nLab
Klein geometry

Context

Geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

The notion of Klein geometry is essentially that of homogeneous space (coset space) G/HG/H 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.

For the generalization of Klein geometry to higher category theory see higher Klein geometry.

Definition

A Klein geometry is a pair (G,H)(G, H) where GG is a Lie group and HH is a closed Lie subgroup of GG such that the (left) coset space

XG/H X \coloneqq G/H

is connected. GG acts transitively on the homogeneous space XX. We may think of HGH\hookrightarrow G as the stabilizer subgroup of a point in XX.

See there at Examples – Stabilizers of shapes / Klein geometry.

History

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 GG is the given group acting on a given space XX, and if SXS \hookrightarrow X is a given subspace (a configuration), then the “body” generated by this is the coset G/Stab G(S)G/Stab_G(S) of GG by the stabilizer subgroup Stab G(X)Stab_G(X) of that configuration. See also there at Stabilizer of shapes – Klein geometry.

The text goes on to argue that spaces of this form G/Stab G(S)G/Stab_G(S) 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.

Examples

gauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
generalLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group SO(d)SO(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz groupMinkowski space d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
orthochronous Lorentz groupconformal geometryconformal connectionconformal gravity
generalsmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

References

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

  • Vladimir Kisil, Erlangen Programme at Large: An Overview (arXiv:1106.1686)

Revised on December 18, 2014 18:35:33 by Anonymous Coward (127.0.0.1)