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Frame bundle

In differential geometry, the frame bundle of a smooth vector bundle \pi: E \to M of rank n over a smooth manifold M is the principal \mathrm{GL}(n, \mathbb{R})-bundle F(E) \to M whose fiber over each point x \in M is the set of all ordered bases (frames) of the fiber E_x.^[1] The total space F(E) is constructed as the disjoint union \bigcup_{x \in M} F(E_x), where F(E_x) \cong \mathrm{GL}(n, \mathbb{R}), and the group \mathrm{GL}(n, \mathbb{R}) acts freely and transitively on the right on each fiber by change of basis.^[1] Local trivializations of E induce those of F(E) via the standard trivialization of \mathrm{GL}(n, \mathbb{R}), ensuring F(E) inherits a smooth structure.^[2] For the tangent bundle TM of an n-dimensional manifold M, the associated frame bundle FM \to M (also called the linear frame bundle or tangent frame bundle) has fibers consisting of ordered bases of tangent vectors at each point, providing a canonical example central to the study of Riemannian and pseudo-Riemannian geometry.^[1] The frame bundle encodes the linear structure of E globally, allowing the reconstruction of E as an associated vector bundle F(E) \times_{\mathrm{GL}(n, \mathbb{R})} \mathbb{R}^n.^[2] Reductions of the structure group to subgroups like \mathrm{O}(n) yield orthonormal frame bundles, which are principal bundles supporting metric-compatible connections and G-structures on M.^[1] Frame bundles are fundamental for defining connections: a principal connection on F(E) (specified by a \mathfrak{gl}(n, \mathbb{R})-valued 1-form) induces a covariant derivative on sections of E, enabling parallel transport along curves in M and the study of curvature as an obstruction to flatness.^[2] In particular, the linear frame bundle FM carries a canonical soldering form, a tautological \mathbb{R}^n-valued 1-form (taking values in the pullback of TM) that, together with the differential of the projection, identifies the tangent spaces of FM with the pullback of TM \oplus TM, facilitating the integration of local coordinate frames into global geometric objects.^[3] In higher-dimensional settings, frame bundles support tensorial structures like almost product tensors, decomposing the tangent bundle into horizontal and vertical components for adapted frames.^[3]

Definition and Construction

Principal bundle formulation

The frame bundle P(M) of an n-dimensional smooth manifold M is defined as the principal \mathrm{GL}(n, \mathbb{R})-bundle over M, where the fiber over each point x \in M consists of all ordered bases (frames) of the tangent space T_x M.^[4] Each such frame is an ordered n-tuple of linearly independent vectors in T_x M, forming the set \mathrm{GL}(T_x M, \mathbb{R}^n), which is acted upon freely and transitively by the general linear group \mathrm{GL}(n, \mathbb{R}).^[4] The structure group \mathrm{GL}(n, \mathbb{R}) acts on the right on P(M) by composition of linear maps: for a frame u = (v_1, \dots, v_n) at x and A \in \mathrm{GL}(n, \mathbb{R}), the action is u \cdot A = (v_1, \dots, v_n) A, preserving the base point x and ensuring the projection \pi: P(M) \to M satisfies \pi(u \cdot A) = \pi(u) = x.^[4] This right action is smooth, free, and proper, establishing P(M) as a principal bundle in the category of smooth manifolds.^[4] Local trivializations of P(M) arise from an atlas \{ (U_\alpha, \psi_\alpha) \} on M, where \psi_\alpha: U_\alpha \to \mathbb{R}^n provides a local coordinate basis for T_x M with x \in U_\alpha; the trivialization map \phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times \mathrm{GL}(n, \mathbb{R}) sends a frame u to (\pi(u), [\psi_\alpha]_*(u)), with [\psi_\alpha]_* the pushforward expressing u in coordinates, and is \mathrm{GL}(n, \mathbb{R})-equivariant.^[4] On overlaps U_\alpha \cap U_\beta, the transition functions are g_{\alpha\beta}: U_\alpha \cap U_\beta \to \mathrm{GL}(n, \mathbb{R}), given by the change-of-basis matrices g_{\alpha\beta}(x) = (\psi_\alpha \circ \psi_\beta^{-1})'(\psi_\beta(x)), which are smooth and invertible, satisfying the cocycle condition g_{\alpha\beta} g_{\beta\gamma} = g_{\alpha\gamma}.^[4] This formulation addresses the challenge of defining frames globally on M, where local coordinate charts provide bases but fail to extend consistently across the entire manifold due to topological obstructions.^[4] As a consequence, the tangent bundle TM emerges as the vector bundle associated to P(M) via the standard representation of \mathrm{GL}(n, \mathbb{R}) on \mathbb{R}^n.^[4]

Local coordinate construction

The frame bundle of an n-dimensional smooth manifold M can be explicitly constructed using a coordinate atlas \{(U_\alpha, \phi_\alpha)\} that covers M, where each \phi_\alpha: U_\alpha \to \mathbb{R}^n is a diffeomorphism onto an open subset \Omega_\alpha \subset \mathbb{R}^n. For a chart (U_\alpha, \phi_\alpha), the local trivialization \Phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times \mathrm{GL}(n, \mathbb{R}) identifies the fiber over x \in U_\alpha with \mathrm{GL}(n, \mathbb{R}) by expressing frames relative to the coordinate frame \{\partial / \partial x^i \mid i=1,\dots,n\} at x, where x^i = \phi_\alpha^i(x). Specifically, an element of the fiber consists of an ordered n-tuple of linearly independent tangent vectors at x, represented in local coordinates by an invertible n \times n matrix A = (a^i_j), whose columns are the coordinate components of these vectors with respect to the basis \partial / \partial x^i; the trivialization maps this frame to (x, A).^[5]^[1] On overlaps U_\alpha \cap U_\beta \neq \emptyset, the trivializations \Phi_\alpha and \Phi_\beta are compatible via the transition map g_{\beta\alpha}: U_\alpha \cap U_\beta \to \mathrm{GL}(n, \mathbb{R}), defined by

\Phi_\beta \circ \Phi_\alpha^{-1}(x, v) = (x, g_{\beta\alpha}(x) v)

for x \in U_\alpha \cap U_\beta and v \in \mathbb{R}^n. Here, g_{\beta\alpha}(x) is the Jacobian matrix of the coordinate change, given explicitly by g_{\beta\alpha}(x) = \frac{\partial \phi_\beta}{\partial \phi_\alpha}(x) = d(\phi_\beta \circ \phi_\alpha^{-1})_{\phi_\alpha(x)}, which belongs to \mathrm{GL}(n, \mathbb{R}) and ensures the bundle structure glues smoothly across charts. This construction yields a principal \mathrm{GL}(n, \mathbb{R})-bundle, as the transition functions satisfy the required cocycle condition g_{\gamma\beta} g_{\beta\alpha} = g_{\gamma\alpha} on triple overlaps.^[5]^[1] For the Euclidean space \mathbb{R}^n with the standard coordinate chart \phi: \mathbb{R}^n \to \mathbb{R}^n given by the identity, the frame bundle is trivialized globally as \mathbb{R}^n \times \mathrm{[GL](/page/GL)}(n, \mathbb{R}), where each fiber over x \in \mathbb{R}^n consists of n-tuples of linearly independent vectors in \mathbb{R}^n, represented by invertible matrices relative to the standard basis \{\partial / \partial x^i\}. Similarly, for the circle S^1 parametrized by \theta \in (0, 2\pi) with chart U = S^1 \setminus \{\pi\} and \phi(\theta) = \theta, the frame bundle over U is trivialized to U \times \mathbb{R}^* (since \mathrm{[GL](/page/GL)}(1, \mathbb{R}) = \mathbb{R}^*), with fiber elements as scalar multiples f(\theta) \cdot \frac{\partial}{\partial \theta} for f(\theta) \neq 0; a second chart covers the remaining point, and the transition function on the overlap is the identity map on \mathbb{R}^*, confirming the global triviality. These examples illustrate fibers as collections of basis vectors adapted to local coordinates.^[1]^[5] This coordinate-based construction generalizes the atlas of the tangent bundle TM, where local trivializations map tangent vectors to \mathbb{R}^n via d\phi_\alpha, to the frame bundle by instead mapping bases of tangent spaces to \mathrm{[GL](/page/GL)}(n, \mathbb{R}) via change-of-basis matrices, thereby encoding the full linear structure at each point while preserving the smooth gluing via the same Jacobian transitions.^[5]^[1]

Associated Vector Bundles

General construction from frame bundles

The general construction of associated vector bundles from a frame bundle proceeds via a linear representation of the structure group. Let P(M) denote the frame bundle of an n-dimensional smooth manifold M, which is a principal \mathrm{GL}(n,\mathbb{R})-bundle over M. Given a representation \rho: \mathrm{GL}(n,\mathbb{R}) \to \mathrm{GL}(V) of \mathrm{GL}(n,\mathbb{R}) on a finite-dimensional real vector space V, the associated vector bundle E = P(M) \times_\rho V is defined as the quotient space of the product P(M) \times V by the equivalence relation

(p, v) \sim (p g, \rho(g^{-1}) v)

for all p \in P(M), v \in V, and g \in \mathrm{GL}(n,\mathbb{R}).^[6] The projection map \pi: E \to M sends the equivalence class [p, v] to the base point x = \pi_{P(M)}(p) \in M, yielding a vector bundle over M with typical fiber V.^[6] This construction is standard in differential geometry and applies to any principal bundle with a compatible representation.^[7] The fibers of E over each x \in M consist of equivalence classes [p, v] where p \in P_x(M) is a frame at x, forming a vector space isomorphic to V. In particular, elements of the fiber E_x can be identified fiberwise with linear combinations of the frame vectors in p = (e_1, \dots, e_n) with coefficients determined by the action of \rho, effectively transforming vectors in V relative to the local frame.^[6] For the standard (defining) representation \rho: \mathrm{GL}(n,\mathbb{R}) \to \mathrm{GL}(\mathbb{R}^n) where \rho(g) w = g w for w \in \mathbb{R}^n, the associated bundle recovers the tangent bundle: TM \cong P(M) \times_{\mathrm{GL}(n,\mathbb{R})} \mathbb{R}^n.^[6] More generally, natural representations of \mathrm{GL}(n,\mathbb{R}) on tensor spaces, such as the dual representation on (\mathbb{R}^n)^* yielding the cotangent bundle T^*M or the adjoint representation on \mathrm{End}(\mathbb{R}^n) yielding the bundle of endomorphisms \mathrm{End}(TM), produce tensor bundles associated to P(M).^[7] This construction is unique in the sense that any smooth vector bundle over M with typical fiber V and structure group a closed subgroup of \mathrm{GL}(\dim V, \mathbb{R}) arises as an associated bundle to its own frame bundle via the standard representation on V; moreover, when the representation factors through \mathrm{GL}(n,\mathbb{R}), such bundles can be realized from the frame bundle P(M) of the tangent space.^[6] This framework unifies the study of various geometric bundles on M under the principal bundle structure of P(M).^[7]

Identification with the tangent bundle

The tangent bundle TM of an n-dimensional smooth manifold M is canonically isomorphic to the associated vector bundle obtained from the frame bundle P(M), which is the principal \mathrm{GL}(n, \mathbb{R})-bundle of linear frames over M, via the defining (standard) representation of \mathrm{GL}(n, \mathbb{R}) on \mathbb{R}^n.^[8] Specifically, TM \cong P(M) \times_{\mathrm{GL}(n, \mathbb{R})} \mathbb{R}^n, where the equivalence class [p, \xi] for a frame p = (e_1, \dots, e_n) at a point x \in M and \xi = (\xi^1, \dots, \xi^n) \in \mathbb{R}^n corresponds to the tangent vector v = \sum_{i=1}^n \xi^i e_i \in T_x M.^[8] This association is well-defined because the right action of g \in \mathrm{GL}(n, \mathbb{R}) transforms [p g, g^{-1} \xi] to the same vector \sum_i (g^{-1} \xi)^i (p g)_i = \sum_i \xi^i e_i.^[8] This construction applies the general method of forming associated bundles from principal bundles, specialized here to recover the tangent bundle globally over M.^[9] The zero section of TM, which maps each x \in M to the zero vector (x, 0) \in T_x M, corresponds under this isomorphism to the equivalence class [p, 0] for any frame p at x, independent of the choice of frame and thus lifting uniquely to the frame bundle in a canonical manner.^[9] In contrast, a general tangent vector v \in T_x M requires a choice of frame p to express its coordinates \xi such that v = \sum \xi^i e_i, highlighting the frame bundle's role in coordinatizing tangent spaces locally.^[8] Tangent vectors can be intrinsically defined as derivations on the space of smooth functions C^\infty(M), acting as v(f) = \frac{d}{dt}\big|_{t=0} f(\gamma(t)) for curves \gamma with \gamma(0) = x and \gamma'(0) = v, providing a frame-independent characterization.^[9] This contrasts with the frame expansion v = \sum \xi^i e_i, which relies on a local basis from the frame bundle to decompose v into components, bridging abstract derivations to concrete linear combinations.^[8] This identification ensures a global isomorphism between TM and the associated bundle, resolving potential inconsistencies arising from local coordinate choices in frame selections by gluing trivializations consistently via the principal bundle structure.^[9]

Linear Frame Bundle

Role of smooth frame fields

A smooth frame field on an n-dimensional smooth manifold M is defined as a smooth section s: M \to P(M) of the linear frame bundle P(M), the principal \mathrm{GL}(n, \mathbb{R})-bundle over M whose fiber at each point x \in M comprises all ordered bases of the tangent space T_x M. This section assigns to every x \in M an ordered basis (e_1(x), \dots, e_n(x)) of T_x M such that each component map x \mapsto e_i(x) is smooth as a vector field on M. Equivalently, a smooth frame field consists of n smooth vector fields on M that are linearly independent at every point, thereby providing a smooth pointwise basis for the tangent bundle TM.^[10]^[5] Such frame fields always exist locally on any smooth manifold M, as the standard coordinate vector fields \left( \frac{\partial}{\partial x^1}, \dots, \frac{\partial}{\partial x^n} \right) in a local chart (U, \phi) form a smooth local frame field over U. Globally, however, the existence of a smooth frame field requires the tangent bundle TM to be trivializable, meaning M must be parallelizable. Parallelizable manifolds admit a global smooth frame field spanning TM everywhere; prominent examples include Euclidean space \mathbb{R}^n, where the constant basis (\partial_1, \dots, \partial_n) serves as a global frame, and Lie groups, which possess global frames given by left-invariant vector fields derived from a basis of the Lie algebra. In contrast, most compact manifolds lack global frame fields due to topological constraints.^[10]^[5] Smooth frame fields establish local trivializations of the frame bundle P(M) via \mathrm{GL}(n, \mathbb{R})-equivariant diffeomorphisms to the trivial bundle M \times \mathrm{GL}(n, \mathbb{R}), enabling the representation of automorphisms of the tangent bundle as linear transformations in \mathrm{GL}(n, \mathbb{R}) over coordinate patches. These equivariant maps facilitate the analysis of bundle structure through transition functions in \mathrm{GL}(n, \mathbb{R}), and the basis vector fields of the frame generate local one-parameter subgroups of diffeomorphisms—i.e., flows—whose linear approximations describe infinitesimal automorphisms near each point.^[5]^[10] Topological obstructions to global frame fields arise from non-triviality of TM, often detected by characteristic classes like the Stiefel-Whitney classes. A classic example is provided by the hairy ball theorem, which asserts that no smooth nowhere-vanishing vector field exists on the even-dimensional sphere S^{2k} for k \geq 1; consequently, since a global frame field would yield n linearly independent smooth vector fields (including at least one nowhere zero), even spheres S^{2k} are not parallelizable. More broadly, only the spheres S^1, S^3, and S^7 among the low-dimensional spheres admit global frame fields, highlighting the rarity of parallelizability in higher dimensions.^[10]^[11]

Solder form and canonical identification

The solder form on the linear frame bundle P(M) of an n-dimensional smooth manifold M is a canonical \mathbb{R}^n-valued 1-form \theta: TP(M) \to \mathbb{R}^n. For u \in P(M) corresponding to a frame p: \mathbb{R}^n \to T_{\pi(u)}M at the base point \pi(u) \in M, and for \xi \in T_u P(M), it is defined by

\theta_u(\xi) = p^{-1} \left( \mathrm{d}\pi_u(\xi) \right),

where \pi: P(M) \to M denotes the bundle projection and \mathrm{d}\pi_u: T_u P(M) \to T_{\pi(u)}M its differential.^[12]^[13] This form exhibits \mathrm{GL}(n,\mathbb{R})-equivariance under the right action R_g of the structure group, satisfying

\theta_{u g} \left( \mathrm{d}R_g (\xi) \right) = g^{-1} \theta_u(\xi)

for all g \in \mathrm{GL}(n,\mathbb{R}) and \xi \in T_u P(M).^[12] It vanishes on vertical tangent vectors—those in the kernel of \mathrm{d}\pi—rendering it horizontal with respect to the trivial vertical distribution, though it requires a choice of horizontal subbundle (via a connection) for full horizontality in the principal bundle sense.^[13] The solder form induces a canonical surjective map \mathrm{d}\pi: TP(M) \to \pi^* TM with kernel the vertical subbundle VP(M), yielding an isomorphism TP(M)/VP(M) \cong \pi^* TM between the quotient by the vertical directions and the pullback of the tangent bundle. Under this identification, a tangent vector \xi \in T_u P(M) projects to the base vector \mathrm{d}\pi_u(\xi) \in T_{\pi(u)}M, with coordinates in the frame p given by \theta_u(\xi), effectively "soldering" the fibers of P(M) to the tangent spaces of M via the inverse frame weighting.^[12]^[13] This isomorphism highlights the frame bundle's role in linearizing the tangent structure of M. In differential geometry, the solder form provides the foundational "soldering" mechanism for Cartan connections on frame bundles, where it pairs with a \mathrm{GL}(n,\mathbb{R})-valued principal connection form to yield a full Cartan geometry modeling the manifold locally on a homogeneous space.^[15] It also forms the core of Élie Cartan's method of moving frames, enabling the normalization of frames along submanifolds to compute differential invariants and infinitesimal symmetries systematically.^[16]

Orthonormal Frame Bundle

Reduction of structure group to O(n)

A Riemannian metric g on an n-dimensional smooth manifold M induces a reduction of the structure group of the linear frame bundle P(M), which is a principal \mathrm{GL}(n,\mathbb{R})-bundle, to the orthogonal group \mathrm{O}(n). This reduction yields the orthonormal frame bundle P^O(M), defined as the subbundle of P(M) whose fibers consist precisely of the orthonormal frames with respect to g. Specifically, for each point x \in M, the fiber P^O_x(M) comprises all ordered bases \{e_1, \dots, e_n\} of T_x M such that g(e_i, e_j) = \delta_{ij}, and \mathrm{O}(n) acts on these fibers by right multiplication, preserving orthonormality.^[1]^[17] The construction proceeds locally: over an open cover \{U_\alpha\} of M, the metric g admits local orthonormal coframes \{\theta^i_\alpha\} on each U_\alpha, which are \mathbb{R}^n-valued 1-forms satisfying g = \sum_{i=1}^n \theta^i \otimes \theta^i (in the standard Euclidean metric on \mathbb{R}^n). The dual frames to these coframes provide local sections of P^O(M), and the transition functions between overlapping trivializations take values in \mathrm{O}(n), ensuring compatibility with the metric. Globally, P^O(M) is a principal \mathrm{O}(n)-bundle over M, obtained as the subbundle of P(M) consisting of orthonormal frames, whose transition functions take values in \mathrm{O}(n).^[1]^[17] There exists a bijective correspondence between \mathrm{O}(n)-reductions of P(M) and Riemannian metrics on M: given such a reduction P^O(M) \hookrightarrow P(M), the metric g is uniquely determined by, given an orthonormal frame u over x \in M with associated isomorphism u: \mathbb{R}^n \to T_x M, g_x(\xi, \eta) = \langle u^{-1} \xi, u^{-1} \eta \rangle_{\mathbb{R}^n} for tangent vectors \xi, \eta \in T_x M, where \langle \cdot, \cdot \rangle_{\mathbb{R}^n} is the standard inner product on \mathbb{R}^n. Conversely, any Riemannian metric yields a unique \mathrm{O}(n)-reduction. This equivalence holds globally on M, as the metric provides compatible local orthonormal trivializations that glue smoothly. If M is orientable, the reduction can be further specialized to the special orthogonal group \mathrm{SO}(n) by selecting oriented orthonormal frames, corresponding to oriented Riemannian metrics.^[18]^[17] The fibers of P^O(M) are diffeomorphic to \mathrm{O}(n), a compact Lie group, in contrast to the non-compact fibers of P(M) diffeomorphic to \mathrm{GL}(n,\mathbb{R}); this reduction thus selects a "smaller" submanifold of frames, though both have infinite cardinality as manifolds. Such reductions are facilitated by an Ehresmann connection on P(M), which allows horizontal lifts to define the subbundle structure.^[1]^[17]

Relation to Riemannian metrics

The orthonormal frame bundle P^{O}(M) of a Riemannian manifold (M, g) is intrinsically tied to the metric structure, as the choice of g determines the reduction of the full frame bundle to the orthogonal group O(n), and conversely, the bundle defines the metric up to the choice of local trivializations. Specifically, given a local section s: U \to P^{O}(M) over an open set U \subset M, the Riemannian metric at a point p \in U is recovered by expressing tangent vectors in terms of the orthonormal frame provided by s(p). For X, Y \in T_p M, the metric is g_p(X, Y) = \langle s(p)^{-1}(X), s(p)^{-1}(Y) \rangle, where \langle \cdot, \cdot \rangle is the standard Euclidean inner product on \mathbb{R}^n and s(p): \mathbb{R}^n \to T_p M is the linear isomorphism given by the frame.^[19] This construction ensures that the frames in the bundle are orthonormal with respect to g, establishing a bijection between Riemannian metrics on M and O(n)-reductions of the frame bundle. The orthonormal frame bundle itself inherits a natural Riemannian metric from g on M and the bi-invariant metric on O(n), often realized as the Sasaki-type metric on principal bundles. This bundle metric is defined such that for tangent vectors at a frame u \in P^{O}(M), it decomposes into horizontal and vertical components: the horizontal part pulls back the metric g via the projection \pi: P^{O}(M) \to M, while the vertical part uses the Killing form or standard metric on the fibers isomorphic to O(n). The O(n)-invariance of this metric ensures compatibility with the right action, making P^{O}(M) a Riemannian submersion over M. Consequently, the Levi-Civita connection of g on M induces a principal connection on P^{O}(M), whose horizontal distribution coincides with the horizontal subbundle defined by the metric orthogonality, thus embedding the geometry of M into that of the frame bundle.^[20]^[21] In Élie Cartan's moving frame approach, the geometry of the Riemannian manifold is captured through differential forms on the orthonormal frame bundle. The Levi-Civita connection corresponds to an \mathfrak{so}(n)-valued connection 1-form \omega on P^{O}(M), which is equivariant under the O(n)-action and reproduces the Christoffel symbols in local coordinates. The curvature of this connection is encoded in the curvature 2-form \Omega = d\omega + \omega \wedge \omega, also \mathfrak{so}(n)-valued, satisfying Cartan's second structure equation \Omega = d\theta + \omega \wedge \theta alongside the torsion-free solder form \theta. This formalism reveals the sectional curvatures of g as the components of \Omega evaluated on adapted frames, providing a coordinate-free description of Riemannian curvature directly on the bundle. A key application of the orthonormal frame bundle lies in parallel transport, which is geometrized as horizontal lifts in the principal bundle setting. For a curve \gamma: [0,1] \to M, parallel transport of a tangent vector from \gamma(0) to \gamma(1) with respect to the Levi-Civita connection is equivalent to lifting \gamma horizontally to a curve in P^{O}(M) starting from a frame containing the initial vector, followed by the O(n)-action to adjust the frame. These O(n)-lifts preserve orthonormality and encode the holonomy of the connection, with closed curves generating the structure group elements that measure infinitesimal rotations induced by curvature.^[22]

G-Structures

General theory of structure group reductions

A G-structure on an n-dimensional manifold M is defined as a principal G-subbundle P^G(M) of the linear frame bundle P(M), where G \subset \mathrm{GL}(n, \mathbb{R}) is a Lie subgroup, equivalently representing a reduction of the structure group of P(M) from \mathrm{GL}(n, \mathbb{R}) to G.^[23]^[24] This reduction endows the tangent bundle TM with additional geometric structure compatible with the action of G, generalizing the principal bundle formulation to closed subgroups.^[24] The construction of a G-structure proceeds either by selecting a smooth section of the quotient bundle P(M)/G \to M, which identifies fibers over each point with G-orbits in the space of frames, or equivalently via a G-valued atlas on M consisting of local G-frames whose transition functions take values in G.^[24] Local sections s: U \to P(M) over open sets U \subset M are required to be G-equivariant, ensuring that the resulting subbundle P^G(M) is smooth and principal with structure group G.^[24] This approach aligns with the general theory of principal bundles, where the reduction is locally trivialized by such atlases.^[23] For an involutive G-structure, Frobenius integrability characterizes the existence of an integrable distribution defined by the G-invariant subbundle: the associated horizontal distribution H \subset TP^G(M) is integrable if and only if it is involutive, meaning [H, H] \subset H.^[24] This condition ensures the local existence of integral submanifolds foliating a neighborhood of each point, corresponding to a Pfaffian system generated by G-invariant forms that is completely integrable.^[23] In the context of G-structures, involutivity often relates to the vanishing of the torsion of a compatible connection, allowing the structure to be locally modeled on the flat G-space.^[25] Global obstructions to the existence or further reduction of a G-structure are detected by elements in Lie algebra cohomology groups or characteristic classes associated to the bundle.^[25] Specifically, the Lie algebra cohomology H^*(\mathfrak{g}, V) of the Lie algebra \mathfrak{g} of G with coefficients in a representation V encodes infinitesimal deformations and integrability obstructions, while characteristic classes in the cohomology H^*(M; \mathbb{R}), such as those induced by invariant polynomials on \mathfrak{g}, provide topological barriers to reducibility; for instance, the Euler class serves as an obstruction for certain orientation-preserving reductions.^[25]^[24] These invariants, independent of choices of connection, arise from the curvature and soldering forms on P^G(M).^[25]

Examples and classifications

Prominent examples of G-structures arise in the study of almost complex, symplectic, and conformal geometries on smooth manifolds. An almost complex structure on a manifold of even dimension $2m corresponds to a reduction of the frame bundle's structure group to G = \mathrm{GL}(m, \mathbb{C}) \subset \mathrm{GL}(2m, \mathbb{R}). This reduction is equivalent to the existence of a tensor field J, an endomorphism of the tangent bundle satisfying J^2 = -\mathrm{Id}, which endows the manifold with a complex structure locally. A symplectic structure on a $2m-dimensional manifold is given by a reduction to G = \mathrm{Sp}(2m, \mathbb{R}) \subset \mathrm{GL}(2m, \mathbb{R}), the group preserving a constant non-degenerate skew-symmetric bilinear form. Geometrically, this manifests as a closed, non-degenerate 2-form \omega on the tangent bundle, enabling the formulation of Hamiltonian dynamics and phase spaces. Unlike the almost complex case, symplectic structures are typically integrable by definition through the closedness of \omega.^[26] Conformal structures reduce the structure group to G = \mathrm{CO}(n) = \mathbb{R}^+ \times \mathrm{O}(n) \subset \mathrm{GL}(n, \mathbb{R}), preserving angles but not lengths. This corresponds to a conformal class of pseudo-Riemannian metrics, where metrics differ by positive scalar functions, and is central to Weyl geometry and scale-invariant theories. In general, each such G-structure is associated with a canonical tensor field on the manifold—for instance, the almost complex tensor J, the symplectic form \omega, or the conformal metric class—whose properties encode the geometric constraints imposed by the group reduction. Integrability conditions, such as the vanishing of the Nijenhuis tensor N_J = 0 for almost complex structures, ensure that the G-structure arises from a more rigid geometric object; for example, integrability of an almost complex structure yields a complex manifold, as established by the Newlander-Nirenberg theorem. Classifications of G-structures often rely on prolongation techniques, as developed in the equivalence methods of Cartan and Weyl. Weyl's classification identifies G-structures of finite type based on the order of their prolongations, where the prolongation process iteratively extends the structure to higher-order jet bundles until invariants determine local equivalence; notably, conformal and projective structures are second-order, admitting unique prolongations to affine structures. Holonomy groups provide another classification framework, acting as the maximal subgroups to which the structure group can be reduced while preserving a compatible connection, thereby capturing the intrinsic symmetries and parallel transport properties of the manifold.^[26] The formalization of G-structures traces to Élie Cartan's development of the moving frames method in the 1920s and 1930s, which unified local solvability of overdetermined partial differential equations with group actions on manifolds, laying the groundwork for modern infinitesimal geometry.^[27]

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... Lee. Introduction to. Smooth Manifolds. Second Edition. Page 6. John M. Lee ... Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218,. DOI ...
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[PDF] Math 396. The C hairy ball theorem 1. Introduction Consider the unit ...
The answer is negative, and is called the hairy ball theorem (since it “explains” why one cannot continuously comb the hair on a ball without a bald spot):.Missing: global frame<|control11|><|separator|>
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vielbein in nLab
May 22, 2024 · A vielbein or solder form on a manifold X X is a linear ... This frame bundle carries a universal “basic” ℝ n \mathbb{R}^n ...Definition · In terms of Cartan geometry · In terms of orthogonal structure
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The tangent bundle and solder form | Mathematics for Physics
The tangent frame bundle is special in that we can relate its tangent vectors to the elements of the bundle as bases. Specifically, we define the solder form ( ...
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### Definition and Properties of the Solder Form on the Frame Bundle
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Cartan connection in nLab
### Summary: Solder Form in Cartan Connections on Frame Bundles
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[2307.16085] Introducing the Classical Method of Moving Frames
Jul 29, 2023 · The method of moving frames (repère mobile) was used by Elie Cartan as a way of organizing the identification of differential invariants and ...
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[PDF] 16 Fibre bundles and structure groups
The set OV of orientations is a Z2-torsor. Let E be a real vector bundle. Recall that a local trivialization Φα : Uα → Uα × Rn determines a local frame,.
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[PDF] Kobayashi-Nomizu.pdf
GL(n; R)) is a principal fibre bundle. We call L(M) the bundle of linear frames over M. In view of Proposition 5.4, a linear frame u at xe M can be defined ...
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[1311.6172] On the frame bundle adapted to a submanifold - arXiv
Nov 24, 2013 · Riemannian metric g induces natural metric on O(N). We study the geometry of a submanifold O(M,N) in O(N).
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On the geometry of orthonormal frame bundles II
Oct 10, 2007 · We study the geometry of orthonormal frame bundles OM over Riemannian manifolds (M, g). The former are equipped with some modifications ...
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Wagner Lift of Riemannian metric to Orthogonal Frame Bundle - arXiv
Jan 3, 2009 · We apply the Wagner construction to the specific case when the distribution is the infinitesimal connection in the orthonormal frame bundle ...
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Parallel transport on the frame bundle | Mathematics for Physics
On any smooth bundle with connection, we consider the horizontal tangent space to define the direction of parallel transport.
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THE GEOMETRY OF G-STRUCTURES1 1. Introduction. Differential ...
Among the important struc- tures on a manifold is the reduction of the structural group of the tangent bundle, which we explain as follows: Let T be an ^- ...
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[PDF] The theory of connections and G-structures. Applications to affine ...
Jun 1, 2015 · This book contains the notes of a short course given by the two authors at the. 14th School of Differential Geometry, ...
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[PDF] characteristic classes for g-structures - Fakultät für Mathematik
Apr 10, 2008 · Obstructions to 1-integrability of G-structures ... The clas- sification of characterictic classes for G − structures with a given Lie group G.Missing: reduction | Show results with:reduction
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G-structure in nLab
Jun 25, 2024 · Equivalently, this means that a G G -structure is a choice of reduction of the canonical structure group GL(n) of the principal bundle to which ...
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Moving Frames: Essential Journey of Élie Cartan - Bhāvanā
Cartan's contributions to differential geometry via moving frames can be regarded as a direct outgrowth of his theory of continuous groups which was developed ...