Appendix - G–structures on manifolds

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/∂x^j 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.