Editing Cartan connection (section)

===Affine connections===
{{main|Affine connection}}
An '''[[affine connection]]''' on a manifold ''M'' is a [[Connection (principal bundle)|connection]] on the [[Frame bundle|frame bundle (principal bundle)]] of ''M'' (or equivalently, a [[Connection (vector bundle)|connection]]  on the [[Tangent bundle|tangent bundle (vector bundle)]] of ''M''). A key aspect of the Cartan connection point of view is to elaborate this notion in the context of [[principal bundle]]s (which could be called the "general or abstract theory of frames").

Let ''H'' be a [[Lie group]], <math>\mathfrak h</math> its [[Lie algebra]]. Then a '''principal ''H''-bundle''' is a [[fiber bundle]] ''P'' over ''M'' with a smooth [[Group action (mathematics)|action]] of ''H'' on ''P'' which is free and transitive on the fibers. Thus ''P'' is a smooth manifold with a smooth map ''π'': ''P'' → ''M'' which looks ''locally'' like the [[trivial bundle]] ''M'' &times; ''H'' → ''M''. The frame bundle of ''M'' is a principal GL(''n'')-bundle, while if ''M'' is a [[Riemannian manifold]], then the [[orthonormal frame bundle]] is a principal O(''n'')-bundle.

Let ''R''<sub>''h''</sub> denote the (right) action of ''h'' ∈ H on ''P''. The derivative of this action defines a '''[[vertical bundle|vertical vector]] field''' on ''P'' for each element ''ξ'' of <math>\mathfrak h</math>: if ''h''(''t'') is a 1-parameter subgroup with ''h''(0)=''e'' (the identity element) and ''h'' '(''0'')=''ξ'', then the corresponding vertical vector field is
:<math>X_\xi=\frac{\mathrm d}{\mathrm dt}R_{h(t)}\biggr|_{t=0}.\,</math>

A '''principal ''H''-connection''' on ''P'' is a [[differential 1-form|1-form]] <math>\omega\colon TP\to \mathfrak h</math>  on ''P'',
with values in the [[Lie algebra]] <math>\mathfrak h</math> of ''H'', such that
# <math>\hbox{Ad}(h)(R_h^*\omega)=\omega</math>
# for any <math>\xi\in \mathfrak h</math>, ''ω''(''X''<sub>''ξ''</sub>) = ''ξ'' (identically on ''P'').

The intuitive idea is that ''ω''(''X'') provides a ''vertical component'' of ''X'', using the isomorphism of the fibers of ''π'' with ''H'' to identify vertical vectors with elements of <math>\mathfrak h</math>.

Frame bundles have additional structure called the [[solder form]], which can be used to extend a principal connection on ''P'' to a trivialization of the tangent bundle of ''P'' called an '''absolute parallelism'''.

In general, suppose that ''M'' has dimension ''n'' and ''H'' acts on '''R'''<sup>''n''</sup> (this could be any ''n''-dimensional real vector space). A '''solder form''' on a principal ''H''-bundle ''P'' over ''M'' is an '''R'''<sup>''n''</sup>-valued 1-form ''θ'': T''P'' → '''R'''<sup>''n''</sup> which is horizontal and equivariant so that it induces a [[bundle homomorphism]] from T''M'' to the [[associated bundle]] ''P'' &times;<sub>''H''</sub> '''R'''<sup>''n''</sup>. This is furthermore required to be a bundle isomorphism. Frame bundles have a (canonical or tautological) solder form which sends a tangent vector ''X'' ∈ T<sub>''p''</sub>''P'' to the coordinates of d''π''<sub>''p''</sub>(''X'') ∈ T<sub>''π''(''p'')</sub>''M'' with respect to the frame ''p''.

The pair (''ω'', ''θ'') (a principal connection and a solder form) defines a 1-form ''η'' on ''P'', with values in the Lie algebra <math>\mathfrak g</math> of the [[semidirect product]] ''G'' of ''H'' with '''R'''<sup>''n''</sup>, which provides an isomorphism of each tangent space T<sub>''p''</sub>''P'' with <math>\mathfrak g</math>. It induces a principal connection ''α'' on the associated principal ''G''-bundle ''P'' &times;<sub>''H''</sub> ''G''. This is a Cartan connection.

Cartan connections generalize affine connections in two ways.
* The action of ''H'' on '''R'''<sup>''n''</sup> need not be effective. This allows, for example, the theory to include spin connections, in which ''H'' is the [[spin group]] Spin(''n'') rather than the [[orthogonal group]] O(''n'').
* The group ''G'' need not be a semidirect product of ''H'' with '''R'''<sup>''n''</sup>.