reduction of structure group

Given a fiber bundleMathworldPlanetmath p:EB with typical fiber F and structure group G (henceforth called an (F,G)-bundle over B), we say that the bundle admits a reductionPlanetmathPlanetmathPlanetmath of its structure group to H, where H<G is a subgroupMathworldPlanetmathPlanetmath, if it is isomorphicPlanetmathPlanetmathPlanetmath to an (F,H)-bundle over B.

Equivalently, E admits a reduction of structure group to H if there is a choice of local trivializations covering E such that the transition functionsMathworldPlanetmathPlanetmath all belong to H.

Remark 1

Here, the action of H on F is the restrictionPlanetmathPlanetmath of the G-action; in particular, this means that an (F,H)-bundle is automatically an (F,G)-bundle. The bundle isomorphismPlanetmathPlanetmathPlanetmath in the definition then becomes meaningful in the category of (F,G)-bundles over B.

Example 1

Let H be the trivial subgroup. Then, the existence of a reduction of structure group to H is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the bundle being trivial.

For the following examples, let E be an n-dimensional vector bundleMathworldPlanetmath, so that Fn with G=GL(n,), the general linear groupMathworldPlanetmath acting as usual.

Example 2

Set H=GL+(n,), the subgroup of GL(n,) consisting of matrices with positive determinantMathworldPlanetmath. A reduction to H is equivalent to an orientation of the vector bundle. In the case where B is a smooth manifold and E=TB is its tangent bundle, this coincides with other definitions of an orientation of B.

Example 3

Set H=O(n), the orthogonal groupMathworldPlanetmath. A reduction to H is called a Riemannian or Euclidean structure on the vector bundle. It coincides with a continuousPlanetmathPlanetmath fiberwise choice of a positive definitePlanetmathPlanetmath inner product, and for the case of the tangent bundle, with the usual notion of a Riemannian metric on a manifold.

When B is paracompact, an argument with partitions of unityMathworldPlanetmath shows that a Riemannian structure always exists on any given vector bundle. For this reason, it is often convenient to start out assuming the structure group to be O(n).

Example 4

Let n=2m be even, and let H=GL(m,), the group of invertiblePlanetmathPlanetmathPlanetmath complex matrices, embedded in GL(n,) by means of the usual identification of with 2. A reduction to H is called a complex structure on the vector bundle, and it is equivalent to a continuous fiberwise choice of an endomorphism J satisfying J2=-I.

A complex structure on a tangent bundle is called an almost-complex structure on the manifold. This is to distinguish it from the more restrictive notion of a complex structure on a manifold, which requires the existence of an atlas with charts in m such that the transition functions are holomorphic.

Example 5

Let H=GL(1,)×GL(n-1,), embedded in GL(n,) by (A,B)AB. A reduction to H is equivalent to the existence of a splitting EE1E2, where E1 is a line bundle. More generally, a reduction to GL(k,)×GL(n-k,) is equivalent to a splitting EE1E2, where E1 is a k-plane bundle.

Remark 2

These examples all have two features in common, namely:

  • the subgroup H can be interpreted as being precisely the subgroup of G which preserves a particular structureMathworldPlanetmath, and,

  • a reduction to H is equivalent to a continuous fiber-by-fiber choice of a structure of the same kind.

For example, O(n) is the subgroup of GL(n,) which preserves the standard inner product of n, and reduction of structure to O(n) is equivalent to a fiberwise choice of inner products.

This is not a coincidence. The intuition behind this is as follows. There is no obstacle to choosing a fiberwise inner product in a neighborhood of any given point xB: we simply choose a neighborhood U on which the bundle is trivial, and with respect to a trivialization p-1(U)n×U, we can let the inner product on each p-1(y) be the standard inner product. However, if we make these choices locally around every point in B, there is no guarantee that they “glue together” properly to yield a global continuous choice, unless the transition functions preserve the standard inner product. But this is precisely what reduction of structure to O(n) means.

The same explanation holds for subgroups preserving other kinds of structure.

Title reduction of structure group
Canonical name ReductionOfStructureGroup
Date of creation 2013-03-22 13:26:06
Last modified on 2013-03-22 13:26:06
Owner antonio (1116)
Last modified by antonio (1116)
Numerical id 12
Author antonio (1116)
Entry type Definition
Classification msc 55R10
Related topic VectorBundle
Related topic FiberBundle
Defines Euclidean structure
Defines Riemannian structure
Defines complex structure
Defines almost-complex structure