nLab Cartan geometry

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 22). The physics literature tends to use the term “Cartan moving frame method” instead of “Cartan geometry”.

Definition

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

Examples

geometric context	gauge 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

Related entries

• super-Cartan geometry

• Cartan geometry, Supergravity and Branes

• G-structure, homothety

• first-order formulation of gravity

• torsion constraints in supergravity

• parabolic geometry

• tractor bundle

• higher Cartan geometry

References

The original articles are

• Élie Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces á torsion, C. R. Acad. Sci. 174, 593-595 (1922).

• Élie Cartan, Comptes rendus hebdomadaires des séances de l’Académie des sciences, 174, 437-439, 593-595, 734-737, 857-860, 1104-1107 (January 1922).

• 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 Čap, Jan Slovák, chapter 1 of Parabolic Geometries I – Background and General Theory, AMS 2009 (ISBN:978-1-4704-1381-1)

• ps

For more see at Cartan connection – References.

Discussion in modal homotopy type theory is in

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory, 2017

• Felix Wellen, Cartan Geometry in Modal Homotopy Type Theory (arXiv:1806.05966)

See also

• wikipedia: Cartan connection

• The blog discussion of Derek Wise, MacDowell-Mansouri gravity and Cartan geometry.