Appendix - G–structures on manifolds
Published online by Cambridge University Press: 05 January 2015
Summary
In this appendix we collect some basic definitions and results on G–structures on manifolds. No proofs are provided. Details may be found in [214].
M is a smooth manifold of dimension n and L(M) the bundle of linear frames over M. L(M) is a principal bundle on M with group GL(n, ℝ). G is a (closed) Lie subgroup of GL(n, ℝ).
Definition A.1 A G–structure on M is a smooth subbundle P of L(M) with structure group G. A G–structure P on M is said to be integrable if every point of M has a neighborhood U with local coordinates x1, …, xn such that the local section (∂/∂x1, …, ∂/∂ xn) of L(M) over U is a local section of P. In this case we say that the local coordinate system x1, …, xn is admissible for the G–structure P.
If xi and yi are two admissible coordinate systems in the charts U and V the Jacobian matrix ∂yi/∂xj is in G at all points in U ∩ V.
G is a closed subgroup of GL(n, ℝ). The embedding in GL(n, ℝ) defines a representation ρ of G in ℝn. Let be the rank n vector bundle (with structure group G) associated to the principal bundle P through ρ. The definition of the G–structure is equivalent to the statement that we have an isomorphism of vector bundles
θ : TM → ρ(P), (A.1)
which reduces the structure group from GL(n, ℝ) to its subgroup G. Conversely, any isomorphism of TM with a rank n vector bundle V with structure group G defines a G–structure.
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- Supersymmetric Field TheoriesGeometric Structures and Dualities, pp. 390 - 393Publisher: Cambridge University PressPrint publication year: 2015