nLab torsion of a G-structure

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Cohomology

cohomology

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Contents

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 GG-structure that obstruct its integrability (local flatness) (Guillemin 65).

The torsion of a GG-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 GG-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 kk-torsion of a GG-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-kk-integrable G-structure to order k+1k+1 (Guillemin 65, theorem 4.1).

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

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 2G_2-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf

Discussion specifically for kinematical groups:

See also

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

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 March 17, 2024 at 10:58:51. See the history of this page for a list of all contributions to it.