Editing Cartan connection (section)

===Motivation===

Consider a smooth surface ''S'' in 3-dimensional Euclidean space '''R'''<sup>3</sup>. Near to any point, ''S'' can be approximated by its tangent plane at that point, which is an [[affine subspace]] of Euclidean space. The affine subspaces are ''model'' surfaces&mdash;they are the simplest surfaces in '''R'''<sup>3</sup>, and are homogeneous under the Euclidean group of the plane, hence they are ''Klein geometries'' in the sense of [[Felix Klein]]'s [[Erlangen programme]]. Every smooth surface ''S'' has a unique affine plane tangent to it at each point. The family of all such planes in '''R'''<sup>3</sup>, one attached to each point of ''S'', is called the '''congruence''' of tangent planes. A tangent plane can be "rolled" along ''S'', and as it does so the point of contact traces out a curve on ''S''. Conversely, given a curve on ''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 by affine (in fact Euclidean) transformations, and is an example of a Cartan connection called an [[affine connection]].

Another example is obtained by replacing the planes, as model surfaces, by spheres, which are homogeneous under the Möbius group of conformal transformations. There is no longer a unique sphere tangent to a smooth surface ''S'' at each point, since the radius of the sphere is undetermined. This can be fixed by supposing that the sphere has the same [[mean curvature]] as ''S'' at the point of contact. Such spheres can again be rolled along curves on ''S'', and this equips ''S'' with another type of Cartan connection called a [[conformal connection]].

Differential geometers in the late 19th and early 20th centuries were very interested in using model families such as planes or spheres to describe the geometry of surfaces. A family of model spaces attached to each point of a surface ''S'' is called a '''congruence''': in the previous examples there is a canonical choice of such a congruence. A Cartan connection provides an identification between the model spaces in the congruence along any curve in ''S''. An important feature of these identifications is that the point of contact of the model space with ''S'' ''always moves'' with the curve. This generic condition is characteristic of Cartan connections.

In the modern treatment of affine connections, 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, and so Cartan connections are not needed. However, there is no canonical way to do this in general: in particular for the conformal connection of a sphere congruence, it is not possible to separate the motion of the point of contact from the rest of the motion in a natural way.

In both of these examples the model space is a homogeneous space ''G''/''H''.
* In the first case, ''G''/''H'' is the affine plane, with ''G'' = Aff('''R'''<sup>2</sup>) the [[affine group]] of the plane, and ''H'' = GL(2) the corresponding general linear group.
* In the second case, ''G''/''H'' is the conformal (or [[celestial sphere|celestial]]) sphere, with ''G'' = O<sup>''+''</sup>(3,1) the [[Lorentz group|(orthochronous) Lorentz group]], and ''H'' the [[Group action (mathematics)#Orbits and stabilizers|stabilizer]] of a null line in '''R'''<sup>3,1</sup>.
The Cartan geometry of ''S'' consists of a copy of the model space ''G''/''H'' at each point of ''S'' (with a marked point of contact) together with a notion of "parallel transport" along curves which identifies these copies using elements of ''G''. This notion of parallel transport is generic in the intuitive sense that the point of contact always moves along the curve.

In general, let ''G'' be a group with a subgroup ''H'', and ''M'' a manifold of the same dimension as ''G''/''H''. Then, roughly speaking, a Cartan connection on ''M'' is a ''G''-connection which is generic with respect to a reduction to ''H''.