Editing Cartan connection (section)

===As principal connections===

Another way in which to define a Cartan connection is as a [[connection (principal bundle)|principal connection]] on a certain principal ''G''-bundle.  From this perspective, a Cartan connection consists of
* a principal ''G''-bundle ''Q'' over ''M''
* a principal ''G''-connection ''α'' on ''Q'' (the Cartan connection)
* a principal ''H''-subbundle ''P'' of ''Q'' (i.e., a reduction of structure group)
such that the [[pullback (differential geometry)|pullback]] ''η'' of ''α'' to ''P'' satisfies the Cartan condition.

The principal connection ''α'' on ''Q'' can be recovered from the form ''η'' by taking ''Q'' to be the associated bundle ''P'' &times;<sub>''H''</sub> ''G''.  Conversely, the form η can be recovered from α by pulling back along the inclusion ''P'' ⊂ ''Q''.

Since ''α'' is a principal connection, it induces a [[Ehresmann connection|connection]] on any [[associated bundle]] to ''Q''. In particular, the bundle ''Q'' &times;<sub>''G''</sub> ''G''/''H'' of homogeneous spaces over ''M'', whose fibers are copies of the model space ''G''/''H'', has a connection. The reduction of structure group to ''H'' is equivalently given by a section ''s'' of ''E'' = ''Q'' &times;<sub>''G''</sub> ''G''/''H''. The fiber of <math>P\times_H \mathfrak g/\mathfrak h</math> over ''x'' in ''M'' may be viewed as the tangent space at ''s''(''x'') to the fiber of ''Q'' &times;<sub>''G''</sub> ''G''/''H'' over ''x''. Hence the Cartan condition has the intuitive interpretation that the model spaces are tangent to ''M'' along the section ''s''. Since this identification of tangent spaces is induced by the connection, the marked points given by ''s'' always move under parallel transport.