Editing Cartan connection (section)

== Pseudogroups ==
Cartan connections are closely related to [[pseudogroup]] structures on a manifold.  Each is thought of as ''modelled on'' a Klein geometry ''G''/''H'', in a manner similar to the way in which [[Riemannian geometry]] is modelled on [[Euclidean space]].  On a manifold ''M'', one imagines attaching to each point of ''M'' a copy of the model space ''G''/''H''.  The symmetry of the model space is then built into the Cartan geometry or pseudogroup structure by positing that the model spaces of nearby points are related by a transformation in ''G''.  The fundamental difference between a Cartan geometry and pseudogroup geometry is that the symmetry for a Cartan geometry relates ''infinitesimally'' close points by an ''infinitesimal'' transformation in ''G'' (i.e., an element of the Lie algebra of ''G'') and the analogous notion of symmetry for a pseudogroup structure applies for points that are physically separated within the manifold.

The process of attaching spaces to points, and the attendant symmetries, can be concretely realized by using special [[coordinate system]]s.<ref>This appears to be Cartan's way of viewing the connection. Cf. {{Harvnb|Cartan|1923|p=362}}; {{Harvnb|Cartan|1924|p=208}} especially ''..un repère définissant un système de coordonnées projectives...''; {{Harvnb|Cartan|1951|p=34}}.  Modern readers can arrive at various interpretations of these statements, cf. Hermann's 1983 notes in {{Harvnb|Cartan|1951|pp=384&ndash;385, 477}}.</ref>  To each point ''p'' ∈ ''M'', a [[neighbourhood (mathematics)|neighborhood]] ''U''<sub>p</sub> of ''p'' is given along with a mapping φ<sub>p</sub> : ''U''<sub>p</sub> → ''G''/''H''.  In this way, the model space is attached to each point of ''M'' by realizing ''M'' locally at each point as an open subset of ''G''/''H''.  We think of this as a family of coordinate systems on ''M'', parametrized by the points of ''M''.  Two such parametrized coordinate systems φ and φ&prime; are ''H''-related if there is an element ''h''<sub>p</sub> ∈ ''H'', parametrized by ''p'', such that
: φ&prime;<sub>p</sub> = ''h''<sub>p</sub> φ<sub>p</sub>.<ref>More precisely, ''h''<sub>p</sub> is required to be in the [[isotropy group]] of φ<sub>p</sub>(''p''), which is a group in ''G'' isomorphic to ''H''.</ref>
This freedom corresponds roughly to the physicists' notion of a [[gauge fixing|gauge]].

Nearby points are related by joining them with a curve.  Suppose that ''p'' and ''p''&prime; are two points in ''M'' joined by a curve ''p''<sub>t</sub>. Then ''p''<sub>t</sub> supplies a notion of transport of the model space along the curve.<ref>In general, this is not the rolling map described in the motivation, although it is related.</ref>  Let τ<sub>t</sub> : ''G''/''H'' → ''G''/''H'' be the (locally defined) composite map
:τ<sub>t</sub> = φ<sub>p<sub>t</sub></sub> o φ<sub>p<sub>0</sub></sub><sup>−1</sup>.
Intuitively, τ<sub>t</sub> is the transport map.  A pseudogroup structure requires that τ<sub>t</sub> be a ''symmetry of the model space'' for each ''t'':  τ<sub>t</sub> ∈ ''G''.  A Cartan connection requires only that the [[derivative]] of τ<sub>t</sub> be a symmetry of the model space: τ&prime;<sub>0</sub> ∈ '''g''', the Lie algebra of ''G''.

Typical of Cartan, one motivation for introducing the notion of a Cartan connection was to study the properties of pseudogroups from an infinitesimal point of view.  A Cartan connection defines a pseudogroup precisely when the derivative of the transport map τ&prime; can be [[integral|integrated]], thus recovering a true (''G''-valued) transport map between the coordinate systems.  There is thus an [[integrability condition]] at work, and Cartan's method for realizing integrability conditions was to introduce a [[differential form]].

In this case, τ&prime;<sub>0</sub> defines a differential form at the point ''p'' as follows.  For a curve γ(''t'') = ''p''<sub>t</sub> in ''M'' starting at ''p'', we can associate the [[tangent vector]] ''X'', as well as a transport map τ<sub>t</sub><sup>γ</sup>.  Taking the derivative determines a linear map
: <math> X \mapsto \left.\frac{d}{dt}\tau_t^\gamma\right|_{t=0} = \theta(X) \in \mathfrak{g}.</math>
So θ defines a '''g'''-valued differential 1-form on ''M''.

This form, however, is dependent on the choice of parametrized coordinate system.  If ''h'' : ''U'' → ''H'' is an ''H''-relation between two parametrized coordinate systems φ and φ&prime;, then the corresponding values of θ are also related by
:<math>\theta^\prime_p = Ad(h^{-1}_p)\theta_p + h^*_p\omega_H,</math>
where ω<sub>H</sub> is the Maurer-Cartan form of ''H''.