---
tag: WV4S47X6QC
kind: statement
role: theorem
name: Cartan's test for involutivity
msc: [58A15, 58A17]
uses: [901CCKQHVY, F45VQ47GPV, CYPA882RB9, VKFW27WXEZ]
about: [901CCKQHVY]
sources:
  - url: https://en.wikipedia.org/wiki/Cartan%E2%80%93K%C3%A4hler_theorem
---

## Statement

Let $\mathcal{I}$ be an [[exterior differential system|901CCKQHVY]] on a [[smooth manifold|F45VQ47GPV]] $M$, and assume $\mathcal{I}$ contains no nonzero functions.
For $p \in M$, a linear subspace $E \subseteq T_pM$ is an *integral element* if $\varphi|_E = 0$ for every $\varphi \in \mathcal{I}$.
Let $V_k(\mathcal{I})$ be the set of $k$-dimensional integral elements, a subset of the bundle $G_k(TM)$ of $k$-planes in tangent spaces, whose fibers are [[Grassmannians|CYPA882RB9]].
For an integral element $E$ with basis $e_1, \dots, e_k$, its *polar space* is
$$H(E) = \set{ v \in T_pM : \varphi(v, e_1, \dots, e_k) = 0 \text{ for all } \varphi \in \mathcal{I} \text{ of degree } k + 1 },$$
and $c(E) = \dim M - \dim H(E)$ is its codimension.
An integral element $E$ is *regular* if, near $E$, the set $V_k(\mathcal{I})$ is a smooth [[submanifold|VKFW27WXEZ]] of $G_k(TM)$ and $\dim H(E')$ is constant for $E' \in V_k(\mathcal{I})$.

Let $E \in V_n(\mathcal{I})$ and let $\set{0} = E_0 \subset E_1 \subset \dots \subset E_n = E$ be a flag with $\dim E_k = k$.
Put $c_k = c(E_k)$.
Then:

1. Near $E$, the set $V_n(\mathcal{I})$ lies in a submanifold of $G_n(TM)$ of codimension $c_0 + c_1 + \dots + c_{n-1}$.
2. The following are equivalent. Near $E$, $V_n(\mathcal{I})$ is a smooth submanifold of $G_n(TM)$ of codimension exactly $c_0 + \dots + c_{n-1}$. Each $E_k$ with $k < n$ is a regular integral element.

An integral element $E$ that has a flag as in 2 is called *ordinary*.

## Remarks

The idea is that each extension $E_k \subset E_{k+1}$ is a choice of a vector in the polar space $H(E_k)$, and each such choice imposes at least $c_k$ independent conditions.
Counting these conditions gives the lower bound for the codimension of $V_n(\mathcal{I})$, and equality means that no hidden conditions appear.

When the system comes with an independence condition, an integral element that passes this test is called involutive.
Ordinary integral elements are the starting points of the Cartan–Kähler theorem, which produces integral manifolds through them in the real-analytic case.

Example: let $\mathcal{I}$ be generated by $\theta = dz - y\, dx$ on $\R^3$, and let $n = 1$.
For $E_0 = \set{0}$, the polar space is $\ker \theta$, of dimension $2$, so $c_0 = 1$.
The $1$-dimensional integral elements at a point are the lines in the plane $\ker \theta$, a circle's worth inside the projective plane of lines, so $V_1(\mathcal{I})$ has codimension $1 = c_0$ in $G_1(T\R^3)$.
The test is passed, and every integral line is ordinary.

## Explanations

### L1 human

To decide whether a system of differential equations has many solutions, Cartan counts how many conditions each step of building a solution imposes.
The test compares this count with the actual number of conditions defining the possible tangent planes of solutions.
If the two agree, the system is in good shape: no hidden conditions will appear later.

### L2 human

Cartan's test is a purely algebraic, pointwise computation with polar spaces.
It is the practical way to check the hypothesis of the Cartan–Kähler theorem.
When a system fails the test, one passes to its prolongation, adding the derivatives as new variables, and tests again.
The Cartan–Kuranishi prolongation theorem says that, under regularity assumptions, this process ends with an involutive system after finitely many steps.
The Cartan characters, read off from the numbers $c_k$, then describe how many functions of how many variables the general solution depends on.