PlanetMath: reduction of structure group

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reduction of structure group

(Definition)

Given a fiber bundle $p:E\rightarrow B$ with typical fiber and structure group (henceforth called an -bundle over ), we say that the bundle admits a reduction of its structure group to , where is a subgroup, if it is isomorphic to an -bundle over

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

Remark 1 Here, the action of

is the restriction of the

-action; in particular, this means that an

-bundle is automatically an

-bundle. The bundle isomorphism in the definition then becomes meaningful in the category of

-bundles over

Example 1 Let

be the trivial subgroup. Then, the existence of a reduction of structure group to

is equivalent to the bundle being trivial.

For the following examples, let be an -dimensional vector bundle, so that $F\cong {\mathbb{R}}^n$ with $G=GL(n,{\mathbb{R}}),$ the general linear group acting as usual.

Example 2 Set $H=GL^+(n,{\mathbb{R}}),$ the subgroup of $GL(n,{\mathbb{R}})$ consisting of matrices with positive determinant. A reduction to

is equivalent to an orientation of the vector bundle. In the case where

is a smooth manifold and

is its tangent bundle, this coincides with other definitions of an orientation of

Example 3 Set

, the orthogonal group. A reduction to

is called a Riemannian or Euclidean structure on the vector bundle. It coincides with a continuous fiberwise choice of a positive definite inner product, and for the case of the tangent bundle, with the usual notion of a Riemannian metric on a manifold.

When is paracompact, an argument with partitions of unity 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

Example 4 Let

be even, and let $H=GL(m,{\mathbb{C}}),$ the group of invertible complex matrices, embedded in $GL(n,{\mathbb{R}})$ by means of the usual identification of ${\mathbb{C}}$ with ${\mathbb{R}}^2.$ A reduction to

is called a complex structure on the vector bundle, and it is equivalent to a continuous fiberwise choice of an endomorphism

satisfying

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 ${\mathbb{C}}^m$ such that the transition functions are holomorphic.

Example 5 Let $H=GL(1,{\mathbb{R}})\times GL(n-1,{\mathbb{R}}),$ embedded in $GL(n,{\mathbb{R}})$ by $\left(A,B\right)\mapsto A\oplus B.$ A reduction to

is equivalent to the existence of a splitting $E\cong E_1\oplus E_2,$ where

is a line bundle. More generally, a reduction to $GL(k,{\mathbb{R}})\times GL(n-k,{\mathbb{R}})$ is equivalent to a splitting $E\cong E_1\oplus E_2,$ where

is a

-plane bundle.

Remark 2 These examples all have two features in common, namely:

the subgroup can be interpreted as being precisely the subgroup of which preserves a particular structure, and,
a reduction to is equivalent to a continuous fiber-by-fiber choice of a structure of the same kind.

For example, is the subgroup of $GL(n,{\mathbb{R}})$ which preserves the standard inner product of ${\mathbb{R}}^n,$ and reduction of structure to 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 $x\in B$ : we simply choose a neighborhood on which the bundle is trivial, and with respect to a trivialization $p^{-1}(U)\cong {\mathbb{R}}^n\times U$ , 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 , 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 means.