Editing Affine connection (section)

{{Short description|Construct allowing differentiation of tangent vector fields of manifolds}}
{{More footnotes|date=February 2017}}
[[File:Parallel transport sphere.svg|right|thumb|An affine connection on the sphere rolls the affine tangent plane from one point to another. As it does so, the point of contact traces out a curve in the plane: the [[development (differential geometry)|development]].]]
In [[differential geometry]], an '''affine connection'''{{efn|a ''linear connection'' is also frequently called affine connection or simply ''connection'',{{sfn|Lee|1997|p=51|ignore-err=yes}} So that there is no agreement on the precise definitions of these terms (John M. Lee simply calls it '''connection''').{{sfn|Lee|2018|p=91}}}} is a geometric object on a [[smooth manifold]] which ''connects'' nearby [[tangent space]]s, so it permits [[vector field|tangent vector fields]] to be [[derivative|differentiated]] as if they were functions on the manifold with values in a fixed [[vector space]]. Connections are among the simplest methods of defining differentiation of the [[Section (fiber bundle)|sections]] of [[Vector bundle|vector bundles]].{{sfn|Lee|2018|p=88|loc=Connections}}

The notion of an affine connection has its roots in 19th-century geometry and [[tensor calculus]], but was not fully developed until the early 1920s, by [[Élie Cartan]] (as part of his general theory of [[Cartan connection|connections]]) and [[Hermann Weyl]] (who used the notion as a part of his foundations for [[general relativity]]). The terminology is due to Cartan{{efn|Cartan explains that he has borrowed this term (i.e. "affine connection") from the H. Weyl's book and referred to it (''Space-Time-Matter''), although he used it in more general context.{{sfn|Akivis|Rosenfeld|1993|p=213}}}} and has its origins in the identification of tangent spaces in [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an [[affine space]].

On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a [[metric tensor]] then there is a natural choice of affine connection, called the [[Levi-Civita connection]]. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties ([[linearity]] and the [[product rule|Leibniz rule]]). This yields a possible definition of an affine connection as a [[covariant derivative]] or (linear) [[connection (vector bundle)|connection]] on the [[tangent bundle]]. A choice of affine connection is also equivalent to a notion of [[parallel transport]], which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the [[frame bundle]]. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a [[Cartan connection]] for the [[affine group]] or as a [[connection (principal bundle)|principal connection]] on the frame bundle.

The main invariants of an affine connection are its [[torsion tensor|torsion]] and its [[curvature]]. The torsion measures how closely the [[Lie derivative|Lie bracket]] of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine) [[geodesics]] on a manifold, generalizing the ''straight lines'' of Euclidean space, although the geometry of those straight lines can be very different from usual [[Euclidean geometry]]; the main differences are encapsulated in the curvature of the connection.