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.
See Cartan connection.
|local model||global geometry|
|Klein geometry||Cartan geometry|
|Klein 2-geometry||Cartan 2-geometry|
|higher Klein geometry||higher Cartan geometry|
A standard textbook is
wikipedia: Cartan connection