# 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 $\mathrm{\Gamma}(A;\xi ),$ or by $\mathrm{\Gamma}(\xi )$ for sections defined on all of $B.$ Elements of $\mathrm{\Gamma}(\xi )$ are sometimes called 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 smooth
sections of the bundle. According to the context, the notation $\mathrm{\Gamma}(\xi )$ 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^{} $B\to F.$

###### 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 of an 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 to
make up a vector bundle $\nu (B)$ over $B,$ called the 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

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

###### Remark 2

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

Title | section of a fiber bundle |

Canonical name | SectionOfAFiberBundle |

Date of creation | 2013-03-22 13:26:43 |

Last modified on | 2013-03-22 13:26:43 |

Owner | antonio (1116) |

Last modified by | antonio (1116) |

Numerical id | 10 |

Author | antonio (1116) |

Entry type | Definition |

Classification | msc 55R10 |

Synonym | section |

Synonym | cross section |

Synonym | cross-section |

Related topic | FiberBundle |

Defines | smooth section |

Defines | global section |

Defines | local section |

Defines | zero section |