derived smooth geometry
The notion of Klein geometry is essentially that of homogeneous space (coset space) in the context of differential geometry. This is named “Klein geometry” due to its central role in Felix Klein’s Erlangen program, see below at History.
Klein geometries form the local models for Cartan geometries.
See there at Examples – Stabilizers of shapes / Klein geometry.
In (Klein 1872) (the “Erlangen program”) is first of all, in section 1, considered the general idea of (what in modern language one would call) the action of a Lie group “of transformations” on a smooth manifold. The group of all such transformations
by which the geometric properties of configurations in space remain entirely unchanged
is called the Hauptgruppe, principal group.
Then in (Klein 1872, end of section 5) it says:
Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body.
This means in modern language, that if is the given group acting on a given space , and if is a given subspace (a configuration), then the “body” generated by this is the coset of by the stabilizer subgroup of that configuration. See also there at Stabilizer of shapes – Klein geometry.
The text goes on to argue that spaces of this form are of fundamental importance:
If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies.
|gauge group||stabilizer subgroup||local model space||local geometry||global geometry||differential cohomology||first order formulation of gravity|
|general||Lie group/algebraic group||subgroup (monomorphism)||quotient (“coset space”)||Klein geometry||Cartan geometry||Cartan connection|
|examples||Euclidean group||rotation group||Cartesian space||Euclidean geometry||Riemannian geometry||affine connection||Euclidean gravity|
|Poincaré group||Lorentz group||Minkowski space||Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|super Poincaré group||spin group||super Minkowski spacetime||Lorentzian supergeometry||supergeometry||superconnection||supergravity|
|linear algebraic group||parabolic subgroup/Borel subgroup||flag variety||parabolic geometry|
|orthochronous Lorentz group||conformal geometry||conformal connection||conformal gravity|
|general||smooth 2-group||2-monomorphism||homotopy quotient||Klein 2-geometry||Cartan 2-geometry|
|cohesive ∞-group||∞-monomorphism (i.e. any homomorphism)||homotopy quotient of ∞-action||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
Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)
translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)
in the context of what came to be known as the Erlangen program.
A review is for instance in