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/Part 1: Diffusion Processes/Chapter 2: Integral Manifolds/Shenanigans

2.2 Frobenius’ Theorem

Theorem 2.2.1 (Frobenius).label Let $X$ be a $C^{p}$ ($p \ge 2$) manifold and $E \subset TX$ be a subbundle, then the following are equivalent:

  1. (1)

    For each pair of vector fields $\xi, \eta: X \to E$, $[\xi, \eta]$ lies in $E$.

  2. (2)

    For each $\omega \in \Lambda^{1}(TX)$ vanishing on $E$, $\xi, \eta: X \to E$, $d\omega(\xi, \eta) = 0$.

  3. (3)

    $E$ is integrable.

Proof, [Section VI.1, Lan12]. Let $\omega \in \Lambda^{1}(TX)$, then

\[d\omega(\xi, \eta) = \omega([\xi, \eta]) - \eta\omega(\xi) - \xi\omega(\eta)\]

(1) $\Rightarrow$ (2): If $\omega$ vanishes on $E$ and $[\xi, \eta]$ lies in $E$, then $d\omega$ vanishes on $(\xi, \eta)$.

(2) $\Rightarrow$ (1): If $[\xi, \eta]$ does not lie in $E$, then there exists $\omega \in \Lambda^{1}(TX)$ that vanishes on $E$ but not $[\xi, \eta]$, which contradicts (2).

(1) $\Leftrightarrow$ (3): For any point on $X$, since $E$ is a subbundle, there exists Banach spaces $F, G$ and a coordinate neighbourhood $U \times V$ such that $E$ is given by a $C^{p-1}$-mapping

\[f: U \times V \times F \to U \times V \times (F \times G)\]

such that $\pi_{1}f(x, y)(z) = z$ for all $(x, y) \in U \times V$ and $z \in F$. Let

\[g: U \times V \to L(F; G) \quad g(x, y)(z) = \pi_{2}f(x, y, z)\]

then $g$ is also a $C^{p - 1}$-mapping. Let

\[\Xi: U \times V \to F \times G\]

be the local representation of a vector field, then the vector field lies in $E$ if and only if

\[\pi_{2}\Xi(x, y) = g(x, y)\pi_{1}\Xi(x, y)\]

for all $(x, y) \im U \times V$. Let $H: U \times V \to F \times G$ be another vector field, then by assumption (1), $[\Xi, H]$ lies in $E$. Denote $\xi = \pi_{1}\Xi$ and $\eta = \pi_{1} H$, then

\begin{align*}[\xi, \eta]&= D\eta \cdot \xi - D\xi \cdot \eta \\ g \cdot (D\eta \cdot \xi - D\xi \cdot \eta)&= (Dg \cdot \xi) \cdot \eta + g \cdot D\eta \cdot \xi - (Dg \cdot \eta) \cdot \xi - g \cdot D\eta \cdot \xi \\ (Dg \cdot \xi) \cdot \eta&= (Dg \cdot \eta) \cdot \xi\end{align*}

Let $(x_{0}, y_{0}) \in U \times V$, then by Theorem 2.1.3, there exists $U_{0} \in \cn_{F}(x_{0})$ and $V_{0} \in \cn_{G}(y_{0})$ and $\alpha \in C^{p-1}(U_{0} \times V_{0}; V)$ such that

\[\partial_{x}\alpha(x, y) = g(x, \alpha(x, y))\]

Let

\[\varphi: U_{0} \times V_{0} \to U \times V \quad (x, y) \mapsto (x, \alpha(x, y))\]

then

\[D\varphi(x_{0}, y_{0}) = \begin{bmatrix}Id&0 \\ g(x, \alpha(x, y))&Id\end{bmatrix}\]

so $\varphi$ is a local diffeomorphism at $(x_{0}, y_{0})$. Since for any $(u, v) \in F \times G$,

\[\partial_{x}\varphi(x, y) \cdot (u, v) = (u, g(x, \alpha(x, y)) \cdot u)\]

the bundle $E$ is integrable.$\square$

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