Editing Cartan connection (section)

{{Short description|Generalization of affine connections}}
In the mathematical field of [[differential geometry]], a '''Cartan connection''' is a flexible generalization of the notion of an [[affine connection]]. It may also be regarded as a specialization of the general concept of a [[connection (principal bundle)|principal connection]], in which the geometry of the [[principal bundle]] is tied to the geometry of the base manifold using a [[solder form]]. Cartan connections describe the geometry of manifolds modelled on [[homogeneous space]]s.

The theory of Cartan connections was developed by [[Élie Cartan]], as part of (and a way of formulating) his [[method of moving frames]] ('''''repère mobile'''''<!--boldface per WP:R#PLA-->).<ref>Although Cartan only began formalizing this theory in particular cases in the 1920s {{Harv|Cartan|1926}}, he made much use of the general idea much earlier. The high point of his remarkable 1910 paper on [[Pfaffian system]]s in five variables is the construction of a Cartan connection modelled on a 5-dimensional homogeneous space for the [[exceptional Lie group]] [[G2 (mathematics)|G<sub>2</sub>]], which he and Engels had discovered independently in 1894.</ref> The main idea is to develop a suitable notion of the [[connection form]]s and [[curvature]] using moving frames adapted to the particular geometrical problem at hand. In relativity or Riemannian geometry, [[orthonormal frame]]s are used to obtain a description of the [[Levi-Civita connection]] as a Cartan connection. For Lie groups, [[Maurer–Cartan form|Maurer–Cartan frame]]s are used to view the [[Maurer–Cartan form]] of the group as a Cartan connection.

Cartan reformulated the differential geometry of ([[pseudo-Riemannian manifold|pseudo]]) [[Riemannian geometry]], as well as the differential geometry of [[manifold]]s equipped with some non-metric structure, including [[Lie group]]s and [[homogeneous space]]s. The term 'Cartan connection' most often refers to Cartan's formulation of a (pseudo-)Riemannian, [[affine connection|affine]], [[projective connection|projective]], or [[conformal connection]]. Although these are the most commonly used Cartan connections, they are special cases of a more general concept.

Cartan's approach seems at first to be coordinate dependent because of the choice of frames it involves. However, it is not, and the notion can be described precisely using the language of principal bundles. Cartan connections induce covariant derivatives and other differential operators on certain associated bundles, hence a notion of parallel transport. They have many applications in geometry and physics: see the [[method of moving frames]], [[Cartan formalism (physics)|Cartan formalism]] and [[Einstein–Cartan theory]] for some examples.