section of a fiber bundleSectionOfAFiberBundle2003-02-10 01:52:192004-06-19 05:05:51Definitionsmooth sectionglobal sectionlocal sectionzero section% used for TeXing text within eps files
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Let $\funcdef{p}{E}{B}$ be a fiber bundle, denoted by $\xi.$
A {\em section\/} of $\xi$
is a continuous map $\funcdef{s}{B}{E}$ such that the composition $p\comp s$ equals the identity.
That is, for every $b\in B,$ $s(b)$ is an element of the fiber over $b.$
More generally, given a topological subspace $A$ of $B,$ a section of $\xi$ over $A$ is a section of the restricted bundle
$\funcdef{\restr{p}{A}}{\inv{p}(A)}{A}.$
The set of sections of $\xi$ over $A$ is often denoted by $\Gamma(A;\xi),$ or
by $\Gamma(\xi)$ for sections defined on all of $B.$ Elements of $\Gamma(\xi)$ are sometimes
called {\em global sections,\/} in contrast with the {\em local sections\/} $\Gamma(U;\xi)$ defined on an open set $U.$
\begin{rmk}
If $E$ and $B$ have, for example, smooth structures, one can talk about smooth
sections of the bundle. According to the context, the notation $\Gamma(\xi)$ often
denotes smooth sections, or some other set of suitably restricted sections.
\end{rmk}
\begin{exm}
If $\xi$ is a trivial fiber bundle with fiber $F,$ so that $E=F\cross B$ and
$p$ is projection to $B,$ then sections of $\xi$ are in a natural bijective correspondence with continuous functions $\funcsig{B}{F}.$
\end{exm}
\begin{exm}
If $B$ is a smooth manifold and $E=TB$ its tangent bundle, a (smooth) section of this bundle is precisely a (smooth) tangent vector field.
In fact, any tensor field on a smooth manifold $M$ is a section of
an appropriate vector bundle. For instance, a contravariant $k$-tensor field is a section of the bundle $TM^{\otimes k}$ obtained by repeated tensor product from the tangent bundle, and similarly for covariant and mixed tensor fields.
\end{exm}
\begin{exm}
If $B$ is a smooth manifold which is smoothly embedded in a Riemannian manifold
$M,$ we can let the fiber over $b\in B$ be the orthogonal complement in $T_b M$ of the tangent space $T_b B$ of $B$ at $b$. These choices of fiber turn out to
make up a vector bundle $\nu(B)$ over $B,$ called the {\em
\PMlinkescapetext{normal bundle}\/} of $B$. A section of $\nu(B)$ is a normal
vector field on $B.$
\end{exm}
\begin{exm}
If $\xi$ is a vector bundle, the {\em zero section\/} is defined simply by
$s(b)=0,$ the zero vector on the fiber.
It is interesting to ask if a vector bundle admits a section which is
nowhere zero. The answer is yes, for example, in the case of a trivial vector
bundle, but in general it depends on the topology of the spaces involved.
A well-known case of this question is the {\em hairy ball theorem,} which
says that there are no nonvanishing tangent vector fields on the sphere.
\end{exm}
\begin{exm}
If $\xi$ is a \PMlinkname{principal}{PrincipalBundle} $G$-\PMlinkname{bundle}{PrincipalBundle}, the existence of {\em any\/} section is
equivalent to the bundle being trivial.
\end{exm}
\begin{rmk}
The correspondence taking an open set $U$ in $B$ to $\Gamma(U;\xi)$ is an example
of a sheaf on $B.$
\end{rmk}