Editing Cartan connection (section)

===Klein geometries as model spaces===

Klein's [[Erlangen programme]] suggested that geometry could be regarded as a study of [[homogeneous space]]s: in particular, it is the study of the many geometries of interest to geometers of 19th century (and earlier).   A Klein geometry consisted of a space, along with a law for motion within the space (analogous to the [[Euclidean transformation]]s of classical [[Euclidean geometry]]) expressed as a [[Lie group]] of [[transformation group|transformations]].  These generalized spaces turn out to be homogeneous [[smooth manifold]]s diffeomorphic to the [[Quotient space (topology)|quotient space]] of a Lie group by a [[Lie subgroup]]. The extra differential structure that these homogeneous spaces possess allows one to study and generalize their geometry using calculus.

The general approach of Cartan is to begin with such a ''smooth Klein geometry'', given by a Lie group ''G'' and a Lie subgroup ''H'', with associated Lie algebras <math>\mathfrak g</math> and <math>\mathfrak h</math>, respectively.  Let ''P'' be the underlying [[principal homogeneous space]] of ''G''.  A Klein geometry is the homogeneous space given by the quotient ''P''/''H'' of ''P'' by the right action of ''H''.  There is a right ''H''-action on the fibres of the canonical projection
:''π'': ''P'' &rarr; ''P''/''H''
given by ''R''<sub>''h''</sub>''g'' = ''gh''.  Moreover, each [[fibre bundle|fibre]] of ''π'' is a copy of ''H''. ''P'' has the structure of a [[principal bundle|principal ''H''-bundle]] over ''P''/''H''.<ref>{{Harvnb|Chevalley|1946| p=110}}.</ref>

A vector field ''X'' on ''P'' is ''vertical'' if d''π''(''X'') = 0.  Any ''ξ'' ∈ <math>\mathfrak h </math> gives rise to a canonical vertical vector field ''X''<sub>''ξ''</sub> by taking the derivative of the right action of the 1-parameter subgroup of ''H'' associated to ξ.  The [[Maurer-Cartan form]] ''η'' of ''P'' is the <math>\mathfrak g</math>[[Lie algebra-valued form|-valued one-form]] on ''P'' which identifies each tangent space with the Lie algebra. It has the following properties:

# Ad(''h'') ''R''<sub>''h''</sub><sup>*</sup>''η'' = ''η'' for all ''h'' in ''H''
# ''η''(''X''<sub>''ξ''</sub>) = ''ξ'' for all ''ξ'' in <math>\mathfrak h</math>
# for all ''g''∈''P'', ''η'' restricts a linear isomorphism of T<sub>''g''</sub>''P'' with <math>\mathfrak g</math>  (η is an '''absolute parallelism''' on ''P'').

In addition to these properties, ''η'' satisfies the '''structure''' (or '''structural''') '''equation'''
: <math> d\eta+\tfrac{1}{2}[\eta,\eta]=0. </math>

Conversely, one can show that given a manifold ''M'' and a principal ''H''-bundle ''P'' over ''M'', and a 1-form ''η'' with these properties, then ''P'' is locally isomorphic as an ''H''-bundle to the principal homogeneous bundle ''G''→''G''/''H''.  The structure equation is the [[integrability condition]] for the existence of such a local isomorphism.

A Cartan geometry is a generalization of a smooth Klein geometry, in which the structure equation is not assumed, but is instead used to define a notion of [[curvature]]. Thus the Klein geometries are said to be the '''flat models''' for Cartan geometries.<ref>See R. Hermann (1983), Appendix 1&ndash;3 to {{Harvtxt|Cartan|1951}}.</ref>