Definition

In the following, let ${\displaystyle X}$ be a topological space, ${\displaystyle G,H}$ topological groups and a group homomorphism ${\displaystyle \phi \colon H\to G}$.

Properties and examples

Reductions of the structure group do not always exist. If they exist, they are usually not essentially unique, since the isomorphism ${\displaystyle \phi }$ is an important part of the data.

As a concrete example, every even-dimensional real vector space is isomorphic to the underlying real space of a complex vector space: it admits a linear complex structure. A real vector bundle admits an almost complex structure if and only if it is isomorphic to the underlying real bundle of a complex vector bundle. 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.

More abstractly, "G-bundles over X" is a functor^[2] in G: Given a Lie group homomorphism H → G, one gets a map from H-bundles to G-bundles by inducing (as above). 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. For example, not every manifold is orientable, and those that are orientable admit exactly two orientations.

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 B_H = σ⁻¹B is a reduction of B.^[3]

Examples

The following examples are defined for real vector bundles, particularly the tangent bundle of a smooth manifold.

Group homomorphism	Group ${\displaystyle G}$	${\displaystyle G}$-structure	Obstruction
${\displaystyle GL^{+}(n)<GL(n)}$	General linear group of positive determinant	Orientation	Bundle must be orientable
${\displaystyle SL(n)<GL(n)}$	Special linear group	Volume form	Bundle must be orientable (${\displaystyle SL\to GL^{+}}$ is a deformation retract)
${\displaystyle SL^{\pm }(n)<GL(n)}$	Determinant ${\displaystyle \pm 1}$	Pseudo-volume form	Always possible
${\displaystyle O(n)<GL(n)}$	Orthogonal group	Riemannian metric	Always possible (${\displaystyle O(n)}$ is the maximal compact subgroup, so the inclusion is a deformation retract)
${\displaystyle O(1,n-1)<GL(n)}$	Indefinite orthogonal group	Pseudo-Riemannian metric	Topological obstruction[4]
${\displaystyle GL(n,\mathbf {C} )<GL(2n,\mathbf {R} )}$	Complex general linear group	Almost complex structure	Topological obstruction
${\displaystyle GL(n,\mathbf {H} )\cdot Sp(1)<GL(4n,\mathbf {R} )}$	${\displaystyle GL(n,\mathbf {H} )}$: quaternionic general linear group acting on ${\displaystyle \mathbf {H} ^{n}\cong \mathbf {R} ^{4n}}$ from the left ${\displaystyle Sp(1)=Spin(3)}$: group of unit quaternions acting on ${\displaystyle \mathbf {H} ^{n}}$ from the right	almost quaternionic structure[5]	Topological obstruction[5]
${\displaystyle GL(k)\times GL(n-k)<GL(n)}$	General linear group	Decomposition as a Whitney sum (direct sum) of sub-bundles of rank ${\displaystyle k}$ and ${\displaystyle n-k}$.	Topological obstruction

Some ${\displaystyle G}$-structures are defined in terms of others: Given a Riemannian metric on an oriented manifold, a ${\displaystyle G}$-structure for the 2-fold cover ${\displaystyle {\mbox{Spin}}(n)\to {\mbox{SO}}(n)}$ is a spin structure. (Note that the group homomorphism here is not an inclusion.)

Principal bundles

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. For example, consider two Riemannian metrics on Rⁿ. The associated O(n)-structures are isomorphic if and only if the metrics are isometric. But, since Rⁿ is contractible, the underlying O(n)-bundles are always going to be isomorphic as principal bundles because the only bundles over contractible spaces are trivial bundles.

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. The solder form is what ties the underlying principal bundle of the G-structure to the local geometry of the manifold itself by specifying a canonical isomorphism of the tangent bundle of M to an associated vector bundle. Although the solder form is not a connection form, it can sometimes be regarded as a precursor to one.

In detail, suppose that Q is the principal bundle of a G-structure. If Q is realized as a reduction of the frame bundle of M, then the solder form is given by the pullback of the tautological form of the frame bundle along the inclusion. Abstractly, if one regards Q as a principal bundle independently of its realization as a reduction of the frame bundle, then the solder form consists of a representation ρ of G on Rⁿ and an isomorphism of bundles θ : TM → Q ×_ρ Rⁿ.

Torsion of a G-structure

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 Ad_Q. That is to say, the space A^Q of adapted connections is an affine space for Ω¹(Ad_Q).

The torsion of an adapted connection defines a map

${\displaystyle A^{Q}\to \Omega ^{2}(TM)\,}$

to 2-forms with coefficients in TM. This map is linear; its linearization

${\displaystyle \tau$ :\Omega ^{1}(\mathrm {Ad} _{Q})\to \Omega ^{2}(TM)\,}

is called the algebraic torsion map. Given two adapted connections ∇ and ∇′, their torsion tensors T_∇, T_∇′ differ by τ(∇−∇′). Therefore, the image of T_∇ in coker(τ) is independent from the choice of ∇.

The image of T_∇ in coker(τ) for any adapted connection ∇ is called the torsion of the G-structure. A G-structure is said to be torsion-free if its torsion vanishes. This happens precisely when Q admits a torsion-free adapted connection.

Example: Torsion for almost complex structures

An example of a G-structure is an almost complex structure, that is, a reduction of a structure group of an even-dimensional manifold to GL(n,C). Such a reduction is uniquely determined by a C^∞-linear endomorphism J ∈ End(TM) such that J² = −1. In this situation, the torsion can be computed explicitly as follows.

An easy dimension count shows that

${\displaystyle \Omega ^{2}(TM)=\Omega ^{2,0}(TM)\oplus \mathrm {im} (\tau )}$,

where Ω^2,0(TM) is a space of forms B ∈ Ω²(TM) which satisfy

${\displaystyle B(JX,Y)=B(X,JY)=-JB(X,Y).\,}$

Therefore, the torsion of an almost complex structure can be considered as an element in Ω^2,0(TM). It is easy to check that the torsion of an almost complex structure is equal to its Nijenhuis tensor.