---
tag: WWX7H5W1RP
kind: concept
name: Section of a vector bundle
msc: [55R25, 58A05]
sources:
  - url: https://en.wikipedia.org/wiki/Section_(fiber_bundle)
---

## Summary

A section of a vector bundle chooses one vector in the fiber over each point of the base.
The choice varies smoothly with the point.
Vector fields are sections of the tangent bundle, and differential forms are sections of bundles of alternating forms.
The sections of a vector bundle form a vector space, and in fact a module over the ring of smooth functions on the base.
Many objects of geometry and physics, such as fields in gauge theory, are sections of suitable bundles.

## Explanations

### L1 human

A vector bundle attaches a vector space to each point of a space.
A section picks one vector from each of these spaces, without jumps.
Wind directions on the surface of the Earth form a section of the tangent bundle of the sphere: at each place you get an arrow tangent to the surface.
On a trivial bundle $M \times \R^k$, a section is just a smooth function from $M$ to $\R^k$.