nLab torsion of a G-structure

Idea

The integrability of G-structures exists to first order, precisely if a certain torsion obstruction vanishes. This is the first in an infinite tower of tensor invariants in Spencer cohomology associated with a $G$-structure that obstruct its integrability (local flatness) (Guillemin 65).

The torsion of a $G$-structure is defined to be the space in which the invariant part of the torsion of a Cartan connection takes values, for any Cartan connection compatible with the $G$-structure (see at Cartan connection – Examples – G-Structure) (Sternberg 64, from p. 317 on, Guillemin 65, section 4), for review see also (Lott 90, p.10, Joyce 00, section 2.6).

The order $k$-torsion of a $G$-structure (counting may differ by 1) is an element in a certain Spencer cohomology group (Guillemin 65, prop. 4.2) and is the obstruction to lifting an order-$k$-integrable G-structure to order $k+1$ (Guillemin 65, theorem 4.1).

Related concepts

• torsion of a Cartan connection

References

General

Historical origin of the notion in Cartan geometry:

• 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, 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, 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]

Textbook accounts include

• Shlomo Sternberg, section VII of Lectures on differential geometry, Prentice Hall 1964; Russian transl. Mir 1970

Discussion including the higher order obstructions in Spencer cohomology to integrability of G-structures is in

• Victor Guillemin, The integrability problem for $G$-structures, Trans. Amer. Math. Soc. 116 (1965), 544–560. (jstor:1994134)

Discussion with an eye towards torsion constraints in supergravity is in

• John Lott, The Geometry of Supergravity Torsion Constraints, Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)

Discussion with an eye towards special holonomy is in

• Dominic Joyce, section 2.6 of Compact manifolds with special holonomy, Oxford University Press 2000

Further mentioning of the higher order torsion invariants includes

• Robert Bryant, section 4.2 of Some remarks on $G_2$-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf

Discussion specifically for kinematical groups:

• José Figueroa-O'Farrill, On the intrinsic torsion of spacetime structures (arXiv:2009.01948)

See also

• Wikipedia, Torsion of a G-structure

Formalization in homotopy type theory (so far only for infinite-order torsion):

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory (2017)

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]

• 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]

• 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]