nLab Cartan geometry

Idea

“In the wake of the movement of ideas which followed the general theory of relativity, I was led to introduce the notion of new geometries, more general than Riemannian geometry, and playing with respect to the different Klein geometries the same role as the Riemannian geometries play with respect to Euclidean space. The vast synthesis that I realized in this way depends of course on the ideas of Klein formulated in his celebrated Erlangen programme while at the same time going far beyond it since it includes Riemannian geometry, which had formed a completely isolated branch of geometry, within the compass of a very general scheme in which the notion of group still plays a fundamental role.”

[Élie Cartan 1939, as quoted in Sharpe 1997, p. 171]

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

Example

(Riemann-Cartan geometry of $S^3$) We discuss the Riemannian geometry of the round 3-sphere $S^3$ as a $ISO(3)/O(3)$-Cartan geometry, hence via frame field and spin connection with vanishing torsion tensor (as one encounters it in the first-order formulation of gravity).

Among all $n$-spheres, this is particularly easy for the $S^3$, since its underlying smooth manifold is that of the Lie group $SU(2)$. This means that the frame field can be chosen to be globally defined

$$\big( e^i \big)_{i = 1} \,\in\, \Omega^1_{dR}\big( S^3 ;\, \mathbb{R}^3 \big)$$

and to be given by left-invariant differential forms that satisfy the Maurer-Cartan equation

$$\mathrm{d} e^i \;=\; \epsilon^{i j k } \, e_j \, e_k \,,$$

where $\epsilon^{i j k}$ is the Levi-Civita symbol in 3 dimensions, and we use the Einstein summation convention throughout.

In next defining the “spin connection” form

$$\big( \omega^{i j} = - \omega^{j i} \big)_{i,j = 1}^3 \,\in\, \Omega^1_{dR}\big(S^3;\, \mathfrak{so}(3)\big)$$

we shall use the convention where the torsion-tensor $\big(T^i\big)_{i = 1}^3$ and curvature-tensor $\big(R^{i j}\big)_{i,j=1}^3$ are formed with a relative minus sign as

(1)$$\begin{aligned} T^i & \coloneqq\; \mathrm{d} \, e^i - \omega^{i j} \, e_j \,, \\ R^{i j} & \coloneqq\; \mathrm{d}\, \omega^{i j} - \omega^{i k}\, \omega_k{}^j \,. \end{aligned}$$

This implies that the unique torsion-free spin connection is given by

$$\omega^{i j} \;\equiv\; - \epsilon^{i j k} e_k \,,$$

since, by the above:

$$\begin{array}{l} \mathrm{d} e^i - \omega^{i j} e_j \\ \;=\; \epsilon^{i j k } e_j e_k + \epsilon^{i j k} e_k\, e_j \\ \;=\; 0 \,. \end{array}$$

To find the curvature of this connection we compute

$$\begin{array}{l} \mathrm{d} \omega^{i j} \\ \;=\; - \epsilon^{i j k} \epsilon_{k l m} e^l \, e^m \\ \;=\; -2 \delta^{ i j }_{l m} e^{l}\, e^m \\ \;=\; -2 e^{i}\, e^{j} \,, \end{array}$$

where we take the multi-Kronecker symbol is normalized as

$$\delta^{i j}_{k l} \;\coloneqq\; \delta^{[i j]}_{[k l]} \;=\; \delta^{[i j]}_{k l} \;=\; \tfrac{1}{2} \Big( \delta^i_k\, \delta^j_l - \delta^j_k\, \delta^i_l \Big) \,,$$

and

$$\begin{array}{l} \omega^{i k} \, \omega_{k}{}^j \\ \;=\; \epsilon^{i k l} \epsilon_{k j m} e_l \, e^m \\ \;=\; -2\delta^{i l}_{j m} e_l \, e^m \\ \;=\; - e^i \, e^j \end{array}$$

to obtain:

$$\begin{array}{l} R^{i j} \\ \;=\; \mathrm{d}\omega^{i j} - \omega^{i k}\, \omega_k{}^j \\ \;=\; -2 e^{i}\, e^{j} + e^i \, e^j \\ \;=\; - e^{i}\, e^{j} \,, \end{array}$$

hence

$$R^{i j}{}_{k l} \;=\; - \delta^{i j}_{k l} \,.$$

Notice that, therefore, with the conventions (1) the scalar curvature $\mathrm{R}$ of the $S^3$ comes out with a negative sign, since the Ricci tensor of $S^3$ is

$$Ric_{i j} \;\equiv\; R^{i k}{}_{j k} \;=\; - \delta_{i j}$$

so that

$$\mathrm{R} \;\equiv\; Ric^i{}_i \;=\; - 3 \,.$$

This may seem undesirable, but it is a common choice in practice (cf. e.g. Freund & Rubin 1980, below (4b)).

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 re-formulation of Riemannian geometry via coframe fields:

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

• 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) 325-412 [doi:ASENS_1923_3_40__325_0]

• Élie Cartan, La géométrie des espaces de Riemann, Mémorial des sciences mathématiques 9 (1925) [numdam:MSM_1925__9__1_0]

and English translation of original lectures:

• Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]

• Élie Cartan (translated by Robert Hermann from Cartan’s lectures in 1951): Geometry of Riemannian Spaces, Lie Groups: History, Frontiers and Applications XIII, Math Sci Press (1983) [ark:/13960/s28rzmj9xrv]

Historical review:

• Erhard Scholz, E. Cartan’s attempt at bridge-building between Einstein and the Cosserats – or how translational curvature became to be known as “torsion”, The European Physics Journal H 44 (2019) 47-75 [doi:10.1140/epjh/e2018-90059-x]

Introductions:

• Richard W. Sharpe, An introduction to Cartan Geometries, Proceedings of the 21st Winter School “Geometry and Physics”, Circolo Matematico di Palermo (2002) 61-75 [eudml:220395, dml:701688]

• Benjamin McKay, An introduction to Cartan geometries, Proceedings of the 21st Winter School “Geometry and Physics”, Palermo (2002) [arXiv:2302.14457, eudml:220395]

• Raphaël Alexandre, Elisha Falbel, Introduction to Cartan geometry (2023) [pdf, pdf]

Monographs:

• Richard W. Sharpe, Differential geometry – Cartan’s generalization of Klein’s Erlagen program, Graduate Texts in Mathematics 166, Springer (1997) [ISBN:9780387947327]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Chapter I.3.7 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.3: pdf]

  (these authors refer to Cartan geometries as “soft group manifolds”)

• 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.