Editing Parallel transport (section)

{{Use American English|date=March 2019}}{{Short description|System of moving vectors in differential geometry}}
[[File:Parallel Transport.svg|thumb|Parallel transport of a vector around a closed loop (from A to N to B and back to A) on the sphere. The angle by which it twists, <math>\alpha</math>, is proportional to the area inside the loop.]]

In [[differential geometry]], '''parallel transport''' (or '''parallel translation'''{{efn|In some sources like Spivak{{sfn|Spivak|1999|p=234|loc=Vol. 2, Ch. 6}}}}) is a way of transporting geometrical data along smooth curves in a [[manifold]]. If the manifold is equipped with an [[affine connection]] (a [[covariant derivative]] or [[connection (vector bundle)|connection]] on the [[tangent bundle]]), then this connection allows one to transport vectors of the manifold along curves so that they stay ''[[parallel (geometry)|parallel]]'' with respect to the connection.

The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of ''connecting'' the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one way of connecting up the geometries of points on a curve is tantamount to providing a ''connection''. In fact, the usual notion of connection is the [[infinitesimal]] analog of parallel transport. Or, ''vice versa'', parallel transport is the local realization of a connection.

As parallel transport supplies a local realization of the connection, it also supplies a local realization of the [[curvature]] known as [[holonomy]]. The [[Holonomy#Ambrose–Singer theorem|Ambrose–Singer theorem]] makes explicit this relationship between the curvature and holonomy.

Other notions of [[connection (mathematics)|connection]] come equipped with their own parallel transportation systems as well. For instance, a [[Koszul connection]] in a [[vector bundle]] also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An [[Ehresmann connection|Ehresmann]] or [[Cartan connection]] supplies a ''lifting of curves'' from the manifold to the total space of a [[principal bundle]]. Such curve lifting may sometimes be thought of as the parallel transport of [[frame of reference|reference frames]].