nLab Cartan geometry

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Riemannian geometry

Riemannian geometry

Contents

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 contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)$rotation group $O(d)$Cartesian space $\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d-1,1)$Lorentz group $O(d-1,1)$Minkowski spacetime $\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein 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 groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group $O(d,t+1)$conformal parabolic subgroupMöbius space $S^{d,t}$conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$super Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group $G$2-monomorphism $H \to G$homotopy quotient $G//H$Klein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) $H \to G$homotopy quotient $G//H$ of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

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:

Monographs:

For more see at Cartan connection – References.

Discussion in modal homotopy type theory is in