nLab Cartan structural equations

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.

Related concepts

• Bianchi identities

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:

• Shiing-Shen Chern, p. 748 of: A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals of Mathematics, Second Series, 45 4 (1944) 747-752 [doi:10.2307/1969302]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §I.2 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.2: pdf]

• C. C. Briggs, A Sequence of Generalizations of Cartan’s Conservation of Torsion Theorem [arXiv:gr-qc/9908034]

• Loring Tu, §22 in: Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer (2017) [ISBN:978-3-319-55082-4]

• Thoan Do, Geoff Prince, An intrinsic and exterior form of the Bianchi identities, International Journal of Geometric Methods in Modern Physics 14 01 (2017) 1750001 [doi:10.1142/S0219887817500013, arXiv:1501.01123]

• Ivo Terek Couto, Cartan Formalism and some computations [pdf, pdf]

• Cadabra, Cartan structural equations and Bianchi identity

Generalization to supergeometry (motivated by supergravity):

• Julius Wess, Bruno Zumino, p. 362 of: Superspace formulation of supergravity, Phys. Lett. B 66 (1977) 361-364 [doi:10.1016/0370-2693(77)90015-6]

• Richard Grimm, Julius Wess, Bruno Zumino, §2 in: A complete solution of the Bianchi identities in superspace with supergravity constraints, Nuclear Phys. B 152 (1979) 255-265 [doi:10.1016/0550-3213(79)90102-0]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §III.3.2 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch III.3: pdf]