Connection (principal bundle)

In mathematics, 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 π:P→M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.

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