Editing Affine connection (section)

===Approaches===

The complex history has led to the development of widely varying approaches to and generalizations of the affine connection concept.

The most popular approach is probably the definition motivated by covariant derivatives. On the one hand, the ideas of Weyl were taken up by physicists in the form of [[gauge theory]] and [[gauge covariant derivative]]s. On the other hand, the notion of covariant differentiation was abstracted by [[Jean-Louis Koszul]], who defined (linear or Koszul) [[connection (vector bundle)|connections]] on [[vector bundle]]s. In this language, an affine connection is simply a [[covariant derivative]] or (linear) [[connection (vector bundle)|connection]] on the [[tangent bundle]].

However, this approach does not explain the geometry behind affine connections nor how they acquired their name.{{efn|As a result, many mathematicians use the term ''linear connection'' (instead of ''affine connection'') for a connection on the tangent bundle, on the grounds that [[parallel transport]] is linear and not affine. However, the same property holds for any (Koszul or linear Ehresmann) [[connection (vector bundle)|connection on a vector bundle]]. Originally the term ''affine connection'' is short for an affine ''[[Cartan connection|connection]]'' in the sense of Cartan, and this implies that the connection is defined on the tangent bundle, rather than an arbitrary vector bundle. The notion of a linear Cartan connection does not really make much sense, because linear representations are not transitive.}} The term really has its origins in the identification of tangent spaces in Euclidean space by translation: this property means that Euclidean {{mvar|n}}-space is an [[affine space]]. (Alternatively, Euclidean space is a [[principal homogeneous space]] or [[torsor]] under the group of translations, which is a subgroup of the affine group.) As mentioned in the introduction, there are several ways to make this precise: one uses the fact that an affine connection defines a notion of [[parallel transport]] of vector fields along a curve. 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 {{math|Aff(''n'')}} or as a principal {{math|GL(''n'')}} connection on the frame bundle.