synthetic differential geometry
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Cartan geometry (super, higher)
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 and the curvature of a Cartan moving frame with (Cartan-)connection 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):
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
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.
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:
Historical review:
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]
Sigurdur Helgason, §I.8 in: Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]
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]
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]
Last revised on June 26, 2024 at 09:06:24. See the history of this page for a list of all contributions to it.