Connection form

In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
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. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative.
A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita connection. The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle with a structure group.

Vector bundles

Frames on a vector bundle

Let be a vector bundle of fibre dimension over a differentiable manifold. A local frame for is an ordered basis of local sections of. It is always possible to construct a local frame, as vector bundles are always defined in terms of local trivializations, in analogy to the atlas of a manifold. That is, given any point on the base manifold, there exists an open neighborhood of for which the vector bundle over is locally trivial, that is isomorphic to projecting to. The vector space structure on can thereby be extended to the entire local trivialization, and a basis on can be extended as well; this defines the local frame.
Let be a local frame on. This frame can be used to express locally any section of. For example, suppose that is a local section, defined over the same open set as the frame. Then
where denotes the components of in the frame. 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 .

Exterior connections

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 module 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.

Connection forms

The connection form arises when applying the exterior connection to a particular frame e. Upon applying the exterior connection to the e_α, it is the unique k × k matrix of one-forms on M such that
In terms of the connection form, the exterior connection of any section of E can now be expressed. For example, suppose that ξ = Σ_α e_αξ^α. 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.

Change of frame

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. Write ω_α^β = ω_α^β to indicate the dependence on the choice of e.
Suppose that e is a different choice of local basis. 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.

Global connection forms

If is an open covering of M, and each U_p is equipped with a trivialization e_p 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 of 1-forms defined on each U_p 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.

Curvature

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. Specifically, if e → e g is a change of frame, then the curvature two-form transforms by
One interpretation of this transformation law is as follows. Let e^* be the dual basis corresponding to the frame e. Then the 2-form
is independent of the choice of frame. In particular, Ω is a vector-valued two-form on M with values in the endomorphism ring Hom. Symbolically,
In terms of the exterior connection D, the curvature endomorphism is given by
for v ∈ E. Thus the curvature measures the failure of the sequence
to be a chain complex.

Soldering and torsion

Suppose that the fibre dimension k of E is equal to the dimension of the manifold M. In this case, the vector bundle E is sometimes equipped with an additional piece of data besides its connection: a solder form. A solder form is a globally defined vector-valued one-form θ ∈ Ω¹ such that the mapping
is a linear isomorphism for all x ∈ M. If a solder form is given, then it is possible to define the torsion of the connection as
The torsion Θ is an E-valued 2-form on M.
A solder form and the associated torsion may both be described in terms of a local frame e of E. If θ is a solder form, then it decomposes into the frame components
The components of the torsion are then
Much like the curvature, it can be shown that Θ behaves as a contravariant tensor under a change in frame:
The frame-independent torsion may also be recovered from the frame components:

Bianchi identities

The Bianchi identities relate the torsion to the curvature. The first Bianchi identity states that
while the second Bianchi identity states that

Example: the Levi-Civita connection

As an example, suppose that M carries a Riemannian metric. If one has a vector bundle E over M, then the metric can be extended to the entire vector bundle, as the bundle metric. One may then define a connection that is compatible with this bundle metric, this is the metric connection. For the special case of E being the tangent bundle TM, the metric connection is called the Riemannian connection. Given a Riemannian connection, one can always find a unique, equivalent connection that is torsion-free. This is the Levi-Civita connection on the tangent bundle TM of M.
A local frame on the tangent bundle is an ordered list of vector fields, where, defined on an open subset of M that are linearly independent at every point of their domain. The Christoffel symbols define the Levi-Civita connection by
If θ =, denotes the dual basis of the cotangent bundle, such that θⁱ = δⁱ_j, then the connection form is
In terms of the connection form, the exterior connection on a vector field is given by
One can recover the Levi-Civita connection, in the usual sense, from this by contracting with e_i:

Curvature

The curvature 2-form of the Levi-Civita connection is the matrix given by
For simplicity, suppose that the frame e is holonomic, so that. Then, employing now the summation convention on repeated indices,
where R is the Riemann curvature tensor.