Editing Affine connection (section)

==Motivation and history==

A [[smooth manifold]] is a [[mathematical]] object which looks locally like a smooth deformation of Euclidean space {{math|'''R'''<sup>''n''</sup>}}: for example a smooth curve or surface looks locally like a smooth deformation of a line or a plane. [[Smooth function]]s and [[vector field]]s can be defined on manifolds, just as they can on Euclidean space, and [[scalar (mathematics)|scalar]] functions on manifolds can be differentiated in a natural way. However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point {{mvar|p}} can be identified naturally (by translation) with the tangent space at a nearby point {{mvar|q}}. On a general manifold, there is no such natural identification between nearby tangent spaces, and so tangent vectors at nearby points cannot be compared in a well-defined way. The notion of an affine connection was introduced to remedy this problem by ''connecting'' nearby tangent spaces. The origins of this idea can be traced back to two main sources: [[Differential geometry of surfaces|surface theory]] and [[tensor calculus]].

===Motivation from surface theory===

{{see also|Cartan connection}}

Consider a smooth surface {{mvar|S}} in a 3-dimensional Euclidean space. Near any point, {{mvar|S}} can be approximated by its [[tangent plane]] at that point, which is an [[affine subspace]] of Euclidean space. Differential geometers in the 19th century were interested in the notion of [[development (differential geometry)|development]] in which one surface was ''rolled'' along another, without ''slipping'' or ''twisting''. In particular, the tangent plane to a point of {{mvar|S}} can be rolled on {{mvar|S}}: this should be easy to imagine when {{mvar|S}} is a surface like the 2-sphere, which is the smooth boundary of a [[convex set|convex]] region. As the tangent plane is rolled on {{mvar|S}}, the point of contact traces out a curve on {{mvar|S}}. Conversely, given a curve on {{mvar|S}}, the tangent plane can be rolled along that curve. This provides a way to identify the tangent planes at different points along the curve: in particular, a tangent vector in the tangent space at one point on the curve is identified with a unique tangent vector at any other point on the curve. These identifications are always given by [[affine transformation]]s from one tangent plane to another.

This notion of parallel transport of tangent vectors, by affine transformations, along a curve has a characteristic feature: the point of contact of the tangent plane with the surface ''always moves'' with the curve under parallel translation (i.e., as the tangent plane is rolled along the surface, the point of contact moves). This generic condition is characteristic of [[Cartan connection]]s. In more modern approaches, the point of contact is viewed as the ''origin'' in the tangent plane (which is then a vector space), and the movement of the origin is corrected by a translation, so that parallel transport is linear, rather than affine.

In the point of view of Cartan connections, however, the affine subspaces of Euclidean space are ''model'' surfaces &mdash; they are the simplest surfaces in Euclidean 3-space, and are homogeneous under the affine group of the plane &mdash; and every smooth surface has a unique model surface tangent to it at each point. These model surfaces are ''Klein geometries'' in the sense of [[Felix Klein]]'s [[Erlangen programme]]. More generally, an {{mvar|n}}-dimensional affine space is a [[Klein geometry]] for the [[affine group]] {{math|Aff(''n'')}}, the stabilizer of a point being the [[general linear group]] {{math|GL(''n'')}}. An affine {{mvar|n}}-manifold is then a manifold which looks infinitesimally like {{mvar|n}}-dimensional affine space.

===Motivation from tensor calculus===
{{see also|Covariant derivative}}

[[File:Affine connection example.svg|thumbnail|Historically, people used the covariant derivative (or Levi-Civita connection given by the metric) to describe the variation rate of a vector along the direction of another vector. Here on the punctured 2-dimensional Euclidean space, the blue vector field {{mvar|X}} sends the [[one-form]] {{math|d''r''}} to 0.07 everywhere. The red vector field {{mvar|Y}} sends the one-form {{math|''r''d''θ''}} to {{Math|0.5''r''}} everywhere. Endorsed by the metric {{math|d''s''<sup>2</sup> {{=}} d''r''<sup>2</sup> + ''r''<sup>2</sup>d''θ''<sup>2</sup>}}, the Levi-Civita connection {{math|∇<sub>''Y''</sub>''X''}} is 0 everywhere, indicating {{mvar|X}} has no change along {{mvar|Y}}. In other words, {{mvar|X}} [[parallel transport]]s along each [[concentric]] circle. {{math|1=∇<sub>''X''</sub>''Y'' = ''Y''/''r''}} everywhere, which sends {{math|''r''d''θ''}} to 0.5 everywhere, implying {{mvar|Y}} has a "constant" changing rate on the radial direction.]]
The second motivation for affine connections comes from the notion of a [[covariant derivative]] of vector fields. Before the advent of coordinate-independent methods, it was necessary to work with vector fields by [[embedding]] their respective [[Euclidean vector]]s into an [[Atlas (topology)|atlas]]. These components can be differentiated, but the derivatives do not transform in a manageable way under changes of coordinates.{{citation needed|date=June 2016}} Correction terms were introduced by [[Elwin Bruno Christoffel]] (following ideas of [[Bernhard Riemann]]) in the 1870s so that the (corrected) derivative of one vector field along another transformed [[covariant transformation|covariantly]] under coordinate transformations &mdash; these correction terms subsequently came to be known as [[Christoffel symbol]]s.

This idea was developed into the theory of ''absolute differential calculus'' (now known as [[tensor calculus]]) by [[Gregorio Ricci-Curbastro]] and his student [[Tullio Levi-Civita]] between 1880 and the turn of the 20th century.

Tensor calculus really came to life, however, with the advent of [[Albert Einstein]]'s theory of [[general relativity]] in 1915. A few years after this, Levi-Civita formalized the unique connection associated to a Riemannian metric, now known as the [[Levi-Civita connection]]. More general affine connections were then studied around 1920, by [[Hermann Weyl]],<ref>{{Harvnb|Weyl|1918}}, 5 editions to 1922.</ref> who developed a detailed mathematical foundation for general relativity, and [[Élie Cartan]],<ref name="Cartan-affine">{{Harvnb|Cartan|1923}}.</ref> who made the link with the geometrical ideas coming from surface theory.

===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.