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 $B$ .

Example 1   Let $H$ be the trivial subgroup. Then, the existence of a reduction of structure group to $H$ is equivalent to the bundle being trivial.

Example 2   Set $H=GL^+(n,\reals),$ the subgroup of $GL(n,\reals)$ 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   Set $H=O(n)$ , the orthogonal group. A reduction to $H$ 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 $B$ 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 $H=GL(m,\cpxs),$ the group of invertible complex matrices, embedded in $GL(n,\reals)$ by means of the usual identification of $\cpxs$ with $\reals^2.$ 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 $\cpxs^m$ such that the transition functions are holomorphic.

Example 5   Let $H=GL(1,\reals)\cross GL(n-1,\reals),$ embedded in $GL(n,\reals)$ by $\tuple{A,B}\mapsto A\oplus B.$ A reduction to $H$ is equivalent to the existence of a splitting $E\isom E_1\oplus E_2,$ where $E_1$ is a line bundle. More generally, a reduction to $GL(k,\reals)\cross GL(n-k,\reals)$ is equivalent to a splitting $E\isom E_1\oplus E_2,$ where $E_1$ 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 structure, 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 $GL(n,\reals)$ which preserves the standard inner product of $\reals^n,$ 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 $\inv{p}(U)\homeo\reals^n\cross U$ , we can let the inner product on each $\inv{p}(y)$ 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.