An intuitive definition of G structures

A quite intuitive definition and description of G-structures can be given in terms of the tangent frame bundle of the concerned manifold M.1 _{Let us therefore remind the reader}

of the concept of a bundle first. A generic bundle E consists of a base B and a fibre F

such that locally it looks like the direct product of B and F. However, globally it does not

need to have a direct product structure. This is taken into account by transition functions that determine how the fiber transforms if one goes from one patch on the base to another patch. Only when all transition functions equal the identity, one obtains globally the direct product manifold B×F. Moreover, the bundle comes with a smooth projection π to the base. Assigning an element of the fiber to every point of the base, defines a section s of the bundle. The map s should of course satisfy the relation π(s(p)) = p for every point

p∈B.

A very simple example for a trivial bundle is the cylinder. There, the base space is taken to be the circle S1 _{and the fiber is a finite line. A section of this bundle is given}

by a closed line that winds exactly once around the cylinder. A well known non-trivial generalization of this is the moebius strip. Here, the base and the fiber are the same as for the cylinder, but one has two patches whose transition function is an inversion of the line, resulting in the non-orientedness of the moebius strip. But still, a section is given by a closed line that winds once around the strip like in the case of the cylinder.

Another prominent class of examples for bundles, which is more important for our discussion, is provided by vector bundles. Here, the fiber is a vector space and a section of the bundle is given by a vector field. In particular, to every manifoldM one can associate its tangent vector bundle (or tangent bundle, for short) T M. The fiber of this bundle at a point p ∈ M is given by the vector space TpM, that contains all vectors tangent to the point p. A basis for this space is given by the partial derivatives in every direction

∂A. Clearly, for a d-dimensional manifold TpM is a d-dimensional vector space and the transition functions of the fiber are given by elements of the group GL(d,_R). A section assigns to every point p one vector of the space TpM and gives hence a vector field over

M.

The tangent frame bundle associated to T M can then be defined as the bundle whose fiber for a given point p∈M is the set of ordered bases of the vector spaceTpM. Locally, as we explained above, the bundle looks like the direct product (p, eA) with eA = eAA∂A a set of d independent vectors forming a base of TpM. Note that the matrix eAA is only restricted by the condition that the eA form a base of TpM. Hence, eA should really be understood as the set of all (ordered) bases.

What can one say about the transition functions of the tangent frame bundle between two different overlapping patchesUα andUβ with coordinatesxAand x0A, respectively? In the overlap region one can represent eA in terms of both coordinates

eA = (eα)AA∂A = (eβ)AA∂ 0 A = (eβ)BA ∂ xA ∂ x0B ∂A = (eβ) B A(tβα)BA∂A . (3.1.1) Moreover, the transition functions tβα must satisfy the consistency condition

tαβtβα = 1 , (3.1.2)

and the transitivity relation on an overlap region of three patchesUα, Uβ, Uγ

All these requirements provide a group structure for the transition functions tαβ. This group is called the structure group of M. In general it will be the group of general lin- ear transformations in d dimensions GL(d,_R), and the same group as for the transition functions of the tangent bundleT M.

After this preliminary work, it is quite easy to define G-structure. A manifold is said to be aG-structure manifold if its structure group can be reduced to the groupG. Differently put: a manifold will provide a G-structure when the transition functions of the tangent frame bundle belong to the groupG. But since every vector l from the tangent bundle can be decomposed as l = lA_e

A this means that the vectors of the tangent bundle transform under the group G, too. This in turn implies that also one-forms and generic tensors transform under G when one considers a G-structure manifold.

The next question is then of course what the structure group of a given manifold is. Again, one can find the answer by considering tensors and not the full tangent frame bundle. As a matter of fact a tensor (or a spinor) that is globally well defined (and non-degenerate) onM will reduce the structure group. The simplest way to imagine this is to assume that one has already found as structure group the d dimensional rotations. If one finds now in addition a nowhere vanishing vector l the group will reduce to the (d−1)-dimensional rotations, since one can find frames such thatl points always in the same direction.

Another well known case with reduced structure group are Riemannian manifolds, e.g. manifolds that admit a metric g. The structure group is then reduced to the orthogonal transformations, as the metric fixes the length of vectors in all patches. Also important is the case where one can define an almost complex structure on the manifold M. This is a map J : T M → T M, that squares to minus one: J2 _{= −1. This means that J has}

eigenvalues of +i and −i and hence the structure group will reduce to GL(d/2,_C). If one has both, a metric g and an almost complex structure J, that satisfy J gJ =g one speaks of a hermitian metric. The structure group is then given by the unitary group U(d/2). In this case one can also define a two-form J as

Jij = gikJkj . (3.1.4)

This two-form is called pre-symplectic structure2 and it would reduce the structure group to the symplectic group Sp(d,_R) if considered solely.