section of a fiber bundle

Let $p:E\to B$ be a fiber bundle, denoted by $\xi .$

A section of $\xi$is a continuous map $s:B\to E$ such that the composition $p\circ s$ 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${p|}_A:p^{-1}(A)\to A.$

The set of sections of $\xi$ over $A$ is often denoted by $\mathrm{\Gamma }(A;\xi ),$ orby $\mathrm{\Gamma }(\xi )$ for sections defined on all of $B.$ Elements of $\mathrm{\Gamma }(\xi )$ are sometimescalled global sections, in contrast with the local sections $\mathrm{\Gamma }(U;\xi )$ defined on an open set $U.$

Remark 1

If $E$ and $B$ have, for example, smooth structures, one can talk about smoothsections of the bundle. According to the context, the notation $\mathrm{\Gamma }(\xi )$ oftendenotes smooth sections, or some other set of suitably restricted sections.

Example 1

If $\xi$ is a trivial fiber bundle with fiber $F,$ so that $E=F\times B$ and$p$ is projection to $B,$ then sections of $\xi$ are in a natural bijective correspondence with continuous functions $B\to F.$

Example 2

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 ofan appropriate vector bundle. For instance, a contravariant $k$-tensor field is a section of the bundle $T{M}^{\otimes k}$ obtained by repeated tensor product from the tangent bundle, and similarly for covariant and mixed tensor fields.

Example 3

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 tomake up a vector bundle $\nu (B)$ over $B,$ called the of $B$. A section of $\nu (B)$ is a normalvector field on $B.$

Example 4

If $\xi$ is a vector bundle, the 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 isnowhere zero. The answer is yes, for example, in the case of a trivial vectorbundle, but in general it depends on the topology of the spaces involved.A well-known case of this question is the hairy ball theorem, whichsays that there are no nonvanishing tangent vector fields on the sphere.

Example 5

If $\xi$ is a principal (http://planetmath.org/PrincipalBundle) $G$-bundle (http://planetmath.org/PrincipalBundle), the existence of any section isequivalent to the bundle being trivial.

Remark 2

The correspondence taking an open set $U$ in $B$ to $\mathrm{\Gamma }(U;\xi )$ is an exampleof a sheaf on $B.$