Cartan geometry is a common generalization of Riemannian geometry, conformal geometry and Klein geometry, which generalizes the linear tangent spaces of the former to the more general homogeneous spaces of the latter.
Intuitively, Cartan geometry studies the geometry of a manifold by ‘rolling’ another manifold, the ‘model geometry’ on it. The model geometry may be any Klein geometry.
See Cartan connection.
A standard textbook is
See also
wikipedia: Cartan connection
The blog discussion of Derek Wise, MacDowell-Mansouri gravity and Cartan geometry.