Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames.
The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object.
Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work.
Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them.
As a matrix equation, this reads In general relativity, such frame fields are referred to as tetrads.
The tetrad specifically relates the local frame to an explicit coordinate system on the base manifold
A connection in E is a type of differential operator where Γ denotes the sheaf of local sections of a vector bundle, and Ω1M is the bundle of differential 1-forms on M. For D to be a connection, it must be correctly coupled to the exterior derivative.
Specifically, if v is a local section of E, and f is a smooth function, then where df is the exterior derivative of f. Sometimes it is convenient to extend the definition of D to arbitrary E-valued forms, thus regarding it as a differential operator on the tensor product of E with the full exterior algebra of differential forms.
Given an exterior connection D satisfying this compatibility property, there exists a unique extension of D: such that where v is homogeneous of degree deg v. In other words, D is a derivation on the sheaf of graded modules Γ(E ⊗ Ω*M).
Then Taking components on both sides, where it is understood that d and ω refer to the component-wise derivative with respect to the frame e, and a matrix of 1-forms, respectively, acting on the components of ξ. Conversely, a matrix of 1-forms ω is a priori sufficient to completely determine the connection locally on the open set over which the basis of sections e is defined.
In order to extend ω to a suitable global object, it is necessary to examine how it behaves when a different choice of basic sections of E is chosen.
Then there is an invertible k × k matrix of functions g such that Applying the exterior connection to both sides gives the transformation law for ω: Note in particular that ω fails to transform in a tensorial manner, since the rule for passing from one frame to another involves the derivatives of the transition matrix g. If {Up} is an open covering of M, and each Up is equipped with a trivialization ep of E, then it is possible to define a global connection form in terms of the patching data between the local connection forms on the overlap regions.
In detail, a connection form on M is a system of matrices ω(ep) of 1-forms defined on each Up that satisfy the following compatibility condition This compatibility condition ensures in particular that the exterior connection of a section of E, when regarded abstractly as a section of E ⊗ Ω1M, does not depend on the choice of basis section used to define the connection.
The curvature two-form of a connection form in E is defined by Unlike the connection form, the curvature behaves tensorially under a change of frame, which can be checked directly by using the Poincaré lemma.
Symbolically, In terms of the exterior connection D, the curvature endomorphism is given by for v ∈ E (we can extend v to a local section to define this expression).
Thus the curvature measures the failure of the sequence to be a chain complex (in the sense of de Rham cohomology).
This is the Levi-Civita connection on the tangent bundle TM of M.[2][3] A local frame on the tangent bundle is an ordered list of vector fields e = (ei | i = 1, 2, ..., n), where n = dim M, defined on an open subset of M that are linearly independent at every point of their domain.
[4] Then, employing now the summation convention on repeated indices, where R is the Riemann curvature tensor.
The torsion of the connection is given by Θ = Dθ, or in terms of the frame components of the solder form by Assuming again for simplicity that e is holonomic, this expression reduces to which vanishes if and only if Γikj is symmetric on its lower indices.
A more specific type of connection form can be constructed when the vector bundle E carries a structure group.
Informally, the vector bundle E has the structure of a G-bundle if a preferred class of frames is specified, all of which are locally G-related to each other.
A connection is compatible with the structure of a G-bundle on E provided that the associated parallel transport maps always send one G-frame to another.
Formally, along a curve γ, the following must hold locally (that is, for sufficiently small values of t): for some matrix gαβ (which may also depend on t).
Under a change of frame where g is a G-valued function defined on an open subset of M, the connection form transforms via Or, using matrix products: To interpret each of these terms, recall that g : M → G is a G-valued (locally defined) function.
With this in mind, where ωg is the Maurer-Cartan form for the group G, here pulled back to M along the function g, and Ad is the adjoint representation of G on its Lie algebra.
The language of principal bundles, pioneered by Charles Ehresmann in the 1940s, provides a manner of organizing these many connection forms and the transformation laws connecting them into a single intrinsic form with a single rule for transformation.
The disadvantage to this approach is that the forms are no longer defined on the manifold itself, but rather on a larger principal bundle.
Suppose that E → M is a vector bundle with structure group G. Let {U} be an open cover of M, along with G-frames on each U, denoted by eU.
These are related on the intersections of overlapping open sets by for some G-valued function hUV defined on U ∩ V. Let FGE be the set of all G-frames taken over each point of M. This is a principal G-bundle over M. In detail, using the fact that the G-frames are all G-related, FGE can be realized in terms of gluing data among the sets of the open cover: where the equivalence relation
It can be shown that ω is a principal connection in the sense that it reproduces the generators of the right G action on FGE, and equivariantly intertwines the right action on T(FGE) with the adjoint representation of G. Conversely, a principal G-connection ω in a principal G-bundle P→M gives rise to a collection of connection forms on M. Suppose that e : M → P is a local section of P. Then the pullback of ω along e defines a g-valued one-form on M: Changing frames by a G-valued function g, one sees that ω(e) transforms in the required manner by using the Leibniz rule, and the adjunction: where X is a vector on M, and d denotes the pushforward.