---
tag: 5Q4BHBFXJY
kind: statement
role: theorem
name: Frobenius theorem
msc: [58A30, 53C12]
uses: [F45VQ47GPV, WY6K3P4YCG, 68FGQEBD6J]
about: [68FGQEBD6J]
sources:
  - url: https://en.wikipedia.org/wiki/Frobenius_theorem_(differential_topology)
same_as:
  wikidata: Q837551
  wikipedia: Frobenius theorem (differential topology)
---

## Statement

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
Let $D \subset TM$ be a smooth subbundle of rank $k$ of the [[tangent bundle|WY6K3P4YCG]], called a distribution.
Call $D$ *involutive* if for all smooth vector fields $X, Y$ with values in $D$, the [[Lie bracket|68FGQEBD6J]] $[X, Y]$ also has values in $D$.
Call $D$ *integrable* if every point of $M$ lies in an integral manifold of $D$.
An integral manifold is an immersed submanifold $N$ with $T_qN = D_q$ for all $q \in N$.
Then the following are equivalent.

1. $D$ is involutive.
2. $D$ is integrable.
3. Every point of $M$ has a chart with coordinates $x^1, \dots, x^n$ in which $D$ is spanned by $\partial_1, \dots, \partial_k$.

## Remarks

Condition 3 implies condition 2, since the slices $x^{k+1} = c_{k+1}, \dots, x^n = c_n$ are integral manifolds.
Condition 2 implies condition 1, since vector fields tangent to a submanifold have a bracket tangent to it.
The main step is that condition 1 implies condition 3.
One picks local vector fields spanning $D$ that pairwise commute, which involutivity allows after a change of basis, and then uses their commuting flows as coordinates.
Example: on $\R^3$ the distribution spanned by $X = \partial_x$ and $Y = \partial_y + x \, \partial_z$ is the kernel of $dz - x \, dy$.
Here $[X, Y] = \partial_z$, which is not in $D$, so no surface is tangent to $D$; this is the standard contact structure.
In the global form of the theorem, an involutive distribution splits $M$ into maximal connected integral manifolds, the leaves of a foliation.

## Explanations

### L1 human

At every point of a space, choose a plane of allowed directions.
Can the space be sliced into surfaces that are tangent to these planes everywhere, like the pages of a book?
The Frobenius theorem says this happens exactly when moving back and forth along allowed directions never produces a forbidden direction, a condition tested by the Lie bracket.

### L2 human

The theorem turns a question about solving a system of first-order PDEs into an algebraic check on brackets.
For $k = 1$ every line field is involutive, and the theorem reduces to the existence of integral curves of a vector field.
A dual form uses differential forms: $D = \ker \theta^1 \cap \dots \cap \ker \theta^{n-k}$ is involutive if and only if each $d\theta^i$ lies in the ideal generated by the $\theta^j$.
The theorem is used to integrate Lie subalgebras to Lie subgroups, to show that flat connections are locally trivial, and as the foundation of the theory of foliations.