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Connection form

In differential geometry, a connection form is a Lie algebra-valued 1-form \omega on a principal G-bundle P over a manifold M, where \mathfrak{g} is the Lie algebra of the Lie group G, designed to define a connection by specifying horizontal subspaces as the kernel of \omega. It satisfies two key axioms: \omega(\hat{X}) = X for fundamental vector fields \hat{X} generated by X \in \mathfrak{g}, and right-invariance under the G-action via the adjoint representation, \omega \circ R_g = \mathrm{Ad}_{g^{-1}} \circ \omega for g \in G. This structure allows for the unique horizontal lifting of curves from the base manifold and facilitates parallel transport along paths.^[1] Connection forms generalize the notion of a covariant derivative, extending it from vector bundles to principal bundles and associated fiber bundles, where they induce compatible connections. For instance, on the frame bundle of a Riemannian manifold, the Levi-Civita connection corresponds to a unique torsion-free connection form that preserves the metric. Key properties include the curvature form \Omega = d\omega + \frac{1}{2}[\omega, \omega], a \mathfrak{g}-valued 2-form measuring the failure of flatness, and the torsion, which vanishes for metric-compatible connections in certain settings. These forms are essential for defining geodesics via the horizontal lift and the exponential map, enabling the study of manifold geometry through parallel transport isomorphisms between fibers.^[1]^[2] Applications of connection forms span gauge theory in physics, where they model electromagnetic and other fundamental forces via principal bundles like the frame bundle SO(n), and algebraic geometry, where they relate to holomorphic structures on complex manifolds. In flat space \mathbb{R}^n, the trivial connection form \omega = 0 yields standard differentiation, while non-flat examples, such as left-invariant connections on Lie groups, illustrate curvature's role in non-Euclidean geometries.^[1]^[2]

Vector bundles

Frames on a vector bundle

A frame on a vector bundle E \to M of rank r over a smooth manifold M is defined as an ordered collection of r smooth sections (s_1, \dots, s_r) of E over an open subset U \subset M such that, for every point p \in U, the vectors s_1(p), \dots, s_r(p) form a basis of the fiber E_p.^[3]^[4] Such frames provide a local basis for the bundle, allowing the geometry of E to be described in coordinates over U. A global frame exists if and only if E is trivializable as a bundle.^[4] The frame bundle F_E \to M associated to E is the principal bundle whose fiber over each point x \in M consists of all ordered bases (frames) of the fiber E_x, which can be identified with the general linear group \mathrm{GL}(r, \mathbb{R}) for real vector bundles.^[4] The group \mathrm{GL}(r, \mathbb{R}) acts on the right on F_E by matrix multiplication: if e = (e_1, \dots, e_r) is a frame at x and A \in \mathrm{GL}(r, \mathbb{R}), then e \cdot A = (e_1, \dots, e_r) A, where the product denotes the linear combination \sum_j (e_j A^j_k) for the k-th component.^[4] Local sections of F_E over U \subset M correspond precisely to frames on E|_U.^[3] Frames induce local trivializations of the vector bundle. Given a frame (s_1, \dots, s_r) over U, there is a bundle isomorphism

\Phi: E|_U \to U \times [\mathbb{R}^r](/page/R&R)

defined by \Phi(p, \xi) = \sum_{i=1}^r \xi^i s_i(p) for p \in U and

\xi = (\xi^1, \dots, \xi^r) \in [\mathbb{R}^r](/page/R&R)

, which is a diffeomorphism respecting the vector space structure on each fiber.^[3]^[4] On overlaps U_\alpha \cap U_\beta between two such trivializations \Phi_\alpha and \Phi_\beta induced by frames over U_\alpha and U_\beta, the transition functions are smooth maps g_{\alpha\beta}: U_\alpha \cap U_\beta \to \mathrm{GL}(r, \mathbb{R}) satisfying \Phi_\beta \circ \Phi_\alpha^{-1}(p, v) = (p, g_{\alpha\beta}(p) v) for p \in U_\alpha \cap U_\beta and v \in \mathbb{R}^r.^[3]^[4] These functions encode how bases change between overlapping charts and satisfy the cocycle condition g_{\alpha\beta} g_{\beta\gamma} = g_{\alpha\gamma} on triple overlaps.^[4] Using a local frame (s_1, \dots, s_r) over U, any smooth section \sigma \in \Gamma(E|_U) admits a coordinate expression \sigma(p) = \sum_{i=1}^r \sigma^i(p) s_i(p) for p \in U, where the component functions \sigma^i: U \to \mathbb{R} are smooth.^[3] This representation facilitates the coordinate description of bundle morphisms and other structures on E.^[4]

Exterior connections

An exterior connection on a vector bundle E \to M over a smooth manifold M is defined as a family of maps \nabla: \Omega^k(M, E) \to \Omega^{k+1}(M, E) for each k \geq 0, where \Omega^k(M, E) denotes the space of smooth E-valued k-forms on M. For k=0, this reduces to a covariant derivative \nabla: \Gamma(E) \to \Omega^1(M, E) on sections of E, satisfying the Leibniz rule \nabla(f \sigma) = df \otimes \sigma + f \nabla \sigma for any smooth function f \in C^\infty(M) and section \sigma \in \Gamma(E).^[5]^[6] The operator \nabla is \mathbb{R}-linear in the sections, meaning \nabla(a \sigma + b \tau) = a \nabla \sigma + b \nabla \tau for scalars a, b \in \mathbb{R} and sections \sigma, \tau \in \Gamma(E), and extends naturally to higher-degree forms while preserving the algebraic structure of the exterior algebra. Specifically, it is compatible with the wedge product, satisfying \nabla(\alpha \wedge \sigma) = d\alpha \wedge \sigma + (-1)^{\deg \alpha} \alpha \wedge \nabla \sigma for a scalar k-form \alpha \in \Omega^k(M) and E-valued form \sigma \in \Omega^l(M, E). The explicit formula for the extension to k-forms is given by

(\nabla \omega)(X_0, \dots, X_k) = \sum_{i=0}^k (-1)^i (\nabla_{X_i} \omega)(X_0, \dots, \hat{X}_i, \dots, X_k) + \sum_{0 \leq i < j \leq k} (-1)^{i+j} \omega([X_i, X_j], X_0, \dots, \hat{X}_i, \dots, \hat{X}_j, \dots, X_k),

where \omega \in \Omega^k(M, E) and X_0, \dots, X_k are vector fields on M. This extension to higher forms is uniquely determined by the connection on 0-forms (sections), ensuring a consistent geometric structure across degrees.^[5] Geometrically, an exterior connection provides an intrinsic notion of differentiation that aligns with parallel transport along curves in M. For a smooth curve \gamma: [0,1] \to M, parallel transport is defined by solving the ordinary differential equation induced by \nabla, yielding a linear isomorphism P_t^\gamma: E_{\gamma(0)} \to E_{\gamma(t)} that maps a vector v \in E_{\gamma(0)} to a curve \xi(t) \in E_{\gamma(t)} satisfying \nabla_{\dot{\gamma}(t)} \xi(t) = 0. This process relies on the horizontal lift: the connection identifies a horizontal subbundle H \subset [TE](/page/TE) complementary to the vertical subbundle V = \ker(d\pi), allowing unique lifts of tangent vectors in TM to horizontal vectors in TE such that \pi_*(H_p) = T_{\pi(p)} M for each p \in E. Parallel sections along \gamma are precisely those constant under this horizontal transport.^[7] In local frames for E, the exterior connection admits an expression in terms of matrix-valued forms, as detailed in subsequent sections.^[8]

Connection forms

In differential geometry, a connection form on a vector bundle E \to M provides a local coordinate expression for an exterior connection, facilitating computations of parallel transport and covariant differentiation. Over a local trivialization U \subset M, where E|_U \cong U \times \mathbb{R}^n, the connection form \omega is defined as a smooth \mathfrak{gl}(E)-valued 1-form on U, with \mathfrak{gl}(E) denoting the Lie algebra of endomorphisms of the fibers, isomorphic to \mathfrak{gl}(n, \mathbb{R}).^[9]^[8] Given a local frame \{e_a\}_{a=1}^n for E|_U and a smooth section \sigma = \sigma^a e_a with coordinate functions \sigma^a: U \to \mathbb{R}, the covariant derivative \nabla_X \sigma induced by the exterior connection \nabla takes the explicit form

(\nabla_X \sigma)^a = X(\sigma^a) + \omega(X)_b^a \sigma^b

in these coordinates, where X is a tangent vector on U and \omega(X)_b^a are the matrix components of \omega(X) \in \mathfrak{gl}(n, \mathbb{R}).^[9] This formula expresses the connection as a correction term to the directional derivative, capturing how the frame varies along X.^[10] The connection form \omega also determines the horizontal distribution in the pullback of the frame bundle over U. Specifically, pulling back the frame bundle P_U \to U (associated to the local trivialization), \omega identifies the horizontal subspace at each point as the kernel of \omega, which is complementary to the vertical subspace and consists of lifts of tangent vectors from U that preserve parallelism in the bundle fibers.^[8] This horizontal structure enables the definition of parallel transport along curves in U. There is a one-to-one correspondence between exterior connections on E and families of local connection forms \{\omega_U\} over a trivializing cover of M, where the forms agree on overlaps up to the transformation law under change of frame.^[10] Conversely, any such compatible local connection forms define a global exterior connection on E.^[9] Under a change of local frame, the connection form transforms via the adjoint action combined with the derivative of the transition matrix, ensuring consistency across trivializations.

Change of frame

In the setting of a vector bundle equipped with a reduced structure group G \subset \mathrm{GL}(r, \mathbb{R}), the frame bundle is reduced to a principal G-bundle, and local changes of frame are described by smooth maps g: U \to G, where U is an open subset of the base manifold and G acts on the right. The connection form \omega, which is a \mathfrak{g}-valued 1-form on the frame bundle with \mathfrak{g} the Lie algebra of G, undergoes a specific transformation under such a frame change to ensure consistency across overlapping trivializations.^[11] The explicit transformation law is given by

\omega' = g^{-1} \omega g + g^{-1} \mathrm{d}g,

where the first term g^{-1} \omega g arises from the adjoint action of [G](/page/G) on \mathfrak{g}, and the second term g^{-1} \mathrm{d}g is the pullback of the Maurer-Cartan form on G. The Maurer-Cartan form \theta = g^{-1} \mathrm{d}g encodes the infinitesimal left-invariant structure of the Lie group [G](/page/G) and ensures that the horizontal distribution defined by \omega is preserved under the frame adjustment. This affine combination maintains the \mathfrak{g}-valued nature of the connection form, as both \omega and \theta lie in \Omega^1(U; \mathfrak{g}).^[11]^[12] This transformation preserves the compatibility of the connection with the reduced structure group G, meaning that if \omega satisfies the equivariance condition R_h^* \omega = \mathrm{Ad}(h^{-1}) \omega for all h \in G, then so does \omega'. In contrast to the unrestricted case with the full general linear group, where transition functions can map to all of \mathrm{GL}(r, \mathbb{R}) and the connection form takes values in \mathfrak{gl}(r, \mathbb{R}), the G-reduced setting confines g to G and \omega to \mathfrak{g}, preventing any extension of the structure beyond the subgroup and thereby respecting the geometric constraints imposed by G.^[11]

Global connection forms

A global trivialization of the frame bundle over the manifold M implies the existence of a global frame e for the vector bundle, in which the connection form \omega can be expressed without local restrictions imposed by the topology of the bundle.^[13] This global frame allows the connection to be described uniformly across M, facilitating the analysis of parallel transport and covariant derivatives on the entire space.^[13] In the special case where the connection form vanishes in this global frame, i.e., \omega = 0, the connection is trivial, meaning every section of the bundle is parallel.^[13] Such a trivial connection is necessarily flat, as its curvature form, which involves the exterior covariant derivative of \omega, is zero.^[13] Equivalently, the bundle admits global parallel sections spanning the fibers everywhere, corresponding to a representation of the fundamental group of M that is trivial.^[13] The existence of a global frame in which \omega = 0 is obstructed by the non-vanishing of characteristic classes of the bundle, such as the Chern classes for complex vector bundles, which must vanish for the bundle to admit a flat connection with such a simplification.^[14] These classes detect topological twists that prevent the frame bundle from being trivialized globally while supporting a flat structure.^[14] For flat bundles more generally, the relation to developing maps provides a way to embed the universal cover \tilde{M} into the model space, such as the vector space of the fiber, via a map that is equivariant under the holonomy action and locally an isomorphism, reflecting the local flatness without requiring global triviality.^[15]

Curvature

The curvature form of a connection on a vector bundle E \to M is a \mathfrak{gl}(E)-valued 2-form that measures the failure of the connection to be flat. In a local frame over an open set U \subset M, if \omega denotes the connection form, a \mathfrak{gl}(k,\mathbb{R})-valued 1-form (for \operatorname{rank} E = k), the curvature form \Omega is defined by

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

where the wedge product incorporates the Lie bracket in the matrix Lie algebra: \omega \wedge \omega (X,Y) = [\omega(X), \omega(Y)].^[3] This expression arises from the structure equation of the connection, capturing the quadratic nonlinearity inherent to parallel transport around infinitesimal loops.^[16] Intrinsically, without reference to a local frame, the curvature acts on sections \sigma \in \Gamma(E) and vector fields X, Y on M via the covariant derivative \nabla associated to the connection:

\Omega(X,Y) \sigma = \nabla_X (\nabla_Y \sigma) - \nabla_Y (\nabla_X \sigma) - \nabla_{[X,Y]} \sigma.

This formula reveals \Omega as the commutator of covariant derivatives, adjusted for the Lie bracket of the base fields, and it takes values in \operatorname{End}(E).^[17] The vanishing of \Omega implies that \nabla commutes on sections, allowing local trivializations where parallel transport is path-independent.^[16] Under a change of local frame given by an invertible matrix-valued function g: U \to \mathrm{GL}(k,\mathbb{R}), the curvature transforms tensorially as \Omega' = g^{-1} \Omega g, in contrast to the inhomogeneous transformation of the connection form itself.^[16] This adjoint action ensures \Omega is well-defined globally as a section of the bundle \Lambda^2 T^*M \otimes \operatorname{End}(E), independent of frame choices. The curvature form interprets as the infinitesimal generator of holonomy, quantifying how parallel transport around an infinitesimal parallelogram deviates from the identity; non-zero \Omega obstructs the integrability of the horizontal distribution defined by the connection, leading to non-trivial monodromy along closed paths.^[3] In this sense, \Omega governs geodesic deviation in the bundle, where nearby geodesics (or parallel sections) separate according to the action of \Omega on tangent vectors.^[17]

Soldering and torsion

When specializing a linear connection to the tangent bundle TM of a smooth manifold [M](/page/M), the frame bundle F_M \to [M](/page/M) admits a canonical soldering form \theta: T F_M \to \pi^* T [M](/page/M), which is an equivariant 1-form that vanishes on vertical tangent vectors and identifies horizontal directions with tangent vectors on the base manifold [M](/page/M), thereby "soldering" the fibers of the frame bundle to the tangent spaces.^[18] The torsion of such a connection \nabla on TM is defined as the \mathbb{R}^n-valued (or TM-valued) 2-form T, given by

T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]

for vector fields X, Y on [M](/page/M). This measures the antisymmetric failure of the connection to preserve the Lie bracket, distinguishing it from the symmetric part captured in the curvature.^[19] For metric-compatible connections on a Riemannian manifold (M, g), the torsion relates to the contorsion tensor K, which decomposes the connection as \nabla = \hat{\nabla} + K, where \hat{\nabla} is the Levi-Civita connection. The contorsion components are expressed in terms of the torsion as

K^\lambda_{\ \mu\nu} = \frac{1}{2} \left( T^\lambda_{\ \mu\nu} + T_{\mu\ \lambda}^{\ \ \nu} + T_{\nu\ \lambda}^{\ \ \mu} \right),

ensuring metric preservation while incorporating the torsional effects; note the sign convention may vary, but this form maintains g-compatibility.^[20] Geometrically, nonzero torsion signifies that autoparallel curves—those satisfying \nabla_{\dot{\gamma}} \dot{\gamma} = 0—do not generally coincide with geodesics, which extremize arc length; the torsion quantifies this deviation, reflecting how the connection alters the natural parallelism of paths on M.^[21]

Bianchi identities

The Bianchi identities are fundamental differential relations satisfied by the torsion and curvature forms of a connection on a vector bundle, providing constraints that ensure consistency in the geometry and lead to important invariants in differential geometry.^[22] For a soldered connection on the tangent bundle, equipped with a soldering form \theta, a canonical 1-form on the frame bundle with values in \pi^* TM, the first Bianchi identity relates the covariant exterior derivative of the torsion 2-form T to the curvature 2-form \Omega via \mathrm{Alt}(\nabla T) = \Omega \wedge \theta, where \mathrm{Alt} denotes the alternation operator.^[22] In components, this expresses the cyclic symmetry \sum_{\mathrm{cyc}} R(X,Y)Z + \nabla_X T(Y,Z) - \nabla_Y T(Z,X) + \nabla_Z T(X,Y) = 0 for vector fields X,Y,Z, but vanishes in the torsion-free case where T=0, yielding the algebraic first Bianchi identity \sum_{\mathrm{cyc}} R(X,Y)Z = 0.^[23] A sketch of the proof follows from Cartan's first structure equation d\theta + \omega \wedge \theta = T, where \omega is the connection 1-form. Applying the exterior derivative gives dT = d^2 \theta + d(\omega \wedge \theta) = 0 + d\omega \wedge \theta - \omega \wedge d\theta, and substituting d\theta = - \omega \wedge \theta + T along with the second structure equation d\omega + \omega \wedge \omega = \Omega yields dT + \omega \wedge T = \Omega \wedge \theta, which is the coordinate-free form of \nabla T = \Omega \wedge \theta (up to alternation for the precise identity).^[22] The second Bianchi identity states that the covariant exterior derivative of the curvature vanishes, \nabla \Omega = 0, or locally d\Omega + \omega \wedge \Omega - \Omega \wedge \omega = 0.^[22] This encodes the cyclic relation \sum_{\mathrm{cyc}} (\nabla_X \Omega)(Y,Z) = 0 on the curvature tensor.^[23] The proof derives from the second structure equation d\omega + \omega \wedge \omega = \Omega; exterior differentiation produces d\Omega = -d(\omega \wedge \omega) = - (d\omega \wedge \omega - \omega \wedge d\omega), and substituting d\omega = -\omega \wedge \omega + \Omega twice leads to d\Omega + \omega \wedge \Omega - \Omega \wedge \omega = 0 after collecting terms.^[22] These identities play a key role in physics, particularly in general relativity, where the contracted second Bianchi identity implies the covariant conservation \nabla_\mu G^\mu{}_\nu = 0 of the Einstein tensor G, ensuring consistency with the stress-energy tensor conservation \nabla_\mu T^\mu{}_\nu = 0 via Einstein's field equations.^[24]

Example: the Levi-Civita connection

The Levi-Civita connection is the unique connection on the tangent bundle TM of a Riemannian manifold (M, g) that is both compatible with the metric g and torsion-free.^[25] It provides a canonical way to differentiate vector fields intrinsically on M, extending the notion of directional derivatives while preserving the geometry defined by g.^[26] The connection \nabla satisfies the metric compatibility condition: for all vector fields X, Y, Z on M,

g(\nabla_X Y, Z) + g(Y, \nabla_X Z) = X \cdot g(Y, Z),

where X \cdot g(Y, Z) denotes the directional derivative of the function g(Y, Z) along X.^[25] This ensures that parallel transport along curves preserves lengths and angles defined by g. In the case where M is isometrically embedded in Euclidean space, \nabla_X Y can be realized as the orthogonal projection onto TM of the ambient directional derivative of Y along X.^[25] The uniqueness of the Levi-Civita connection follows from the fundamental theorem of Riemannian geometry, which guarantees the existence of a unique torsion-free, metric-compatible connection.^[26] This is established via the Koszul formula, which explicitly determines \nabla by

$2 g(\nabla_X Y, Z) = X g(Y, Z) + Y g(Z, X) - Z g(X, Y) - g(Y, [X, Z]) - g(Z, [Y, X]) + g(X, [Z, Y]),

for all vector fields X, Y, Z, where [ \cdot, \cdot ] is the Lie bracket.^[27] The formula derives from combining the metric compatibility and torsion-free conditions, yielding a symmetric bilinear expression that defines \nabla pointwise.^[27] By construction, the Levi-Civita connection is torsion-free, meaning the torsion tensor vanishes: T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y] = 0 for all vector fields X, Y.^[26] This symmetry reflects the absence of "twisting" in the connection, aligning it with the coordinate-free nature of the manifold's geometry. The curvature of the Levi-Civita connection is captured by the Riemannian curvature tensor R, defined by

R(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z

for all vector fields X, Y, Z.^[26] This tensor measures the failure of second covariant derivatives to commute and satisfies key symmetries, including antisymmetry R(X, Y) = -R(Y, X), the first Bianchi identity \sum_{\text{cyc}} R(X, Y) Z = 0, and metric compatibility g(R(X, Y) Z, W) = -g(Z, R(X, Y) W).^[26] These properties encode the intrinsic curvature of the Riemannian manifold.^[26]

Structure groups

Compatible connections

In differential geometry, a vector bundle E \to M of rank r over a smooth manifold M admits a reduction of its structure group from \mathrm{GL}(r, \mathbb{R}) to a closed Lie subgroup G \subset \mathrm{GL}(r, \mathbb{R}) if there exists a principal G-subbundle Q \subset F(E), where F(E) is the frame bundle of E, such that the fibers of Q consist of G-frames (bases transforming under the action of G).^[7] Such a reduction equips the bundle with additional structure preserved by the G-action, such as a Riemannian metric when G = \mathrm{O}(r).^[7] A connection \nabla on E is said to be compatible with the G-structure (or G-compatible) if its parallel transport maps along any curve \gamma: [0,t] \to M preserve the structure, meaning that for any vector v \in E_{\gamma(0)}, the transported vector P_t^\gamma(v) \in E_{\gamma(t)} satisfies P_t^\gamma(v) = g(t) \cdot v for some smooth curve g: [0,t] \to G with g(0) = \mathrm{Id}.^[7] Equivalently, the horizontal lifts induced by \nabla map G-frames to G-frames.^[7] Locally, in a G-compatible trivialization \Phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times \mathbb{R}^r, the connection form \omega \in \Omega^1(U_\alpha, \mathfrak{gl}(r, \mathbb{R})) takes values in the Lie algebra \mathfrak{g} \subset \mathfrak{gl}(r, \mathbb{R}) of G, i.e., \omega(X) \in \mathfrak{g} for all X \in \mathfrak{X}(U_\alpha).^[7] This condition ensures that the covariant derivative respects the G-structure in local coordinates.^[7] Every vector bundle with a G-structure admits at least one G-compatible connection, constructed via an Ehresmann connection on the reduced frame bundle Q.^[7] Prominent examples include metric connections, where G = \mathrm{O}(r) and the bundle is equipped with a positive-definite inner product \langle \cdot, \cdot \rangle on fibers; compatibility requires that parallel transport is an isometry, so \langle P_t^\gamma(u), P_t^\gamma(v) \rangle = \langle u, v \rangle for u, v \in E_{\gamma(0)}.^[7] Another example is a complex linear connection on a real vector bundle of even rank $2r with a reduction to G = \mathrm{GL}(r, \mathbb{C}) \subset \mathrm{GL}(2r, \mathbb{R}), induced by a complex structure J on fibers satisfying J^2 = -\mathrm{Id}; here, compatibility means the connection is \mathbb{C}-linear in its second argument, preserving the complex structure under parallel transport.^[28]

Change of frame

In the setting of a vector bundle equipped with a reduced structure group G \subset \mathrm{GL}(r, \mathbb{R}), the frame bundle is reduced to a principal [G](/page/G)-bundle, and local changes of frame are described by smooth maps g: U \to [G](/page/G), where U is an open subset of the base manifold and [G](/page/G) acts on the right. The connection form \omega, which is a \mathfrak{g}-valued 1-form on the frame bundle with \mathfrak{g} the Lie algebra of [G](/page/G), undergoes a specific transformation under such a frame change to ensure consistency across overlapping trivializations.^[11] The explicit transformation law is given by

\omega' = g^{-1} \omega g + g^{-1} \mathrm{d}g,

where the first term g^{-1} \omega g arises from the adjoint action of G on \mathfrak{g}, and the second term g^{-1} \mathrm{d}g is the pullback of the Maurer-Cartan form on G. The Maurer-Cartan form \theta = g^{-1} \mathrm{d}g encodes the infinitesimal left-invariant structure of the Lie group G and ensures that the horizontal distribution defined by \omega is preserved under the frame adjustment. This affine combination maintains the \mathfrak{g}-valued nature of the connection form, as both \omega and \theta lie in \Omega^1(U; \mathfrak{g}).^[11]^[12] This transformation preserves the compatibility of the connection with the reduced structure group G, meaning that if \omega satisfies the equivariance condition R_h^* \omega = \mathrm{Ad}(h^{-1}) \omega for all h \in G, then so does \omega'. In contrast to the unrestricted case with the full general linear group, where transition functions can map to all of \mathrm{GL}(r, \mathbb{R}) and the connection form takes values in \mathfrak{gl}(r, \mathbb{R}), the G-reduced setting confines g to G and \omega to \mathfrak{g}, preventing any extension of the structure beyond the subgroup and thereby respecting the geometric constraints imposed by G.^[11]

Principal bundles

The principal connection for a connection form

Given a smooth vector bundle E \to M of rank r, the frame bundle P \to M is the principal \mathrm{GL}(r, \mathbb{R})-bundle whose fiber over each point x \in M consists of all ordered bases (frames) of the fiber E_x, with the right action of \mathrm{GL}(r, \mathbb{R}) defined by matrix multiplication on the frames: if p \in P_x is a frame represented as column vectors and g \in \mathrm{GL}(r, \mathbb{R}), then p \cdot g is the frame whose columns are p g.^[29] This action is free and transitive on fibers, endowing P with the structure of a principal bundle.^[7] A principal connection on P is specified by a Lie algebra-valued 1-form A \in \Omega^1(P, \mathfrak{gl}(r, \mathbb{R})), called the connection form, which reproduces the generators of the right action (i.e., A(\xi^\#_p) = \xi for \xi \in \mathfrak{gl}(r, \mathbb{R}) and fundamental vector field \xi^\#_p = \frac{d}{dt}\big|_{t=0} p \cdot \exp(t\xi)) and satisfies the equivariance condition

R_g^* A = \mathrm{Ad}_{g^{-1}} \circ A

for all g \in \mathrm{GL}(r, \mathbb{R}), where R_g: P \to P is the right multiplication map and \mathrm{Ad}_{g^{-1}}(\eta) = g^{-1} \eta g is the adjoint action.^[7] This form defines a horizontal subbundle H_p P = \ker A_p \subset T_p P at each p \in P, which is invariant under the right action and complementary to the vertical subbundle \mathrm{Ver}_p P = \{ \xi^\#_p \mid \xi \in \mathfrak{gl}(r, \mathbb{R}) \}.^[29] To construct such a principal connection from a given linear connection \nabla on E, proceed locally using sections of P. Let s: U \to P be a local section over an open set U \subset M (corresponding to a local frame for E|_U); the pullback s^* A then equals the local connection form \omega^s \in \Omega^1(U, \mathfrak{gl}(r, \mathbb{R})) associated to \nabla in this frame, defined such that for vector fields X, Y on U and sections \sigma of E|_U with components \sigma = \sum \sigma^i s_i (where s_i are the frame vectors), the covariant derivative satisfies

\nabla_X \sigma = \sum (\ X \sigma^i + \omega^s_j^i(X) \sigma^j\ ) s_i

.^[3] Under a change of local section s' = s \cdot h with transition function h: U \to \mathrm{[GL](/page/GL)}(r, \mathbb{R}), the corresponding local forms transform as \omega^{s'} = h^{-1} \omega^s h + h^{-1} dh, ensuring that A is well-defined globally on P by gluing these pullbacks, as this matches the equivariance condition for A.^[7] The horizontal subbundle is then \ker A, consisting of tangent vectors whose projections to E are covariantly constant along curves in M. Local connection forms on vector bundles, as referenced briefly here, arise precisely from such pullbacks.^[3] This construction establishes a bijective correspondence between linear connections on the vector bundle E and principal connections on its frame bundle P: every \nabla on E yields a unique A on P via the above procedure, and conversely, any principal connection A on P induces a linear connection on E by declaring a section \sigma of E to be parallel along a curve if its frame representation (via a lift to P) lies in the horizontal subbundle \ker A.^[29] This equivalence preserves parallel transport, with the horizontal lifts in P corresponding to parallel sections in E.^[7]

Connection forms associated to a principal connection

Given a principal G-bundle P \to M equipped with a connection form A, which is a \mathfrak{g}-valued 1-form on P satisfying the standard equivariance and normalization properties, one can induce a connection on associated vector bundles via the group action. For a representation \rho: G \to \mathrm{GL}(V) of the structure group G on a vector space V, the associated vector bundle is constructed as E = P \times_G V, where the equivalence relation identifies (p, v) \sim (p g, \rho(g)^{-1} v) for g \in G. Sections of E correspond to G-equivariant maps from P to V. The induced connection \nabla on E is defined on sections \sigma \in \Gamma(E) using the horizontal lift provided by A: for a vector field X on M, the covariant derivative \nabla_X \sigma at a point m \in M is obtained by lifting X horizontally to a curve in P and differentiating the equivariant map representing the section along this lift, incorporating the infinitesimal action of the connection via the Lie algebra representation \rho_*: \mathfrak{g} \to \mathfrak{gl}(V). Locally, over an open set U \subset M with a section s: U \to P of the principal bundle, the induced connection form on the associated bundle takes the explicit form \omega = s^* A, pulled back to U and valued in \mathfrak{gl}(V) via \rho_*. This \omega acts as the matrix of 1-forms defining parallel transport in the trivialization E|_U \cong U \times V, recovering the standard linear connection form on the vector bundle. For a local section \sigma with components \xi: U \to V, the covariant derivative is then \nabla \xi = d\xi + \omega \cdot \xi, where the dot denotes the matrix action. The curvature of the induced connection on E is likewise obtained by pulling back the principal curvature form. The curvature 2-form on the principal bundle is \Omega_P = dA + \frac{1}{2} [A, A], a \mathfrak{g}-valued horizontal 2-form measuring the integrability failure of the horizontal distribution. The induced curvature form on E is then \Omega_E = s^* \Omega_P, again composed with \rho_* to yield an \mathrm{End}(V)-valued 2-form on U, which governs the curvature operator R(X, Y) \sigma = \rho_*(\Omega_E(X, Y)) \cdot \sigma for vector fields X, Y on M. This establishes the full correspondence between principal connections and their induced linear connections on associated vector bundles.

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