# nLab Cartan structural equations

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 Riemann-Cartan differential geometry, what are called Cartan’s structural equations (équations de structure Cartan 1923, p. 368, see Scholz 2019, p. 53) are expressions for the torsion $T$ and the curvature $R$ of a Cartan moving frame $e$ with (Cartan-)connection $\omega$ via the exterior derivative and wedge product of their differential form-representatives (shown as usual in components on any local chart with respect to a trivialized fiber bundles and using Einstein summation convention):

(1)$\begin{array}{ccccc} T^a &=& \mathrm{d}e^a &+& \omega^a{}_b \wedge e^b \,, \\ R^{ab} &=& \mathrm{d} \omega^{a b} &+& \omega^{a}{}_c \wedge \omega^{c b} \,. \end{array}$

In the historically motivating case relating to the description of the field of gravity in what is now called first-order formulation, the representatives of the frame field and connection

$\begin{array}{rcl} \big(e^a\big)_{a = 0}^{d} &\in& \Omega^1_{dR}\big(-; \mathbb{R}^{1+d}\big) \,, \\ \big( \omega^{a b} \,=\, -\omega^{b a} \big)_{a,b \in 0}^d &\in& \Omega^1_{dR}\big(-; \mathfrak{so}(1,d)\big) \end{array}$

are differential 1-forms which may jointly be understood as taking values in the Poincaré Lie algebra of a given dimension, and the two structural equations (1) jointly express the total curvature 2-form of this connection, broken up into its components.

## References

### Cartan structural equations and Bianchi identities

On Cartan structural equations and their Bianchi identities for curvature and torsion of Cartan moving frames and (Cartan-)connections on tangent bundles (especially in first-order formulation of gravity):

The original account:

• 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]

Historical review:

• Erhard Scholz, §2 in: 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]

Further discussion:

Generalization to supergeometry (motivated by supergravity):

Last revised on June 26, 2024 at 09:06:24. See the history of this page for a list of all contributions to it.