Editing Connection (vector bundle) (section)

{{Short description|Defines a notion of parallel transport on a bundle}}
{{about|connections on vector bundles|other types of connections in mathematics|connection (mathematics)}}

In [[mathematics]], and especially [[differential geometry]] and [[gauge theory (mathematics)|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. The most common case is that of a '''linear connection''' on a [[vector bundle]], for which the notion of parallel transport must be [[linear]]. A linear connection is equivalently specified by a ''[[covariant derivative]]'', an operator that differentiates [[section (fiber bundle)|sections]] of the bundle along [[tangent vector|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. [[Ehresmann connection|Nonlinear connections]] generalize this concept to bundles whose fibers are not necessarily linear.

Linear connections are also called '''Koszul connections''' after [[Jean-Louis Koszul]], who gave an algebraic framework for describing them {{harv|Koszul|1950}}.

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 connection|metric connections]] (the comments made there apply to all vector bundles).