derived smooth geometry
Klein geometries form the local models for Cartan geometries.
A Klein geometry is a pair where is a Lie group and is a closed Lie subgroup of such that the (left) coset space is connected. acts transitively on the homogeneous space . We may think of as the stabilizer of a point in .
|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|
The notion of Klein geometry goes back to articles such as
in the context of what came to be known as the Erlangen program.
A review is for instance in