G-structure

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In differential geometry, a G-structure on a n-manifold M, for a given structure group G (which is a Lie subgroup of the general linear group GL(n,R)) is a G-subbundle of the frame bundle FM (or GL(M)) of M.

The notion of G-structures, introduced by Chern, includes many other structures on manifolds, some of them being defined by 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.

On the other hand a symplectic manifold structure is a stronger concept than a G-structure for the symplectic group; the latter would correspond to specifying a two-form ω on M that is non-degenerate, but the former to the extra condition (an integrability condition) that dω = 0. Subgroups of GL(n,R) defined as types of block matrix define a range of G-structures that include foliations. Complex matrix groups define almost complex manifolds.

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.

1 Principal bundles and G-structures
2 Connections on G-structures and torsion
- 2.1 Torsion for almost complex structures
3 Higher order G-structures
4 See also
5 References

[edit] Principal bundles and G-structures

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 frame bundle, but the fact that the G-structure bundle consists of frames is regarded as part of the data. For example, consider two Riemannian metrics on Rⁿ. The associated SO(n)-structures are isomorphic if and only if the metrics are isometric. But, since Rⁿ is contractible, the underlying SO(n)-bundles are always going to be isomorphic as principal 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. Although this is not a connection form, it can sometimes be regarded as a precursor to one.

[edit] Connections on G-structures and torsion

Let Q be a G-structure on M. A connection on TM is said to be compatible with Q if it is induced from a principal connection on Q. Connections compatible with Q are also called adapted connections.

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

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

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

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

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.

[edit] 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

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

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

$B(JX,Y) = B(X, JY) = - J B(X,Y).\,$

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.

[edit] 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.