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 in the context of differential geometry.

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 X=G/HX = G/H is connected. GG acts transitively on the homogeneous space XX. We may think of HH as the stabilizer of a point in XX.

Examples

local model spaceglobal geometrydifferential cohomologyfirst order formulation of gravity
generalKlein geometryCartan geometryCartan connection
examplesEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
Lorentzian supergeometysupergeometrysuperconnectionsupergravity
generalKlein 2-geometryCartan 2-geometry
higher 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 articles such as

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 September 10, 2013 11:52:04 by Urs Schreiber (82.113.99.141)