Contents

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 $G$-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).

References

The concept goes back to the work of Eli Cartan (Cartan geometry).

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)

Formalization in homotopy type theory 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 $G_2$-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf

Discussion specifically for kinematical groups: