Editing Connection (principal bundle) (section)

{{Short description|Concept in mathematics}}
{{About|connections on principal bundles|information about other types of connections in mathematics|Connection (mathematics)}}
In [[mathematics]], and especially [[differential geometry]] and [[gauge theory (mathematics)|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 bundle|principal G-bundle]] <math>P</math> over a [[smooth manifold]] ''<math>M</math>'' is a particular type of connection that is compatible with the [[Group action (mathematics)|action]] of the group ''<math>G</math>''.

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 ''<math>P</math>'' via the [[associated bundle]] construction. In particular, on any [[associated vector bundle]] the principal connection induces a [[covariant derivative]], an operator that can differentiate [[section (fiber bundle)|sections]] of that bundle along [[tangent vector|tangent directions]] in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a [[Connection (vector bundle)|linear connection]] on the [[frame bundle]] of a [[smooth manifold]].