G-structure on a manifold

In differential geometry, a G-structure on an n-manifold M, for a given structure group 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 example, for the orthogonal group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the same as a volume form. For the trivial group, an {e}-structure consists of an absolute parallelism of the manifold.

Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal <math>G</math>-bundle over a group <math>G</math> "comes from" a subgroup <math>H</math> of <math>G</math>. This is called reduction of the structure group (to <math>H</math>).

Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are G-structures with an additional integrability condition.

Reduction of the structure group

One can ask if a principal <math>G</math>-bundle over a group <math>G</math> "comes from" a subgroup <math>H</math> of <math>G</math>. This is called reduction of the structure group (to <math>H</math>), and makes sense for any map <math>H \to G</math>, which need not be an inclusion map (despite the terminology).

Definition

In the following, let <math>X</math> be a topological space, <math>G, H</math> topological groups and a group homomorphism <math>\phi\colon H \to G</math>.

In terms of concrete bundles

Given a principal <math>G</math>-bundle <math>P</math> over <math>X</math>, a reduction of the structure group (from <math>G</math> to <math>H</math>) is a <math>H</math>-bundle <math>Q</math> and an isomorphism <math>\phi_Q\colon Q \times_H G \to P</math> of the associated bundle to the original bundle.

In terms of classifying spaces

Given a map <math>\pi\colon X \to BG</math>, where <math>BG</math> is the classifying space for <math>G</math>-bundles, a reduction of the structure group is a map <math>\pi_Q\colon X \to BH</math> and a homotopy <math>\phi_Q\colon B\phi \circ \pi_Q \to \pi</math>.

Properties and examples

Reductions of the structure group do not always exist. If they exist, they are usually not essentially unique, since the isomorphism <math>\phi</math> 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 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 BH = σ−1B is a reduction of B.

G-structures

Every vector bundle of dimension <math>n</math> has a canonical <math>GL(n)</math>-bundle, the frame bundle. In particular, every smooth manifold has a canonical vector bundle, the tangent bundle. For a Lie group <math>G</math> and a group homomorphism <math>\phi\colon G \to GL(n)</math>, a <math>G</math>-structure is a reduction of the structure group of the frame bundle to <math>G</math>.

Examples

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

{| class="wikitable"

!Group homomorphism

!Group <math>G</math>

!<math>G</math>-structure

!Obstruction

|<math>GL^+(n) < GL(n)</math>

|General linear group of positive determinant

|Orientation

|Bundle must be orientable

|<math>SL(n) < GL(n)</math>

|Special linear group

|Volume form

|Bundle must be orientable (<math>SL \to GL^+</math> is a deformation retract)

|<math>SL^{\pm}(n) < GL(n)</math>

|Determinant <math>\pm 1</math>

|Pseudo-volume form

|Always possible

|<math>O(n) < GL(n)</math>

|Orthogonal group

|Riemannian metric

|Always possible (<math>O(n)</math> is the maximal compact subgroup, so the inclusion is a deformation retract)

|<math>O(p,n-p) < GL(n)</math>

|Indefinite orthogonal group

|Pseudo-Riemannian metric

|Topological obstruction

|<math>GL(n,\mathbf{C}) < GL(2n,\mathbf{R})</math>

|Complex general linear group

|Almost complex structure

|Topological obstruction

|<math>GL(n,\mathbf{H})\cdot Sp(1) < GL(4n,\mathbf{R})</math>

<math>GL(n,\mathbf{H})</math>: quaternionic general linear group acting on <math>\mathbf{H}^n \cong \mathbf{R}^{4n}</math> from the left
<math>Sp(1)=Spin(3)</math>: group of unit quaternions acting on <math>\mathbf{H}^n</math> from the right

|Almost quaternionic structure

|Topological obstruction 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. For example, a pair of Riemannian manifolds are (locally) equivalent if and only if their bundles of orthonormal frames are (locally) isomorphic 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.

Connections on G-structures

Let Q be a G-structure on M. A principal connection on the principal bundle Q induces a connection on any associated vector bundle: in particular on the tangent bundle. A linear connection ∇ on TM arising in this way is said to be compatible with Q. Connections compatible with Q are also called adapted connections.

Concretely speaking, adapted connections can be understood in terms of a moving frame. Suppose that Vi is a basis of local sections of TM (i.e., a frame on M) which defines a section of Q. Any connection ∇ determines a system of basis-dependent 1-forms ω via

:∇X Vi = ωij(X)Vj

where, as a matrix of 1-forms, ω ∈ Ω1(M)&otimes;gl(n). An adapted connection is one for which ω takes its values in the Lie algebra g of G.

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.

The difference of two adapted connections is a 1-form on M with values in the adjoint bundle AdQ. That is to say, the space AQ of adapted connections is an affine space

for Ω1(AdQ).

The torsion of an adapted connection defines a map

:<math>A^Q \to \Omega^2 (TM)\,</math>

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

:<math>\tau:\Omega^1(\mathrm{Ad}_Q)\to \Omega^2(TM)\,</math>

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

:<math>\Omega^2(TM)= \Omega^{2,0}(TM)\oplus \mathrm{im}(\tau)</math>,

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

:<math>B(JX,Y) = B(X, JY) = - J B(X,Y).\,</math>

Therefore, the torsion of an almost complex structure can be considered as an element in

Ω2,0(TM). It is easy to check that the torsion of an almost complex structure is equal to its Nijenhuis tensor.

Higher order G-structures

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 many cases, however, the prolongation is a principal bundle in its own right, and its structure group can be identified with a subgroup of a higher-order jet group. In which case, it is called a higher order G-structure [Kobayashi]. In general, Cartan's equivalence method applies to such cases.