nLab
Cartan geometry
Context
Differential geometry
Contents
Idea
Cartan geometry is geometry of spaces that are locally (infinitesimally, tangentially) like coset spaces $G/H$ , i.e. like Klein geometries . Intuitively, Cartan geometry studies the geometry of a manifold by ‘rolling without sliding’ the ‘model geometry’ $G/H$ along it. Hence Cartan geometry may be thought of as the globalization of the program of Klein geometry initiated in the Erlangen program .

Cartan geometry subsumes many types of geometry, such as notably Riemannian geometry , conformal geometry , parabolic geometry and many more. As a Cartan geometry is defined by principal connection data (hence by cocycles in nonabelian differential cohomology ) this means that it serves to express all these kinds of geometries in connection data. This is used notably in the first order formulation of gravity , which was the motivating example in the original text (Cartan 32 ).

Definition
A Cartan geometry is a space equipped with a Cartan connection . See there for more.

Examples
geometric contextgauge group stabilizer subgroup local model space local geometry global geometry differential cohomology first order formulation of gravity differential geometry Lie group /algebraic group $G$ subgroup (monomorphism ) $H \hookrightarrow G$ quotient (“coset space ”) $G/H$ Klein geometry Cartan geometry Cartan connection
examples Euclidean group $Iso(d)$ rotation group $O(d)$ Cartesian space $\mathbb{R}^d$ Euclidean geometry Riemannian geometry affine connection Euclidean gravity
Poincaré group $Iso(d-1,1)$ Lorentz group $O(d-1,1)$ Minkowski spacetime $\mathbb{R}^{d-1,1}$ Lorentzian geometry pseudo-Riemannian geometry spin connection Einstein gravity
anti de Sitter group $O(d-1,2)$ $O(d-1,1)$ anti de Sitter spacetime $AdS^d$ AdS gravity
de Sitter group $O(d,1)$ $O(d-1,1)$ de Sitter spacetime $dS^d$ deSitter gravity
linear algebraic group parabolic subgroup /Borel subgroup flag variety parabolic geometry
conformal group $O(d,t+1)$ conformal parabolic subgroup Möbius space $S^{d,t}$ conformal geometry conformal connection conformal gravity
supergeometry super Lie group $G$ subgroup (monomorphism ) $H \hookrightarrow G$ quotient (“coset space ”) $G/H$ super Klein geometry super Cartan geometry Cartan superconnection
examples super Poincaré group spin group super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$ Lorentzian supergeometry supergeometry superconnection supergravity
super anti de Sitter group super anti de Sitter spacetime
higher differential geometry smooth 2-group $G$ 2-monomorphism $H \to G$ homotopy quotient $G//H$ Klein 2-geometry Cartan 2-geometry
cohesive ∞-group ∞-monomorphism (i.e. any homomorphism ) $H \to G$ homotopy quotient $G//H$ of ∞-action higher Klein geometry higher Cartan geometry higher Cartan connection
examples extended super Minkowski spacetime extended supergeometry higher supergravity : type II , heterotic , 11d

References
The original article is

Élie Cartan Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie) . Annales scientifiques de l’École Normale Supérieure, Sér. 3, 40 (1923), p. 325-412 (NUMDAM )
Textbook accounts are in

R. Sharpe, Differential Geometry – Cartan’s Generalization of Klein’s Erlangen program Springer (1997)

Andreas Cap , Jan Slovák , chapter 1 of Parabolic Geometries I – Background and General Theory , AMS 2009

ps

For more see at Cartan connection – References .

See also