Cartan Connection: Geometry & Gauge Theories

• Cartan Connection is a geometric framework that extends principal connections by integrating local symmetry, absolute parallelism, and soldering to model manifold structures.
• It decomposes into an Ehresmann connection and a soldering form, capturing both curvature and torsion to facilitate studies in parabolic, conformal, and CR geometries.
• Applications span from classifying complex geometries to formulating modern gauge theories of gravity, exemplified by teleparallel gravity and Cartan approaches to gauge-diffeomorphism interplay.

A Cartan connection is an enriched geometric structure that generalizes the classical concept of a principal connection by encoding infinitesimal symmetry breaking, absolute parallelism, and local homogeneous modeling. Cartan connections formalize the notion of "rolling" a geometric model space (often a homogeneous space $G/H$) along a manifold $M$ in an infinitesimal, soldered fashion, and serve as the structural backbone of parabolic, conformal, CR, and equivalence theory geometries, as well as modern gauge theories of gravity.

1. Formal Definition and Structure

Given a Lie group $G$ and a closed subgroup $H\subset G$, a Cartan geometry of type $(G,H)$ on a manifold $M$ is defined by:

• A principal $H$-bundle $P\to M$.
• A $\mathfrak{g}$-valued 1-form $\omega\in\Omega^1(P;\mathfrak{g})$ (the Cartan connection) with the following properties:
  1. Absolute parallelism: For each $M$0, $M$1 is a linear isomorphism.
  2. Reproduction of generators: For each $M$2, $M$3, where $M$4 is the vertical vector field associated to $M$5.
  3. $M$6-equivariance: For all $M$7, $M$8.

This formalism enables a pointwise identification of tangent spaces of $M$9 with the Lie algebra $G$0, with the $G$1-bundle structure capturing local homogeneous symmetry and the Cartan 1-form encoding both the principal bundle and "soldering" information (Catren, 2014, Stoica, 2011, Marle, 2014).

2. Soldering Form, Splitting, and Curvature

If $G$2 admits an $G$3-invariant reductive decomposition $G$4 (as in the canonical Klein geometry case), the Cartan connection splits: $G$5

• $G$6 is an Ehresmann connection (horizontal $G$7-connection form).

• $G$8 (the "soldering form") is an $G$9-valued 1-form, horizontal and $H\subset G$0-equivariant, establishing the local identification $H\subset G$1.

The Cartan curvature 2-form is given by

$H\subset G$2

splitting as

$H\subset G$3

where $H\subset G$4 is the $H\subset G$5-curvature, and $H\subset G$6 is the torsion (covariant derivative of the soldering form) (Catren, 2014, Stoica, 2011, Marle, 2014).

3. Cartan Connections in Parabolic and Equivalence Geometries

Cartan connections are canonical in the context of parabolic geometries (modeled on $H\subset G$7 where $H\subset G$8 is parabolic in a semisimple $H\subset G$9), and play a central role in CR, projective, and conformal geometry. Here, certain regularity and normality conditions (such as the vanishing of specific torsion or non-harmonic curvature components via the Kostant codifferential) single out canonical Cartan connections. For instance, regular, normal parabolic geometries of type $(G,H)$0 are equivalent to $(G,H)$1-distributions on 5-manifolds, with the curvature invariant precisely capturing Cartan's "quartic," and the classification of highly symmetric models reduces to explicit algebraic data on the Cartan connection and its curvature (The, 2022).

The structure equations for a Cartan connection incorporate these invariants and enable direct computation of the key geometric quantities: $(G,H)$2 Such approaches underlie the modern understanding of the equivalence problem and the local geometry of distributions, as demonstrated in the classification of multiply transitive $(G,H)$3-distributions (The, 2022).

4. Cartan Connections in Physics and Gauge Theories of Gravity

In contemporary gauge theory frameworks for gravity, the Cartan connection unifies local Lorentz and translational symmetries. For the Poincaré group, the Cartan connection on the orthonormal frame bundle $(G,H)$4 is

$(G,H)$5

with spin connection $(G,H)$6 and soldering form $(G,H)$7 (the coframe or tetrad). The Cartan curvature splits into the $(G,H)$8-curvature and torsion: $(G,H)$9 In teleparallel gravity (TEGR), one sets $M$0 and interprets $M$1 as the physical field strength, yielding the teleparallel lagrangian equivalently to the Einstein-Hilbert action (Huguet et al., 2021, Huguet et al., 2020, Fontanini et al., 2018, Vey, 8 Jan 2026).

This extends to fully covariant multisymplectic (10-plectic) Hamiltonian treatments: Cartan connections arise naturally as the geometrical data on covariant phase space, with equivariance and gauge covariance not as imposed constraints but as necessary consequences of the Hamiltonian formalism (Vey, 8 Jan 2026).

5. Cartan Connections on Lie Algebroids and Groupoids

The notion of a Cartan connection generalizes beyond principal bundles to Lie algebroids and groupoid theory. Here, a linear connection $M$2 on a transitive Lie algebroid $M$3 is Cartan if it is compatible with both the anchor and Lie bracket, in the sense that $M$4 is a Lie algebroid morphism. On groupoids, a Cartan connection is realized as a multiplicatively closed, horizontal $M$5-plane field $M$6 whose infinitesimalization $M$7 provides Cartan parallel translation in $M$8 and whose curvature encodes the (non-)integrability of the induced pseudogroup action (Blaom, 2016, Blaom, 2013, Attard et al., 2019).

6. Canonical Examples and Analytic Applications

Cartan connections admit explicit constructions in a wide variety of settings:

• CR geometry: Canonical Cartan connections encode all biholomorphic invariants of a CR manifold and characterize the flat model (uniqueness up to automorphisms of the base model cubic) (Pocchiola, 2014, Merker et al., 2014).
• Singular and degenerate metrics: For radical-stationary singular metrics, a generalized Cartan connection and curvature can be constructed, extending all classical structural equations and connection forms canonically even in the absence of invertibility for $M$9 (Stoica, 2011).
• Sub-Riemannian and stochastic geometry: Cartan connections enable construction of canonical stochastic developments on sub-Riemannian manifolds and connect the structure of sub-Laplacians (e.g. the Popp Laplacian) to representation-theoretic and curvature data (Beschastnyi et al., 2020).
• Partial differential equations: Cartan connections associated to evolution equations (e.g. Schrödinger) can be constructed on suitable jet spaces, encoding divergence-type (conservation) structures in geometric language (Kycia, 2020).

7. Cartan Connections, Gauge-Diffeomorphism Interplay, and Symmetry Breaking

A Cartan connection unifies internal gauge symmetry with external diffeomorphism invariance: in the Lie algebroid and Atiyah sequence formalism, infinitesimal generators of gauge transformations and diffeomorphisms are encoded simultaneously as derivations, with the curvature of the Cartan connection measuring not only field strength but failure to be a Lie-algebroid morphism. This structure resolves conceptual puzzles in gravitational gauge theories regarding the lift of spacetime vector fields into the bundle and illustrates how the “broken” symmetry of translations is geometrized as soldered diffeomorphisms, with local Lorentz symmetry maintained (Attard et al., 2019, Catren, 2014).

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References

• (Stoica, 2011): "Cartan's Structural Equations for Singular Manifolds"
• (Marle, 2014): "The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles"
• (Catren, 2014): "Geometrical Foundations of Cartan Gauge Gravity"
• (Blaom, 2016): "Cartan Connections on Lie Groupoids and their Integrability"
• (Blaom, 2013): "The Infinitesimalization and Reconstruction of Locally Homogeneous Manifolds"
• (Attard et al., 2019): "Cartan Connections and Atiyah Lie Algebroids"
• (The, 2022): "A Cartan-theoretic classification of multiply-transitive $H$0-distributions"
• (Huguet et al., 2021): "Cartan approach to Teleparallel Equivalent to General Relativity: a review"
• (Huguet et al., 2020): "Teleparallel gravity as a gauge theory: coupling to matter with Cartan connection"
• (Fontanini et al., 2018): "Teleparallel gravity (TEGR) as a gauge theory: Translation or Cartan connection?"
• (Vey, 8 Jan 2026): "10-plectic formulation of gravity and Cartan connections"
• (Pocchiola, 2014, Merker et al., 2014): Cartan connections in CR equivalence theory
• (Beschastnyi et al., 2020): "Cartan connections for stochastic developments on sub-Riemannian manifolds"
• (Kycia, 2020): "Cartan Connection for Schrödinger equation. The nature of vacuum"