In differential geometry, a G-structure on an n-manifold M, for a given structure group[1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.
For the trivial group, an {e}-structure consists of an absolute parallelism of the manifold.
This is then a reduction along the inclusion GL(n,C) → GL(2n,R) In terms of transition maps, a G-bundle can be reduced if and only if the transition maps can be taken to have values in H. Note that the term reduction is misleading: it suggests that H is a subgroup of G, which is often the case, but need not be (for example for spin structures): it's properly called a lifting.
Reduction of the structure group of a G-bundle B is choosing an H-bundle whose image is B.
The inducing map from H-bundles to G-bundles is in general neither onto nor one-to-one, so the structure group cannot always be reduced, and when it can, this reduction need not be unique.
If H is a closed subgroup of G, then there is a natural one-to-one correspondence between reductions of a G-bundle B to H and global sections of the fiber bundle B/H obtained by quotienting B by the right action of H. Specifically, the fibration B → B/H is a principal H-bundle over B/H.
If σ : X → B/H is a section, then the pullback bundle BH = σ−1B is a reduction of B.
-structures are defined in terms of others: Given a Riemannian metric on an oriented manifold, a
Although the theory of principal bundles plays an important role in the study of G-structures, the two notions are different.
A G-structure is a principal subbundle of the tangent frame bundle, but the fact that the G-structure bundle consists of tangent frames is regarded as part of the data.
This fundamental difference between the two theories can be captured by giving an additional piece of data on the underlying G-bundle of a G-structure: the solder form.
Similarly, foliations correspond to G-structures coming from block matrices, together with integrability conditions so that the Frobenius theorem applies.
The set of diffeomorphisms of M that preserve a G-structure is called the automorphism group of that structure.
For an O(n)-structure they are the group of isometries of the Riemannian metric and for an SL(n,R)-structure volume preserving maps.
Automorphisms arise frequently[6] in the study of transformation groups of geometric structures, since many of the important geometric structures on a manifold can be realized as G-structures.
A wide class of equivalence problems can be formulated in the language of G-structures.
In this view, the general procedure for solving an equivalence problem is to construct a system of invariants for the G-structure which are then sufficient to determine whether a pair of G-structures are locally isomorphic or not.
Concretely speaking, adapted connections can be understood in terms of a moving frame.
An adapted connection is one for which ω takes its values in the Lie algebra g of G. Associated to any G-structure is a notion of torsion, related to the torsion of a connection.
Note that a given G-structure may admit many different compatible connections which in turn can have different torsions, but in spite of this it is possible to give an independent notion of torsion of the G-structure as follows.
[8] The difference of two adapted connections is a 1-form on M with values in the adjoint bundle AdQ.
The torsion of an adapted connection defines a map to 2-forms with coefficients in TM.
Given two adapted connections ∇ and ∇′, their torsion tensors T∇, T∇′ differ by τ(∇−∇′).
The image of T∇ in coker(τ) for any adapted connection ∇ is called the torsion of the G-structure.
Such a reduction is uniquely determined by a C∞-linear endomorphism J ∈ End(TM) such that J2 = −1.
It is easy to check that the torsion of an almost complex structure is equal to its Nijenhuis tensor.
Imposing integrability conditions on a particular G-structure (for instance, with the case of a symplectic form) can be dealt with via the process of prolongation.
In such cases, the prolonged G-structure cannot be identified with a G-subbundle of the bundle of linear frames.
In which case, it is called a higher order G-structure [Kobayashi].
In general, Cartan's equivalence method applies to such cases.