The notion of Cartan connection is a special case of that of connection on a bundle which is more general than the notion of affine connection , but more special than the notion of principal connection , in general:
it is an -principal connection subject to the constraint that the connection 1-form linearly identifies each tangent space of the base space with a quotient of the Lie algebra of by a sub-Lie algebra .
Cartan connections are also just called Cartan geometries .
A -Cartan connection over a smooth manifold is;
This appears for instance as (Sharpe, section 5.1).
is Lorentzian spacetime. Therefore an -Cartan connection is equivalently an -connection on a manifold whose tangent spaces look like Minkowski spacetime: this is equivalently a pseudo-Riemannian manifold from the perspective discussed at first-order formulation of gravity:
the -valued part of the connection is the vielbein.
|local model space||global geometry||differential cohomology||first order formulation of gravity|
|general||Klein geometry||Cartan geometry||Cartan connection|
|examples||Euclidean geometry||Riemannian geometry||affine connection||Euclidean gravity|
|Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|general||Klein 2-geometry||Cartan 2-geometry|
|higher Klein geometry||higher Cartan geometry||higher Cartan connection|
|examples||extended super Minkowski spacetime||extended supergeometry||higher supergravity: type II, heterotic, 11d|
A standard textbook reference is