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Homesection of a fiber bundle

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# section of a fiber bundle

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

A section of $\xi$ is a continuous map $s:B\rightarrow E$ 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 ${p}|_{{A}}:p^{{-1}}(A)\rightarrow A.$

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 global sections, in contrast with the local sections $\Gamma(U;\xi)$ defined on an open set $U.$

###### Remark 1.

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.

###### 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 $B\rightarrow F.$

###### 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 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.

###### 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 to make up a vector bundle $\nu(B)$ over $B,$ called the normal bundle of $B$. A section of $\nu(B)$ is a normal vector 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 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 hairy ball theorem, which says that there are no nonvanishing tangent vector fields on the sphere.

###### Example 5.

###### Remark 2.

The correspondence taking an open set $U$ in $B$ to $\Gamma(U;\xi)$ is an example of a sheaf on $B.$

## Mathematics Subject Classification

55R10*no label found*

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