nLab
Cartan geometry

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

Cartan geometry is geometry of spaces that are locally (infinitesimally, tangentially) like coset spaces G/HG/H, i.e. like Klein geometries. Intuitively, Cartan geometry studies the geometry of a manifold by ‘rolling without sliding’ the ‘model geometry’ G/HG/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

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 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 Erlagen 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

Revised on December 19, 2014 11:20:08 by Urs Schreiber (127.0.0.1)