---
tag: GCS702QT54
kind: concept
name: Contact structure on a jet bundle
msc: [58A20, 58A30, 35A30]
sources:
  - url: https://en.wikipedia.org/wiki/Jet_bundle
---

## Summary

The contact structure of a [[jet bundle|HC3VBBCJQP]] is the family of contact forms: the $1$-forms that vanish on every [[prolonged section|BMCCQ8JXNE]].
Their common kernel is a distribution, called the contact or Cartan distribution.
The contact forms tell the holonomic sections apart from all others.
A section of the jet bundle is a prolongation exactly when it pulls back every contact form to zero.
In the simplest case, the first jet bundle of functions of one variable, it is the standard [[contact structure|HBQ8BZ7XB5]] on $\R^3$.

## Explanations

### L1 human

In the jet space with coordinates $(x, u, p)$, the letter $p$ is meant to be the slope $du/dx$.
The contact form $du - p\, dx$ expresses this wish.
Along the curve $x \mapsto (x, s(x), s'(x))$ of a function $s$, this form is zero, since $du = s'(x)\, dx = p\, dx$ there.
Along a curve where $p$ is not the slope, the form is not zero.
So the contact form is a test that a curve in jet space really comes from a function.