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Concept# Connection (principal bundle)

Summary

In mathematics, and especially differential geometry and gauge theory, a connection 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 principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.
A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.
Let be a smooth principal G-bundle over a smooth manifold . Then a principal -connection on is a differential 1-form on with values in the Lie algebra of which is -equivariant and reproduces the Lie algebra generators of the fundamental vector fields on .
In other words, it is an element ω of such that
where denotes right multiplication by , and is the adjoint representation on (explicitly, );
if and is the vector field on P associated to ξ by differentiating the G action on P, then (identically on ).
Sometimes the term principal G-connection refers to the pair and itself is called the connection form or connection 1-form of the principal connection.
Most known non-trivial computations of principal G-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let , be a principal G-bundle over ) This means that 1-forms on the total space are canonically isomorphic to , where is the dual lie algebra, hence G-connections are in bijection with .

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