Editing Affine connection (section)

===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.