reduction of structure group

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

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

Remark 1

Here, the action of $H$ on $F$ is the restriction of the $G$ -action; inparticular, this means that an $(F,H)$ -bundle is automatically an $(F,G)$ -bundle. The bundle isomorphism in the definition then becomes meaningfulin 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.

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 asusual.

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

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 choiceof 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 thata Riemannian structure always exists on any given vector bundle. For this reason, it is often convenient to start out assuming the structure groupto 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, andit 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 themore 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 theexistence 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:

•
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 astructure of the same kind.

For example, $O(n)$ is the subgroup of $GL(n,{\\mathbb{R}})$ which preserves thestandard inner product of ${\\mathbb{R}}^{n},$ and reduction of structure to $O(n)$ isequivalent to a fiberwise choice of inner products.

This is not a coincidence. The intuition behind this is as follows. Thereis 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 globalcontinuous choice, unless the transition functions preserve the standardinner product. But this is precisely what reduction of structure to $O(n)$ means.

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

单词	ReductionOfStructureGroup
释义	reduction of structure group Given a fiber bundle $p:E\\rightarrow B$ with typical fiber $F$ and structure group $G$ (henceforth called an $(F,G)$ -bundle over $B$ ), we say that the bundle admits a reduction of its structure group to $H$ , where $H<G$ is a subgroup, if it is isomorphic to an $(F,H)$ -bundle over $B .$ Equivalently, $E$ admits a reduction of structure group to $H$ if there isa choice of local trivializations covering $E$ such that the transitionfunctions all belong to $H .$ Remark 1 Here, the action of $H$ on $F$ is the restriction of the $G$ -action; inparticular, this means that an $(F,H)$ -bundle is automatically an $(F,G)$ -bundle. The bundle isomorphism in the definition then becomes meaningfulin 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. 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 asusual. 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 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 choiceof 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 thata Riemannian structure always exists on any given vector bundle. For this reason, it is often convenient to start out assuming the structure groupto 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, andit 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 themore 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 theexistence 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: • 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 astructure of the same kind. For example, $O(n)$ is the subgroup of $GL(n,{\\mathbb{R}})$ which preserves thestandard inner product of ${\\mathbb{R}}^{n},$ and reduction of structure to $O(n)$ isequivalent to a fiberwise choice of inner products. This is not a coincidence. The intuition behind this is as follows. Thereis 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 globalcontinuous choice, unless the transition functions preserve the standardinner product. But this is precisely what reduction of structure to $O(n)$ means. The same explanation holds for subgroups preserving other kinds of structure.
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