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Connection (vector bundle)
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, and especially <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> and <a href="/facts/Gauge_theory_(mathematics)/RPnPc4FG">gauge theory</a>, a connection on a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a> is a device that defines a notion of <a href="/facts/Parallel_transport/QX0Wfcte">parallel transport</a> 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 <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a>, for which the notion of parallel transport must be <a href="/facts/Linear/fChtg3KC">linear</a>. A linear connection is equivalently specified by a <i><a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivative</a></i>, an operator that differentiates <a href="/facts/Section_(fiber_bundle)/DLnpXS5a">sections</a> of the bundle along <a href="/facts/Tangent_vector/UEq5yffm">tangent directions</a> in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the <a href="/facts/Levi-Civita_connection/16KGSQUY">Levi-Civita connection</a> on the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> of a <a href="/facts/Pseudo-Riemannian_manifold/VFXsVjqb">pseudo-Riemannian manifold</a>, which gives a standard way to differentiate vector fields. <a href="/facts/Ehresmann_connection/qIpSLmFz">Nonlinear connections</a> generalize this concept to bundles whose fibers are not necessarily linear. </p><p>Linear connections are also called Koszul connections after <a href="/facts/Jean-Louis_Koszul/1bUb0Iz8">Jean-Louis Koszul</a>, who gave an algebraic framework for describing them (Koszul 1950). </p><p>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 <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, vector bundle computations are usually written using indexed tensors; in <a href="/facts/Gauge_theory/fSW3gggv">gauge theory</a>, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on <a href="/facts/Metric_connection/YyGhEDKr">metric connections</a> (the comments made there apply to all vector bundles). </p>