# reduction of structure group

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

###### Remark 1

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

For the following examples, let $E$ be an $n$-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 $H$ 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

When $B$ 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 $O(n).$

###### Example 4

Let $n=2m$ 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 $H$ is called a complex structure on the vector bundle, and it is equivalent to a continuous fiberwise choice of an endomorphism $J$ satisfying $J^{2}=-I.$

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 ${\mathbb{C}}^{m}$ 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 $H$ is equivalent to the existence of a splitting $E\cong E_{1}\oplus E_{2},$ where $E_{1}$ 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 $E_{1}$ is a $k$-plane bundle.

###### Remark 2

These examples all have two features in common, namely:

For example, $O(n)$ is the subgroup of $GL(n,{\mathbb{R}})$ which preserves the standard inner product of ${\mathbb{R}}^{n},$ 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 $x\in B$: we simply choose a neighborhood $U$ 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 $B$, 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