reduction of structure groupReductionOfStructureGroup2003-02-08 01:50:132007-06-22 23:38:48Definitionfiber bundleEuclidean structureRiemannian structurecomplex structurealmost-complex structure% used for TeXing text within eps files
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Given a fiber bundle $\funcdef{p}{E}{B}$ with typical fiber $F$ and structure group $G$ (henceforth called an $(F,G)$-bundle over $B$), we say that the bundle admits a {\em 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 is
a choice of local trivializations covering $E$ such that the transition
functions all belong to $H.$
\begin{rmk}
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 $B$.
\end{rmk}
\begin{exm}
Let $H$ be the trivial subgroup. Then, the existence of a reduction of structure group to $H$ is equivalent to the bundle being trivial.
\end{exm}
For the following examples, let $E$ be an $n$-dimensional vector bundle, so that
$F\isom\reals^n$ with $G=GL(n,\reals),$ the general linear group acting as
usual.
\begin{exm}
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 $B$.
\end{exm}
\begin{exm}
Set $H=O(n)$, the orthogonal group. A reduction to $H$ is called a {\em Riemannian\/} or {\em 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 $O(n).$
\end{exm}
\begin{exm}
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 {\em 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 {\em 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.
\end{exm}
\begin{exm}
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.
\end{exm}
\begin{rmk}
These examples all have two features in common, namely:
\begin{itemize}
\item
the subgroup $H$ can be interpreted as being precisely the subgroup of $G$ which preserves a particular structure, and,
\item
a reduction to $H$ is equivalent to a continuous fiber-by-fiber choice of a
structure of the same kind.
\end{itemize}
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, {\em 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.
\end{rmk}
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