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Cartan connection

A Cartan connection is a differential 1-form on a smooth manifold M taking values in the Lie algebra \mathfrak{g} of a Lie group G, which serves as both an Ehresmann connection on the associated principal G-bundle and a soldering form that identifies the tangent spaces T_m M with the associated homogeneous space G/H for a closed subgroup H \subset G, satisfying equivariance under the right H-action and pointwise isomorphism properties.^[1] This structure encodes infinitesimal models of homogeneous geometries on M, where the connection form \omega reproduces the Maurer-Cartan form along the fibers and allows parallel transport that "rolls" the model space G/H onto tangent planes without slipping.^[2] The concept was introduced by Élie Cartan in the early 1920s as part of his method of moving frames (repères mobiles), initially applied to affine and projective connections in the context of general relativity and conformal structures.^[3] Cartan developed these ideas to generalize Riemannian geometry and address equivalence problems for geometric structures, viewing connections as deformations of flat Lie group geometries.^[4] The modern formulation, integrating principal bundles and characteristic classes, was formalized by Charles Ehresmann in 1950, who connected Cartan's approach to fiber bundle theory.^[1] Cartan connections provide a unified framework for Klein's Erlangen program in curved spaces, encompassing Riemannian, conformal, projective, and other classical geometries through their curvature and torsion forms, which satisfy structure equations analogous to the Maurer-Cartan equations.^[2] For instance, in pseudo-Riemannian geometry, the Cartan connection models spacetime using the Poincaré group \mathrm{ISO}(p,q) with the Lorentz group as stabilizer, incorporating both metric compatibility and affine parallelism.^[1] Flat Cartan connections correspond to locally homogeneous manifolds, while non-flat ones measure deviations via the curvature 2-form \Omega = d\omega + \frac{1}{2} [\omega, \omega], enabling the study of invariants and deformations in higher-dimensional settings.^[5]

Introduction

Overview and motivation

A Cartan connection is a geometric structure on a manifold that generalizes the notion of an affine connection by combining it with a reduction of the structure group to a closed subgroup, thereby modeling the manifold as locally resembling a homogeneous space G/H, where G is a Lie group and H its subgroup. This setup allows for the transport of the model's infinitesimal symmetry structure—defined by the Lie algebra \mathfrak{g} of G and the subalgebra \mathfrak{h} of H—to the manifold via a principal H-bundle equipped with a \mathfrak{g}-valued connection form that also serves as a soldering form.^[2]^[6] The motivation for Cartan connections stems from Felix Klein's Erlangen program, which views geometry as the study of invariants under transformation groups acting on homogeneous spaces, but extends it to curved, non-homogeneous settings by incorporating differential geometry tools like connections to "roll" the model geometry along the manifold without slipping or twisting. This framework reconciles Klein's algebraic approach with Élie Cartan's infinitesimal generalizations, enabling the description of geometries that deviate from flatness through curvature and torsion while preserving local symmetry. Affine connections arise as a special case when the full general linear group acts without reduction to a symmetry-preserving subgroup.^[2]^[6] Cartan connections unify diverse geometric structures; for instance, Riemannian geometry corresponds to the model \mathbb{E}^n = O(n) \ltimes \mathbb{R}^n / O(n), conformal geometry to the Möbius sphere S^n = O(n+1,1)/O(n,1), and projective geometry to \mathbb{RP}^n = PGL(n+1)/PGL(n), each encoded by an appropriate group reduction and connection form. Key advantages include the provision of absolute parallelism, where the connection identifies tangent spaces with the model's Lie algebra, facilitating explicit computations of symmetries, and the seamless incorporation of both metric (e.g., length-preserving) and non-metric (e.g., projective) structures within a single formalism.^[2]^[6]

Historical context

The concept of Cartan connections emerged from foundational work in group theory and geometry during the late 19th century. Sophus Lie developed the theory of continuous transformation groups and infinitesimal transformations in the 1870s and 1880s, providing tools to analyze symmetries through Lie algebras. Building on this, Felix Klein introduced his Erlangen Program in 1872, classifying geometries based on invariance under transformation groups, with homogeneous spaces serving as inspirational model spaces for later generalizations.^[7]^[8]^[8] Élie Cartan advanced these ideas significantly in the 1920s and 1930s, introducing Cartan connections through his method of moving frames and the equivalence problem for geometric structures. In a series of papers from 1922 to 1925, Cartan addressed the local equivalence of manifolds under transformation groups, using moving frames to define connections that generalize affine and Riemannian structures while incorporating torsion. His approach unified differential geometry by treating spaces as modeled on homogeneous Klein geometries, with connections encoding the deviation from flat models. Cartan's 1945 book Les systèmes différentiels extérieurs et leurs applications géométriques, based on lectures from 1936–1937, served as a foundational text, systematizing exterior differential systems central to these connections.^[7]^[7]^[8]^[9] Post-Cartan developments expanded the framework in the mid-20th century. Charles Ehresmann generalized Cartan connections around 1950 by embedding them within the theory of fiber bundles, defining connections via horizontal distributions on principal bundles and viewing Cartan types as special cases where the bundle solders to the tangent space.^[10] In the 1970s and 1980s, Cartan connections influenced modern physics, particularly gauge theories of gravity and unified field theories. In the 1990s, this perspective was further developed, notably in Richard Sharpe's 1997 book Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, which reformulated them as principal connections on bundles modeled after Klein geometries, bridging differential geometry with particle physics and enabling descriptions of spacetime symmetries akin to Yang-Mills gauge fields.^[2]

Background Concepts

Affine connections

An affine connection on a smooth manifold M is a map \nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM), where \Gamma(TM) denotes the space of smooth sections of the tangent bundle TM, satisfying bilinearity in its arguments and the Leibniz product rule \nabla_{fX} Y = f (\nabla_X Y) and \nabla_X (fY) = (X f) Y + f (\nabla_X Y) for vector fields X, Y \in \Gamma(TM) and smooth functions f on M. This structure enables the definition of parallel transport of vector fields along curves in M, providing a way to compare tangent vectors at nearby points by specifying how they are "transported" without rotation relative to the connection.^[11] In local coordinates (x^i) on M, the affine connection is expressed using Christoffel symbols \Gamma^k_{ij}, which are smooth functions on M, via the formula \nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k, where \{\partial_i\} is the coordinate basis for TM. For a general vector field X = X^i \partial_i and Y = Y^j \partial_j, the covariant derivative takes the form \nabla_X Y = (X^i \partial_i Y^j + X^i Y^k \Gamma^j_{ki}) \partial_j. These symbols encode the local behavior of the connection and transform under coordinate changes in a specific non-tensorial way that ensures the overall structure remains well-defined globally.^[11] The torsion tensor T of an affine connection \nabla is the bilinear map T: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM) defined by T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y], where [X, Y] is the Lie bracket of vector fields. In coordinates, its components are T^k_{ij} = \Gamma^k_{ij} - \Gamma^k_{ji}, highlighting the antisymmetric part of the Christoffel symbols. The torsion measures the extent to which the connection fails to preserve the Lie bracket structure, and a connection is said to be torsion-free if T = 0; non-zero torsion arises in contexts where the connection does not symmetrize the differentiation of vector fields.^[11] The curvature tensor R of \nabla is defined for vector fields X, Y, Z \in \Gamma(TM) by R(X, Y) Z = \nabla_X (\nabla_Y Z) - \nabla_Y (\nabla_X Z) - \nabla_{[X, Y]} Z, a tensorial map measuring the integrability of the distribution of parallel vector fields. In local coordinates, the components R^\ell_{ijk} involve derivatives of the Christoffel symbols and their products, such as R^\ell_{ijk} = \partial_i \Gamma^\ell_{jk} - \partial_j \Gamma^\ell_{ik} + \Gamma^\ell_{im} \Gamma^m_{jk} - \Gamma^\ell_{jm} \Gamma^m_{ik}. The curvature vanishes if and only if the connection is flat, meaning parallel transport around closed loops yields the identity; otherwise, it quantifies the "holonomy" or twisting of the manifold's geometry.^[11] A prominent example is the Levi-Civita connection associated to a Riemannian metric g on M, which is the unique torsion-free affine connection satisfying metric compatibility \nabla g = 0, i.e., X g(Y, Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z) for all vector fields X, Y, Z. This connection, introduced by Tullio Levi-Civita, ensures that parallel transport preserves lengths and angles defined by g, making it fundamental for studying geodesic motion and curvature in Riemannian geometry. Its Christoffel symbols are explicitly given by \Gamma^k_{ij} = \frac{1}{2} g^{k\ell} (\partial_i g_{j\ell} + \partial_j g_{i\ell} - \partial_\ell g_{ij}).^[12]

Klein geometries

A Klein geometry is a homogeneous space X = G/H, where G is a Lie group and H is a closed Lie subgroup of G such that G acts transitively on X.^[13] This structure arises from Felix Klein's Erlangen program, which classifies geometries by their symmetry groups, with G representing the group of transformations preserving the geometric structure and H stabilizing a point in X.^[14] The manifold X serves as a flat model space embodying the infinitesimal symmetries of the geometry, where the transitive action ensures every point can be mapped to any other via elements of G. Classic examples of Klein geometries include Euclidean space, modeled as \mathbb{R}^n = E(n)/O(n), where E(n) = \mathbb{R}^n \rtimes [O(n)](/page/orthogonal_group) is the Euclidean group of isometries and O(n) is the orthogonal group stabilizing the origin.^[13] The sphere S^n is realized as SO(n+1)/SO(n), with SO(n+1) acting as rotations in \mathbb{R}^{n+1} and SO(n) fixing the north pole.^[14] Similarly, hyperbolic space H^n takes the form SO^+(n,1)/SO(n), where SO^+(n,1) is the connected Lorentz group preserving the hyperboloid metric, and SO(n) stabilizes a base point.^[13] These models capture constant-curvature geometries, with the quotient structure encoding the local symmetry at each point. Central to the Klein geometry is the Maurer-Cartan form \omega = g^{-1} dg on G, a left-invariant \mathfrak{g}-valued 1-form on the Lie group, where \mathfrak{g} is the Lie algebra of G.^[15] This form provides a linear isomorphism \omega_g: T_g G \to \mathfrak{g} at each point g \in G, satisfying the equivariance condition R_h^* \omega = \mathrm{Ad}(h^{-1}) \omega for h \in H, where R_h denotes right multiplication by h.^[13] When pulled back to the homogeneous space G/H, \omega becomes an invariant connection form that decomposes into a soldering form (identifying the tangent space with \mathfrak{g}/\mathfrak{h}) and a principal connection valued in the subalgebra \mathfrak{h} of H.^[14] The Maurer-Cartan structure equation d\omega + \frac{1}{2} [\omega, \omega] = 0 holds, reflecting the flatness of the model.^[15] In the context of Cartan connections, the Klein geometry G/H provides the infinitesimal model for symmetries on curved manifolds, where a Cartan geometry on a manifold M equips it with a principal H-bundle and a connection form modeled on \omega, allowing M to be viewed as a "gluing" of tangent copies of G/H.^[13] This framework generalizes Riemannian and other structures by measuring deviations from the flat Klein model through curvature, enabling the study of local symmetries akin to those in the homogeneous space.^[14]

Pseudogroups and moving frames

In the context of Cartan connections, pseudogroups provide a framework for understanding local symmetries on manifolds, extending the global action of Lie groups to infinite-dimensional settings. A Lie pseudogroup is a collection of local diffeomorphisms on a manifold M that satisfies group-like axioms—such as containing the identity, inverses, and being closed under local composition—and is defined as the solution set to an involutive system of partial differential equations (PDEs). This structure generalizes finite-dimensional Lie groups by allowing actions that are defined only locally and may involve infinitely many parameters, capturing infinitesimal transformations through their prolongations on jet bundles. Élie Cartan developed this theory in his early work on infinite-dimensional transformation groups, emphasizing their role in differential geometry.^[16] Cartan's moving frame method, or repère mobile, addresses the equivalence problem for geometric structures under pseudogroup actions by constructing normalized frames that adapt to the local symmetries. The moving frame theorem states that, given a free and regular action of a Lie pseudogroup on a jet bundle, one can select a cross-section to the group orbits, leading to a canonical normalization where certain coordinates are set to constants, thereby yielding a system of differential invariants and invariant differential forms. This normalization process solves the equivalence of submanifolds or structures by comparing their invariant signatures, reducing the pseudogroup to a canonical form via successive prolongations. The method, formalized by Cartan in the 1930s, transforms the infinitesimal generators of the pseudogroup into a coframe adapted to the geometry, facilitating the computation of local invariants.^[17] Central to this approach are the structure equations, which encode the intrinsic geometry of the pseudogroup action. For a Cartan connection associated with the moving frame, the coframe \omega satisfies the Maurer-Cartan structure equation

d\omega + \omega \wedge \omega = \Omega,

where \Omega represents the curvature form, measuring the deviation from flatness and capturing the torsion and curvature invariants of the connection. These equations arise from differentiating the normalized coframe and substituting the group action, providing a differential system whose integrability conditions determine the local equivalence class. In Cartan's formulation, they generalize the Maurer-Cartan equations of Lie groups to pseudogroups, enabling the systematic derivation of all invariants through the exterior differential system.^[18] A representative example arises in the classical theory of surfaces in Euclidean space, where the pseudogroup of local isometries acts on the jet space of immersed surfaces. Here, moving frames are adapted to curves on the surface, normalizing the frame so that the tangent vectors align with the curve direction and principal directions, yielding invariants such as Gaussian curvature K and mean curvature H as functions of the normalized parameters. This setup, pioneered by Cartan, reduces the structure group to the orthogonal group while preserving the local metric properties, illustrating how pseudogroups define reductions of the full frame bundle to capture surface symmetries. Local pseudogroups of this type draw inspiration from global Klein geometries, modeling homogeneous spaces as prototypes for their infinitesimal actions. The connection to Cartan geometry lies in how these pseudogroups specify the structure group reduction, ensuring the connection forms respect the local symmetry constraints.^[17]

Formal Definitions

Definition via absolute parallelism

A Cartan connection can be defined using the concept of absolute parallelism on a manifold, which provides a global framing of the tangent bundle. This approach originates from Élie Cartan's work in the 1920s, where he developed it to generalize non-Riemannian geometries beyond the constraints of metric-compatible connections, allowing for the incorporation of torsion and curvature in a unified framework.^[19] Absolute parallelism on an n-dimensional manifold M is established by a global coframe \{e^i\}_{i=1}^n, consisting of n linearly independent 1-forms that form a basis for the cotangent space at every point. This coframe satisfies the structure equation

de^i + \omega^i_j \wedge e^j = T^i,

where \omega^i_j are the components of a connection 1-form \omega with values in the Lie algebra \mathfrak{gl}(n,\mathbb{R}), representing the infinitesimal changes in the frame under parallel transport, and T^i is the torsion 2-form. The absolute parallelism ensures that the coframe defines a soldering form, establishing an isomorphism between the tangent bundle TM and the trivial bundle M \times \mathbb{R}^n, thereby providing a canonical identification of tangent vectors across the manifold without reliance on local coordinates.^[5] The Cartan connection is then specified by the pair (e, \omega), where e = (e^1, \dots, e^n) is the global coframe inducing the soldering isomorphism e: TM \to M \times \mathbb{R}^n, and \omega is the associated connection form on the frame bundle P(M, GL(n,\mathbb{R})) that preserves this structure. More generally, for a Cartan geometry modeled on a Klein geometry G/H with Lie algebra decomposition \mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}, the connection takes values in \mathfrak{g}, with the coframe e valued in \mathfrak{m}^* and satisfying de + \omega \wedge e = T (in matrix notation), where T is the \mathfrak{m}^*-valued torsion 2-form, ensuring the connection is adapted to the model's symmetry. This setup extends affine connections by enforcing the global frame's compatibility, distinguishing it through the absolute parallelism property.^[5] The torsion tensor T of the connection is intrinsically tied to the coframe via

T(e_i, e_j) = (de^k + \omega^k_l \wedge e^l)(e_i, e_j) \, e_k,

where \{e_i\} is the dual frame to \{e^i\}, and (de^k + \omega^k_l \wedge e^l)(e_i, e_j) measures the failure of the coframe to close under exterior differentiation evaluated on the frame vectors, accounting for both the connection and intrinsic torsion. This expression captures the anholonomy of the global frame, providing a geometric interpretation of torsion as the "twist" in the parallel transport defined by \omega.^[19] In the flat case, where the curvature form \Omega = d\omega + \omega \wedge \omega = 0 and torsion T = 0, the Cartan connection implies that the manifold is locally isomorphic to the model space G/H. The absolute parallelism then reduces to a Maurer-Cartan form on a principal H-bundle over M, enabling a transitive action of G that reconstructs the homogeneous model geometry pointwise. This flatness condition underscores the role of Cartan connections in modeling infinitesimal symmetries akin to Klein geometries.^[5]

As principal connections

A Cartan connection can be defined abstractly within the framework of principal bundle theory, where it generalizes the notion of a principal connection to incorporate the structure of a Klein geometry. Specifically, given a model Klein geometry (G, H) consisting of a Lie group G with Lie algebra \mathfrak{g} and a closed subgroup H \subset G with Lie algebra \mathfrak{h} \subset \mathfrak{g}, the base manifold M is equipped with a principal H-bundle P \to M. This bundle represents the frame bundle reduced to the structure group H, where G/H serves as the homogeneous model space determining the geometric type. The Cartan connection is then a \mathfrak{g}-valued 1-form \omega \in \Omega^1(P, \mathfrak{g}) on P that satisfies the defining properties of a principal connection with respect to H, augmented by a normalization condition. First, it reproduces the fundamental vector fields generated by \mathfrak{h}: for every \xi \in \mathfrak{h}, \omega(\xi_P) = \xi, where \xi_P denotes the infinitesimal generator of the H-action on P. Second, it is equivariant under the right H-action: R_h^* \omega = \mathrm{Ad}_{h^{-1}} \omega for all h \in H, ensuring consistency with the group structure. The crucial additional property is absolute parallelism: \omega_p: T_p P \to \mathfrak{g} is a linear isomorphism at each point p \in P, which implies that \omega provides a global framing of the tangent spaces to P. This isomorphism induces a soldering condition that identifies the tangent bundle TM with an associated vector bundle over M. Specifically, the kernel of \omega defines horizontal subspaces, and the projection \tau: \mathfrak{g} \to \mathfrak{g}/\mathfrak{h} (identifying with the tangent space to G/H at the base point) yields a soldering form \theta = \tau \circ \omega \in \Omega^1(P, \mathfrak{g}/\mathfrak{h}), which is H-equivariant. The associated bundle P \times_H (\mathfrak{g}/\mathfrak{h}), where H acts on \mathfrak{g}/\mathfrak{h} via the adjoint representation \mathrm{Ad}: H \to \mathrm{Aut}(\mathfrak{g}/\mathfrak{h}), is canonically isomorphic to TM via the bundle map induced by \theta. This ensures that the Cartan geometry "soldiers" the principal bundle to the base manifold, embedding the infinitesimal model geometry locally. The curvature of the Cartan connection measures the deviation from flatness, analogous to the model Klein geometry where the Maurer-Cartan form on G yields zero curvature. The curvature 2-form is given by

\Omega = d\omega + \frac{1}{2} [\omega, \omega] \in \Omega^2(P, \mathfrak{g}),

where [\cdot, \cdot] is the Lie bracket in \mathfrak{g}, extended bilinearly to forms. This \Omega is horizontal (vanishes on vertical vectors) and equivariant under H, with its vanishing implying local flatness and isomorphism to the model G \to G/H. A representative example arises in conformal geometry, where the Weyl connection can be realized as a Cartan connection modeled on the Klein geometry (\mathrm{PO}(n+1,1), \mathrm{PO}(n,1)) for n-dimensional conformal manifolds. Here, the structure group H = \mathrm{PO}(n,1) preserves a conformal class of metrics, and the \mathfrak{g}-valued form \omega encodes both the Levi-Civita connection and a Weyl gauge field, with the soldering condition ensuring compatibility with the tangent bundle via the adjoint action on \mathfrak{g}/\mathfrak{h}.

Via Ehresmann connections and gauge transitions

In the framework of Ehresmann connections, a connection on a fiber bundle \pi: E \to M is defined by a smooth horizontal subbundle H \subset TE that is complementary to the vertical subbundle VE = \ker(T\pi), so that TE = VE \oplus H. This structure allows for the horizontal lifting of curves in the base manifold M to curves in E, facilitating parallel transport along paths. For principal G-bundles P \to M, the horizontal subbundle is equivalently specified by a G-equivariant \mathfrak{g}-valued 1-form \omega: TP \to \mathfrak{g} with \ker \omega = H, satisfying the equivariance condition r_g^* \omega = \Ad_{g^{-1}} \omega and reproducing the fundamental vector fields via \omega(\xi^\#) = \xi for \xi \in \mathfrak{g}.^[20]^[21] A Cartan connection arises as a special type of Ehresmann connection on a principal G-bundle P \to M modeled on a Klein geometry G/H, where H \subset G is a closed subgroup. Here, the connection form \omega not only defines the horizontal distribution H = \ker \omega but also ensures a soldering condition: the projection \pi_*: H_p \to T_{\pi(p)} M is an isomorphism for each p \in P, identifying the tangent spaces of M with the model tangent space \mathfrak{g}/\mathfrak{h}. This reproducibility property means that infinitesimal displacements along horizontal directions in P faithfully reproduce the infinitesimal structure of the homogeneous model space G/H, distinguishing Cartan connections from general Ehresmann or principal connections.^[20]^[21] Locally, Cartan connections admit a gauge description via an open cover \{U_i\} of M and local sections \sigma_i: U_i \to Q of an H-reduced principal subbundle Q \subset P (with structure group H). The transition functions are then g_{ij}: U_i \cap U_j \to H defined by \sigma_j = \sigma_i \cdot g_{ij}, ensuring the bundle's consistency. The pulled-back connection form on U_i is the \mathfrak{g}-valued 1-form \eta_i = \sigma_i^* \omega. On overlaps U_i \cap U_j, the gauge transformation relates these forms via \eta_j = g_{ij}^{-1} \eta_i g_{ij} + g_{ij}^{-1} dg_{ij}, or equivalently \eta = g^{-1} dg + A where g = g_{ij} is the local transition and A denotes the gauge potential component. This local form captures the Cartan structure, with \eta mapping TU_i isomorphically onto \mathfrak{g}/\mathfrak{h} to enforce reproducibility.^[1]^[21] A concrete example is provided by projective Cartan connections, which model projective geometry on a manifold M^n with Klein model G = \PGL(n+1, \mathbb{R}) and H the stabilizer of a projective point, so G/H \cong \mathbb{RP}^n. Local sections \sigma_i: U_i \to Q yield transition functions g_{ij} \in H preserving the projective structure, and the pulled-back form \eta_i is a projective connection 1-form whose horizontal lifts reproduce projective transformations infinitesimally. This setup ensures that geodesics and developments on M mimic those in the flat projective model.^[20]

Properties and Special Cases

Reductive Cartan connections

A reductive Cartan connection arises in the context of a Cartan geometry modeled on a Klein geometry (G, H), where the Lie algebra \mathfrak{g} of G admits a reductive decomposition \mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m} into the subalgebra \mathfrak{h} of H and a complementary \mathfrak{h}-invariant subspace \mathfrak{m}, satisfying [\mathfrak{h}, \mathfrak{m}] \subset \mathfrak{m}. In typical reductive Cartan geometries, the model also satisfies [\mathfrak{m}, \mathfrak{m}] \subset \mathfrak{h}, ensuring the bracket terms align with the decomposition.^[22] This decomposition ensures that the adjoint action of H on \mathfrak{m} preserves the splitting, allowing the geometry to be modeled on homogeneous spaces like G/H \cong \mathbb{R}^n / \{e\} for Euclidean geometry.^[23] The connection form \eta of a reductive Cartan connection on the principal H-bundle P \to M takes values in \mathfrak{g} and splits as \eta = \omega + \theta, where \omega \in \Omega^1(P, \mathfrak{h}) is an Ehresmann connection form and \theta \in \Omega^1(P, \mathfrak{m}) is the soldering form, which identifies \mathfrak{m} with the cotangent space of the base manifold M.^[22] This form is H-equivariant, meaning R_h^* \eta = \mathrm{Ad}(h^{-1}) \eta for h \in H, and it reproduces the Maurer-Cartan form on vertical vectors while providing an isomorphism T_p P \cong \mathfrak{g} at each point p \in P.^[23] The reductive structure preserves the splitting of \mathfrak{g}, enabling a decomposition of associated geometric objects and simplifying local computations by aligning the connection with the model's symmetry. The curvature 2-form \Omega \in \Omega^2(P, \mathfrak{g}) decomposes accordingly as \Omega = \Omega^\mathfrak{h} + \Omega^\mathfrak{m}, where \Omega^\mathfrak{h} = d\omega + \frac{1}{2} [\omega, \omega] + \frac{1}{2} [\theta, \theta] is the \mathfrak{h}-component (encoding intrinsic curvature, with [\theta, \theta] contributing if [\mathfrak{m}, \mathfrak{m}] \subset \mathfrak{h}), \Omega^\mathfrak{m} = d\theta + [\omega, \theta] is the \mathfrak{m}-component (the torsion).^[23] This separation facilitates analysis of flatness conditions, where \Omega = 0 implies local isomorphism to the model geometry. Prominent examples include Riemannian geometries, modeled on the Euclidean group with \mathfrak{h} = \mathfrak{so}(n) and \mathfrak{m} = \mathbb{R}^n, where \theta serves as the coframe and \omega as the Levi-Civita connection, yielding the standard metric structure.^[23] Similarly, Lorentzian geometries arise from the Poincaré group, with \mathfrak{h} = \mathfrak{so}(1, n-1) and \mathfrak{m} = \mathbb{R}^{1,n-1}, as in the orthonormal frame bundle formulation of general relativity.^[24] These cases are prevalent in physics, where the reductive splitting unifies the spin connection and vielbein into a single form, streamlining derivations in formulations like the MacDowell-Mansouri action for gravity with a cosmological constant.^[24] The approach simplifies tensorial computations and gauge-theoretic interpretations, making it advantageous for applications in teleparallel gravity and equivalence to Einstein's equations.^[22]

Parabolic Cartan connections

Parabolic Cartan connections are a class of Cartan connections modeled on parabolic geometries, which generalize classical structures like projective and conformal geometries through the use of parabolic subgroups of semisimple Lie groups. A parabolic subgroup P \subset G is defined such that its Lie algebra \mathfrak{p} admits a graded filtration \mathfrak{p} = \mathfrak{h} \oplus \bigoplus_{k \geq 1} \mathfrak{m}_k, where \mathfrak{h} is the Levi factor (reductive subalgebra) and the \mathfrak{m}_k form positively graded nilpotent components, preserving the Lie bracket grading. This structure arises from |k|-gradings of the semisimple Lie algebra \mathfrak{g} = \bigoplus_{i=-k}^k \mathfrak{g}_i, with \mathfrak{p} = \bigoplus_{i=0}^k \mathfrak{g}_i, enabling the modeling of filtered manifolds.^[25] The Cartan connection form η for a parabolic geometry on a principal P-bundle over a manifold M is a \mathfrak{g}-valued 1-form, decomposed according to the grading as η = \bigoplus_{i=-k}^k \eta^{(i)}, where each \eta^{(i)} \in \Omega^1(P, \mathfrak{g}_i). The components for i < 0 serve as soldering forms, identifying the graded tangent bundle filtration TM = T^{-k}M \supset \cdots \supset T^{-1}M, with T^{-i}M / T^{-i+1}M isomorphic to the adjoint tractor bundle associated to \mathfrak{g}_{-i}. The components for i ≥ 0 form the connection part. This decomposition ensures that η reproduces the Maurer-Cartan form on G and preserves the grading of the tangent bundle filtration.^[25] The connection thus induces a canonical |A|-structure, where the structure group A = P / P_+ (with P_+ the unipotent radical) acts by transformations preserving the filtration, facilitating equivariant extensions and gauge reductions.^[26] These connections are particularly suited to higher-order geometries beyond the reductive case, as the grading allows for non-trivial interactions across filtration levels. In projective geometry, they model |1|-graded structures on manifolds with projective structure, enabling the definition of projective Weyl tensors.^[25] A prominent example is the conformal Cartan connection on an |A|-bundle for a conformal structure, where the curvature form decomposes into components including the Weyl curvature tensor, which measures the obstruction to local flatness and is conformally invariant. Applications extend to tractor constructions in conformal gravity, where the associated tractor bundle (built from the adjoint representation of \mathfrak{g}) supports natural differential operators like the Paneitz operator, linking geometric invariants to higher-order gravitational theories.^[25]

Curvature and torsion

In Cartan connections, the torsion form quantifies the anholonomy of the associated frame bundle, measuring deviations from integrability of the horizontal distribution. For a reductive Cartan connection, where the Lie algebra \mathfrak{g} decomposes as \mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m} with [\mathfrak{h}, \mathfrak{m}] \subseteq \mathfrak{m} and typically [\mathfrak{m}, \mathfrak{m}] \subseteq \mathfrak{h}, the torsion form \Theta is given by

\Theta = d\theta + \omega \wedge \theta,

where \theta is the soldering form valued in \mathfrak{m}, \omega is the \mathfrak{h}-valued connection form, and the wedge product incorporates the Lie bracket action. This expression arises from the structure equations of the connection, capturing the failure of coordinate frames to close under Lie differentiation. The curvature form \Omega encodes the non-flatness of the geometry, defined for the full connection form \eta = \theta + \omega as

\Omega = d\eta + \frac{1}{2} [\eta, \eta].

This 2-form decomposes into horizontal and vertical components: the \mathfrak{m}-valued part corresponds to torsion \Theta, while the \mathfrak{h}-valued part, often denoted \Omega^\mathfrak{h}, measures intrinsic bending relative to the model Klein geometry. The full \Omega thus unifies both notions, with explicit computation yielding \Omega = \Omega^\mathfrak{h} + \Theta under the reductive splitting. The Bianchi identities provide integrability conditions for these forms. The second Bianchi identity states

d\Omega + [\eta, \Omega] = 0,

ensuring consistency of the curvature under parallel transport, while the first relates torsion and curvature via

d\Theta + \omega \wedge \Theta = \Omega^\mathfrak{h} \wedge \theta.

These identities, derived from the Maurer-Cartan structure equation d\eta + \frac{1}{2} [\eta, \eta] = \Omega, enforce differential closure and are fundamental for deriving field equations in Cartan geometries. When both torsion and curvature vanish (\Theta = 0 and \Omega = 0), the Cartan connection is flat, implying that the manifold is locally isomorphic to the model space of the Klein geometry, allowing global trivialization of the frame bundle. This flatness condition aligns the geometry with the homogeneous model, facilitating explicit coordinate descriptions. In projective geometry, modeled on the Klein pair G/H = \mathrm{PGL}(n+1, \mathbb{R}) / \mathbb{R}^\times \mathrm{PGL}(n, \mathbb{R}), the torsion of the Cartan connection relates directly to the Thomas-Whitehead connection on the bundle of volume elements, where non-vanishing torsion introduces a trace-free component \rho_i = -\frac{1}{n+1} \alpha \wedge dx^i that adjusts the projective equivalence class without altering geodesics.^[27]

Associated Structures and Operators

Covariant differentiation

A Cartan connection on a principal H-bundle P \to M modeled on a Klein geometry G/H induces a covariant derivative on any associated vector bundle E = P \times_H V, where V is a representation space of H. For a vector field X on M and a section \sigma: M \to E, the induced covariant derivative is defined by lifting X horizontally to \tilde{X} \in \mathfrak{X}^{\mathrm{hor}}(P) and applying it to the H-equivariant map from P to V corresponding to \sigma, yielding \nabla^E_X \sigma = [\tilde{X}, \sigma]_E.^[5] In local coordinates, if s: U \subset M \to P is a section and \sigma_a = [s, v_a] for v_a: U \to V, the formula simplifies in the reductive case where \mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m} (with \mathrm{Ad}(H)\mathfrak{m} \subseteq \mathfrak{m}) to \nabla_X \sigma_a = X(v_a) + \rho(A_a(X)) \cdot v_a, where A_a = s^* \omega_{\mathfrak{h}} is the pullback of the \mathfrak{h}-component of the Cartan connection form \omega, and \rho: H \to \mathrm{GL}(V) is the representation. For a local frame \{e_i\} of E induced from a basis of V, this extends tensorially as \nabla_X e_i = \omega^j_i(X) e_j, where \omega^j_i are the components of the connection form, allowing differentiation of tensorial objects over E.^[5] The soldering form \theta: TP \to \mathfrak{m} of the Cartan connection ensures compatibility by identifying TM \cong P \times_H \mathfrak{m}, so that the induced connection on TM acts naturally: for X \in \Gamma(TM), \nabla_X Y = \pr_{\mathfrak{m}} \left( [\tilde{X}, \tilde{Y}] \right) for Y \in \Gamma(TM), preserving the geometric structure modeled on G/H. In the reductive case, if the inner product on \mathfrak{m} is \mathrm{Ad}(H)-invariant, the induced connection on the associated metric bundle is compatible, meaning \nabla g = 0 for the metric g on TM pulled back from \mathfrak{m}, as in Weyl connections for conformal geometry. For parabolic Cartan geometries (where H \subset G is parabolic), the normal Cartan connection induces higher-order tractor connections on associated tractor bundles \mathcal{T} = P \times_H V, where V is a filtered G-module; these connections \nabla^{\mathcal{T}} satisfy normalization conditions like \partial^* (R^{\nabla^{\mathcal{T}}}) = 0, enabling prolongation of differential operators via BGG sequences.^[28]

Fundamental derivative

The fundamental derivative, also known as the universal derivative, is a differential operator intrinsic to Cartan connections that acts on weighted densities on the base manifold. For a density f of conformal weight w along a vector field X, it takes the form D_X f = X(f) + a(X) f, where a(X) is the trace of the connection form induced by X. This operator adjusts the standard directional derivative by a term that accounts for the weight and the geometry encoded in the Cartan connection, ensuring naturality under the structure group action. In the context of conformal Cartan geometries, the fundamental derivative on densities simplifies to a covariant form: for indices i, D_i f = \nabla_i f - w P_i f, where \nabla_i is the Levi-Civita covariant derivative from a chosen metric in the conformal class, and P_i is the Schouten tensor derived from the Ricci curvature.^[29] This adjustment by the Schouten tensor preserves conformal invariance, distinguishing it from the unweighted covariant derivative on tensor fields, which corresponds to the case w = 0. The operator extends naturally to jets of functions and densities, mapping sections of the jet bundle J^k(E) to higher-order jets while preserving the symbol of the differential operator, as induced by the equivariant action on the Cartan bundle.^[30] In parabolic geometries, a graded version D^k acts on weighted jets associated to representations of the Levi factor, enabling the prolongation of overdetermined systems. This graded fundamental derivative plays a central role in the Bernstein-Gelfand-Gelfand (BGG) machinery, where it generates exact sequences of differential complexes resolving generalized Verma modules over the base manifold, facilitating the construction of invariant operators.^[31] Applications include the derivation of conformally invariant higher-order operators, such as the Paneitz operator on four-manifolds and the Graham-Jenne-Mason-Sparling (GJMS) operators in higher dimensions, which arise as compositions involving powers of the fundamental derivative and yield critical tools in conformal geometry and physics.^[32]

References

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Apr 10, 2024 · A Cartan connection is a principal connection on a smooth manifold equipped with a certain compatibility condition with the tangent bundle of the manifold.Idea · History · Definition · Examples
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In this paper Cartan gave the first formal definition of a differential form. Victor Katz writes [26]:- His definition was a "purely symbolic" one; namely ...
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[PDF] A HISTORICAL OVERVIEW OF CONNECTIONS IN GEOMETRY
Sophus Lie who laid the foundations with his work on continuous transformation groups. ... In one article Cartan writes that Lie was interested. “with a ...
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BOOK REVIEWS Les systèmes différentiels extérieurs et leurs ...
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Oct 31, 2003 · The method of moving frames (“rep`eres mobiles”) was forged by Élie Cartan,. [11, 12], into a powerful and algorithmic tool for studying the ...
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[PDF] Moving Frames and Differential Invariants of Lie Pseudo–Groups
♤ The structure equations are on the principal bundle G(∞); if G is a finite-dimensional Lie group, then G(∞) ≃ M × G, and the usual Lie group structure ...
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∗ϑG. With this motivation with us, the Cartan connections are going to be generalisation of these where in the gauge field descriptions these are local 1 ...
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[24]
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As an example of a gauge theory with Cartan connection, let us consider using a Cartan connection in the topological gauge theory known as 'BF theory' [7].
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[29]
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We shall call E[w] the bundle of conformal densities of weight w. Note that each bundle E[w] is trivial, and inherits an orientation from that on R. We.
[30]
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[31]
[PDF] The (relative) BGG machinery lecture 3
Relative BGG sequences. Continuation of lecture 2. We have used the fundamental derivative on Γ(ΛkA∗M ⊗ VM) and the Lie algebra cohomology differential for ...
[32]
[PDF] Conformally Invariant Operators via Curved Casimirs: Examples
Mar 24, 2009 · Dξℓs = ∇ξℓs − P(ξℓ) • s. If T is a tractor bundle, then this follows immediately from the formula for the fundamental derivative in section 1.7 ...