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Connection (mathematics)
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{{Short description|Function which tells how a certain variable changes as it moves along certain points in space}} In [[geometry]], the notion of a '''connection''' makes precise the idea of transporting local geometric objects, such as [[Tangent vector|tangent vectors]] or [[Tensor|tensors]] in the [[tangent space]], along a curve or family of curves in a ''parallel'' and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an [[affine connection]], the most elementary type of connection, gives a means for parallel transport of [[tangent space|tangent vectors]] on a [[manifold]] from one point to another along a curve. An affine connection is typically given in the form of a [[covariant derivative]], which gives a means for taking [[directional derivative]]s of vector fields, measuring the deviation of a [[vector field]] from being parallel in a given direction. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. [[Differential geometry]] embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory. The local theory concerns itself primarily with notions of [[parallel transport]] and [[holonomy]]. The infinitesimal theory concerns itself with the differentiation of geometric data. Thus a covariant derivative is a way of specifying a [[derivative]] of a vector field along another vector field on a manifold. A [[Cartan connection]] is a way of formulating some aspects of connection theory using [[differential forms]] and [[Lie group]]s. An [[Ehresmann connection]] is a connection in a [[fibre bundle]] or a [[principal bundle]] by specifying the allowed directions of motion of the field. A [[Koszul connection]] is a connection which defines directional derivative for sections of a [[vector bundle]] more general than the tangent bundle. Connections also lead to convenient formulations of ''geometric invariants'', such as the [[curvature]] (see also [[Riemann curvature tensor|curvature tensor]] and [[curvature form]]), and [[torsion tensor]].
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