Editing Cartan connection (section)

===Definition via absolute parallelism===

Let ''P'' be a principal ''H'' bundle over ''M''. Then a '''Cartan connection'''<ref>This is the standard definition. Cf. Hermann (1983), Appendix 2 to {{Harvnb|Cartan|1951}}; {{Harvnb|Kobayashi|1970| p=127}}; {{Harvnb|Sharpe|1997}}; {{Harvnb|Slovák|1997}}.</ref> is a <math>\mathfrak g</math>-valued 1-form ''η'' on ''P'' such that

# for all ''h'' in ''H'', Ad(''h'')''R''<sub>''h''</sub><sup>*</sup>''η'' = ''η''
# for all ''ξ'' in <math>\mathfrak h</math>, ''η''(''X''<sub>''ξ''</sub>) = ''ξ''
# for all ''p'' in ''P'', the restriction of ''η'' defines a linear isomorphism from the tangent space T<sub>''p''</sub>''P'' to <math>\mathfrak g</math>.

The last condition is sometimes called the '''Cartan condition''': it means that ''η'' defines an '''absolute parallelism''' on ''P''. The second condition implies that ''η'' is already injective on vertical vectors and that the 1-form ''η'' mod <math>\mathfrak h</math>, with values in <math>\mathfrak g/\mathfrak h</math>, is horizontal. The vector space <math>\mathfrak g/\mathfrak h</math> is a [[group representation|representation]] of ''H'' using the adjoint representation of ''H'' on <math>\mathfrak g</math>, and the first condition implies that ''η'' mod <math>\mathfrak h</math> is equivariant. Hence it defines a bundle homomorphism from T''M'' to the associated bundle <math> P\times_H \mathfrak g/\mathfrak h</math>.
The Cartan condition is equivalent to this bundle homomorphism being an isomorphism, so that ''η'' mod <math>\mathfrak h</math> is a [[solder form]].

The '''curvature''' of a Cartan connection is the <math>\mathfrak g</math>-valued 2-form ''Ω'' defined by
:<math>\Omega=d\eta+\tfrac{1}{2}[\eta\wedge\eta].</math>

Note that this definition of a Cartan connection looks very similar to that of a [[principal connection]].  There are several important differences, however.  First, the 1-form η takes values in <math>\mathfrak g</math>, but is only equivariant under the action of ''H''.  Indeed, it cannot be equivariant under the full group ''G'' because there is no ''G'' bundle and no ''G'' action. Secondly, the 1-form is an absolute parallelism, which intuitively means that η yields information about the behavior of additional directions in the principal bundle (rather than simply being a projection operator onto the vertical space).  Concretely, the existence of a solder form binds (or solders) the Cartan connection to the underlying [[differential topology]] of the manifold.

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An intuitive interpretation of the Cartan connection in this form is that it determines a ''fracturing'' of the tautological principal bundle associated to a Klein geometry. Thus Cartan geometries are deformed analogues of Klein geometries. This deformation is roughly a prescription for attaching a copy of the model space ''G''/''H'' to each point of ''M'' and thinking of that model space as being ''tangent'' to (and ''infinitesimally identical'' with) the manifold at a point of contact. The fibre of the tautological bundle ''G'' → ''G''/''H'' of the Klein geometry at the point of contact is then identified with the fibre of the bundle ''P''. Each such fibre (in ''G'') carries a Maurer-Cartan form for ''G'', and the Cartan connection is a way of assembling these Maurer-Cartan forms gathered from the points of contact into a coherent 1-form η defined on the whole bundle. The fact that only elements of ''H'' contribute to the Maurer-Cartan equation Ad(''h'')''R''<sub>''h''</sub><sup>*</sup>''η'' = ''η'' has the intuitive interpretation that any other elements of ''G'' would move the model space away from the point of contact, and so no longer be tangent to the manifold.

From the Cartan connection, defined in these terms, one can recover a Cartan connection as a system of 1-forms on the manifold (as in the gauge definition) by taking a collection of [[locally trivial|local trivializations]] of ''P'' given as sections ''s''<sub>U</sub> : ''U'' → ''P'' and letting θ<sub>U</sub> = ''s''<sup>*</sup>η be the [[pullback (differential geometry)|pullbacks]] of the Cartan connection along the sections.