian manifold there is:

No well-posed criterion that simultaneously enforces both vanishing torsion and metric compatibility because $q$ is not invertible.
The best one can do is to require “minimal non-metricity,” typically identified with dynamical data such as expansion or shear—quantities that measure the rate at which the spatial metric changes along the degenerate direction (Bruce, 22 Oct 2025).

Explicitly, the intrinsic (preferred) Carrollian connection $D$ is defined such that:

It is torsion-free: $D_{[a} e_{b]} - [e_a, e_b] = 0$ , for a local frame $\{e_a\}$ .
It preserves the degenerate structure in the weakest sense:

$D_a q_{bc} = - k_b \theta_{ac} - k_c \theta_{ab},$

where $\theta_{ab} = \frac{1}{2} \mathcal{L}_\ell q_{ab}$ is the expansion tensor and $k$ is a one-form satisfying $k(\ell) = 1$ .

The evolution of $\ell$ is captured as:

$D_a \ell^b = \theta_a{}^b + \omega_a \ell^b,$

with $\omega_a$ an arbitrary one-form encoding connection freedom.

This defines the most general (minimal) Carrollian connection determined only by the intrinsic geometry (Bruce, 22 Oct 2025). The explicit connection symbols are:

$\Gamma^a_{bc} = \underbrace{[(\underline{b} + \omega_b) k_c + k_b (\pi_c + \phi_c) - \mathscr{A}_{bc}]\,\ell^a}_{\text{vertical part}} + \frac{1}{2} q^{ad} \left(\underline{b} q_{cd} + \underline{c} q_{bd} - \underline{d} q_{bc}\right).$

Here, $\underline{a}$ denotes directional derivative along the horizontal frame, and $\mathscr{A}_{bc}$ , $\pi_c$ , $\phi_c$ parameterize the remaining freedom.

3. Bundle, Cartan, and Algebroid Perspectives

Principal bundle and Cartan-geometric frameworks provide alternative universal constructions:

The tangent bundle is split using an Ehresmann connection, equivalently a one-form $b = b_i dx^i$ so that horizontal vector fields are $E_i = \partial_i + b_i \partial_t$ .
The degenerate Carrollian metric is recovered as $g = a_{ij}(t, x) dx^i dx^j$ with $g(\partial_t, -) = 0$ .
On a principal $\mathbb{R}^\times$ -bundle, a canonical affine connection is constructed by “completing” the degenerate metric with the connection one-form:

$g_\omega = g + \omega^2,$

and then taking its Levi-Civita connection, which is torsionless but only compatible with $g_\omega$ , not $g$ (Bruce, 27 May 2025).

In Cartan geometry, an intrinsic Carrollian connection corresponds to a unique torsion-free “carr”-valued Cartan connection on an appropriate principal bundle, generalizing the geometric data via soldering forms and curvature.
For (conformal) Carrollian geometries, the Cartan connection encodes both the degenerate metric and the “time” direction, unifying extrinsic (null infinity, Bondi gauge) and strictly intrinsic characterizations.
The curvature of the Cartan or tractor connection directly measures the presence of gravitational radiation at null infinity; its flatness characterizes radiative vacua (Herfray, 2021).

When the Carrollian direction becomes singular (e.g., at intersections of null and spacelike surfaces, or under degenerations), the structure lifts to a Carrollian Lie algebroid, where the kernel of the degenerate metric pushes forward to a possibly singular distribution through the anchor.
Compatible (metric and kernel-preserving) connections can always be constructed on the algebroid, allowing for the definition of intrinsic connections even in spaces with singular Carroll vector fields (Bruce, 4 Oct 2025).

4. Differential Structure, Compatible Tensors, and Curvature

An intrinsic Carrollian connection enables the definition and differentiation of Carrollian tensors—tensor fields that are “horizontal,” i.e., vanish if contracted with the degenerate direction.

Examples include the “Carrollian torsion” (vorticity), shear, expansion, and acceleration one-forms.
The connection coefficients enter the definition of derivatives acting on these tensors and underlie dynamic equations for Carrollian fluids and field theories (Ciambelli et al., 2018).
Curvature tensors compatible with the Carrollian structure are constructed using the intrinsic connection, encoding geometric and physical information, such as those entering the flat-space Gauss and Codazzi–Mainardi equations induced by the geometry of embedded null hypersurfaces (Bruce, 22 Oct 2025).

5. Role in Holography, Null Surfaces, and Dynamical Systems

a. Flat Holography and Fluid-Gravity Correspondence

In the zero-cosmological constant limit of AdS/CFT, the asymptotic boundary geometry degenerates to Carrollian form, and the intrinsic Carrollian connection determines the hydrodynamics on null infinity:
- Carrollian hydrodynamic equations—fluid energy, current, and viscous stress tensor evolution—are governed via derivatives defined with the intrinsic connection (Ciambelli et al., 2018).
- Reconstruction of Ricci-flat bulks from boundary data (e.g., via the derivative expansion) hinges on the Carrollian connection through duality relations between friction tensors and descendant Cotton tensors.

b. Null Hypersurfaces and Horizons

The intrinsic Carrollian connection matches the connection induced by projection (via rigging techniques) of the ambient spacetime Levi-Civita connection onto null (or ruled, causal) hypersurfaces (Bruce, 22 Oct 2025, Freidel et al., 10 Jun 2024).
On stretched horizons, the “sCarrollian” connection, specified by non-metricity conditions involving the extrinsic curvature, governs intrinsic dynamics and encodes the Carrollian stress tensor, which appears as the Brown–York stress tensor in the null limit (Freidel et al., 10 Jun 2024).

c. Field Theory and Symmetry

Intrinsic Carrollian connections underlie the construction and conservation of Carrollian symmetry algebras (including conformal Carroll or BMS) in both classical and quantum settings (Saha, 2022).
For scalar fields and sigma models, Carrollian connections determine couplings of kinetic and potential terms, yielding novel wave equations intrinsic to null boundaries and event horizons (Ciambelli, 2023, Bruce, 1 Jul 2025).
The Laplacian, divergence, and Hodge-theoretic objects—otherwise obstructed by metric degeneracy—can be constructed in bundle-enhanced frameworks, such as Carrollian $\mathbb{R}^\times$ -bundles (Bruce, 29 Jul 2025).

6. Singular and Noncommutative Generalizations

The Lie algebroid and almost-commutative ( $\rho$ -Lie-Rinehart) generalizations lift the notion of an intrinsic Carrollian connection to settings where the underlying “Carroll vector field” may be singular or even ill-defined (e.g., vanishing at isolated points, or in noncommutative geometry) (Bruce, 4 Oct 2025, Bruce, 22 Oct 2025).
In these cases, compatible connections always exist, and the Carrollian distribution (the image of the kernel under the anchor) may be a singular Stefan–Sussmann distribution, fluctuating in rank across the manifold.

7. Schematic Comparison of Connection Properties

Property	Pseudo-Riemannian (Levi-Civita)	Intrinsic Carrollian Connection
Torsion	Zero	Zero (in the “preferred” case)
Metric Compatibility	$Dg = 0$	Minimal non-metricity tied to expansion/shear: $Dq \neq 0$ when $\mathcal{L}_\ell q \neq 0$
Kernel Preservation	Not relevant	Connection preserves the kernel of q ( $D\ell \sim \ell$ )
Uniqueness	Unique connection	Non-unique (depends on extension data and choice of k)

8. Connections to Symmetry Algebras

The intrinsic Carrollian connection is inseparably tied to the emergence of infinite-dimensional symmetry algebras on null boundaries:

Carrollian (and conformal Carrollian) symmetry algebras, such as $\mathfrak{ccarr}(d+1)$ , are realized as consistent algebraic structures only when the connection structure is intrinsic and globally defined (Ciambelli et al., 2019, Saha, 2022, Nguyen et al., 2023).
The connection governs transformation laws, the structure of conserved currents, operator product expansions in Carrollian CFTs, and is central to the appearance of BMS symmetries in asymptotically flat holography (Mason et al., 2023).

References (arXiv ids)

"Flat holography and Carrollian fluids" (Ciambelli et al., 2018)
"Carroll Structures, Null Geometry and Conformal Isometries" (Ciambelli et al., 2019)
"Carrollian manifolds and null infinity: A view from Cartan geometry" (Herfray, 2021)
"The gauging procedure and carrollian gravity" (Figueroa-O'Farrill et al., 2022)
"Intrinsic Approach to $1+1$D Carrollian Conformal Field Theory" (Saha, 2022)
"Carrollian conformal fields and flat holography" (Nguyen et al., 2023)
" $p$ -brane Galilean and Carrollian Geometries and Gravities" (Bergshoeff et al., 2023)
"Carroll geodesics" (Ciambelli et al., 2023)
"Dynamics of Carrollian Scalar Fields" (Ciambelli, 2023)
"Induced motions on Carroll geometries" (Marsot, 2023)
"Carrollian Amplitudes and Celestial Symmetries" (Mason et al., 2023)
"Geometry of Carrollian Stretched Horizons" (Freidel et al., 10 Jun 2024)
"Carroll in Shallow Water" (Bagchi et al., 6 Nov 2024)
"Differential Representation for Carrollian Correlators" (Chakrabortty et al., 14 Nov 2024)
"Carrollian $\mathbb{R}^\times$ -bundles: Connections and Beyond" (Bruce, 27 May 2025)
"Carrollian $\mathbb{R}^\times$ -bundles II: Sigma Models on Event Horizons" (Bruce, 1 Jul 2025)
"Carrollian $\mathbb{R}^\times$ -bundles III: The Hodge Star and Hodge--de Rham Laplacians" (Bruce, 29 Jul 2025)
"Carrollian Lie Algebroids: Taming Singular Carrollian Geometries" (Bruce, 4 Oct 2025)
"Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs" (Bruce, 22 Oct 2025)
"Foundations of Carrollian Geometry" (Ciambelli et al., 24 Oct 2025)

The intrinsic Carrollian connection—defined via its torsion, non-metricity, compatibility with the degenerate “time” direction, and often specified in a bundle or Cartan-geometric framework—transcends the limitations of the Levi-Civita paradigm and establishes the geometric bedrock for Carrollian physics, null boundary dynamics, and flat space holography. Its existence and properties are intimately linked to the symplectic, dynamical, and symmetry structures of null hypersurfaces and their holographic avatars, providing a canonical and unifying formalism for gravitational, field-theoretic, and algebraic structures on degenerate metric spaces.