---
tag: DT60HNP41N
kind: definition
name: Pfaffian system
msc: [58A17, 58A15]
defines: RX5MYWE73J
uses: [901CCKQHVY, 02Z5B2HA3F, 0E3G0GRPZ6, F45VQ47GPV, 5Q4BHBFXJY]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_ideal
---

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
A *Pfaffian system* on $M$ is an [[exterior differential system|901CCKQHVY]] $\mathcal{I}$ generated, as a differential ideal, by [[one-forms|02Z5B2HA3F]] $\theta^1, \dots, \theta^s$.
So $\mathcal{I}$ is the algebraic ideal generated by $\theta^1, \dots, \theta^s$ and $d\theta^1, \dots, d\theta^s$.
One usually asks that $\theta^1, \dots, \theta^s$ are linearly independent at each point.
Then $\mathcal{D} = \set{ v \in TM : \theta^a(v) = 0 \text{ for } a = 1, \dots, s }$ is a distribution of rank $n - s$, and an immersed submanifold is an integral manifold of $\mathcal{I}$ exactly when it is tangent to $\mathcal{D}$.

## Remarks

The system is *completely integrable* if $d\theta^a \wedge \theta^1 \wedge \dots \wedge \theta^s = 0$ for $a = 1, \dots, s$, that is, if each $d\theta^a$ lies in the algebraic ideal generated by the $\theta^b$.
By the [[Frobenius theorem|5Q4BHBFXJY]], this holds exactly when every point lies on an integral manifold of dimension $n - s$.

Example: on $\R^3$, the system generated by $\theta = dz$ has $d\theta = 0$, so it is completely integrable.
Its integral surfaces are the planes $z = c$.

Example: on $\R^3$, let $\theta = dz - y\, dx$.
Then $d\theta = dx \wedge dy$ and $\theta \wedge d\theta = dz \wedge dx \wedge dy \ne 0$, so the system is not completely integrable.
It has integral curves, such as $t \mapsto (t, c, ct)$, but no integral surfaces.

Non-example: the exterior differential system on $\R^3$ generated by the [[two-form|0E3G0GRPZ6]] $dx \wedge dy$ is not a Pfaffian system.
It contains no nonzero one-form, so it is not generated by one-forms.