Connection (vector bundle)
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points.
A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero.
Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields.
Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.
This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates.
However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized.
The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).
The problem is resolved by introducing the extra structure of a connection to the vector bundle.
Depending on context, there may be distinguished choices, for instance those which are determined by solving certain partial differential equations.
With this notation, the product rule in the second version of the definition given above is written Remark.
In the case of a complex vector bundle, the above definition is still meaningful, but is usually taken to be modified by changing "real" and "
" This places extra restrictions, as not every real-linear map between complex vector spaces is complex-linear.
the natural pairing between a vector space and its dual (occurring on each fibre between
, the definition of the tensor product connection applies in a straightforward manner to this setting.
is a linear map A connection may then be viewed as a generalization of the exterior derivative to vector bundle valued forms.
Every vector bundle over a manifold admits a connection, which can be proved using partitions of unity.
in terms of the local frame of sections, by using the characteristic product rule for the connection.
is also written in matrix notation as a column vector using the local frame
In pseudo-Riemannian geometry, the Levi-Civita connection is often written in terms of the Christoffel symbols
and the Christoffel symbols, to be given by Contracting this expression with the local coordinate tangent vector
consisting of all parallel transport maps coming from loops based at
The covariant derivative is recovered as This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms
The curvature form has a local description called Cartan's structure equation.
to be wedge product of forms but commutator of endomorphisms as opposed to composition, then
, and with this alternate notation the Cartan structure equation takes the form This alternate notation is commonly used in the theory of principal bundle connections, where instead we use a connection form
In some references (see for example (MadsenTornehave1997)) the Cartan structure equation may be written with a minus sign: This different convention uses an order of matrix multiplication that is different from the standard Einstein notation in the wedge product of matrix-valued one-forms.
A version of the second (differential) Bianchi identity from Riemannian geometry holds for a connection on any vector bundle.
This endomorphism connection has itself an exterior covariant derivative, which we ambiguously call
The group of gauge transformations may be neatly characterised as the space of sections of the capital A adjoint bundle
is a connection, one verifies the product rule It may be checked that this defines a left group action of
