\classheader{2013-02-20}

Everything we're talking about today will be $\Cinfty$.

To finish the proof of the Frobenius theorem from last time, it remains to show the following result:

\begin{lemma-N}
If $w_1,\ldots,w_r$ are linearly independent, commuting vector fields (commuting in the sense that their pairwise Lie brackets are 0), then there is a chart centered at any given point where the $w_i$ are transformed to the coordinate vector fields $\frac{\partial}{\partial x_i}$ for $i=1,\ldots,r$.
\end{lemma-N}
\begin{remark}
Let $v$ and $w$ be vector fields on $M$. Let $\phi_t$ and $\psi_s$ denote the one-parameter groups for $v$ and $w$ respectively (i.e. the flows). Then for all $p\in M$, there is some neighborhood $U(p)$ of $p$ and $(-\epsilon,\epsilon)$ such that $\phi_t(\psi_s(x))$ and $\psi_s(\phi_t(x))$ are defined for all $x\in U(p)$ and $t,s\in(-\epsilon,\epsilon)$.
\end{remark}
\begin{lemma-N}
With notation as above, if $[v,w]=0$, then $\phi_t(\psi_s(x))=\psi_s(\phi_t(x))$ for any $x\in U(p)$ and $s,t\in(-\epsilon,\epsilon)$.
\end{lemma-N}
\begin{proof}[Proof that Lemma 2 $\implies$ Lemma 1]
Let's assume the result of Lemma 2 in the case that $v(p)\neq 0$. Let $\phi_t^i$ denote the one-parameter groups with respect to $w_i$ for each $i=1,\ldots,r$. Let $p\in M$, and select a locally closed $\Cinfty$ submanifold $Z\subset M$ with $p\in Z$ such that $T_pZ\oplus \R w_1(p)\oplus\cdots\oplus\R w_r(p)=T_pM$. Note that by assuming this is true at $p$, we can assume this is true in a neighborhood of $p$.

Let $h:(-\epsilon,\epsilon)^r\times(Z\cap U(p))\to M$ be defined by
\[h(x_1,\ldots,x_r,z)=\phi_{x_1}^1\phi_{x_2}^2\cdots\phi_{x_r}^r(z).\]
We see that $h$ induces an isomorphism from the tangent space at $(0,\ldots,0,z)$ to $T_zM$ for all $z\in Z\cap U(p)$. Note that $h(t,x_2,\ldots,x_r,z)$ is an integral curve for $w_1$, so that $h'(?)\frac{\partial}{\partial x_1}=w(h(?))$ for all $?$ in the domain of $h$ (this is not a ? in the sense of ``I didn't get down what was on the board'', but rather ``?'' itself what was written on the board). This is
\[\phi_{x_2}^2\phi_{x_1}^1\cdots,\]
and thus we see that $h'(?)\frac{\partial}{\partial x_2}=w_2(h(?))$, etc. (not sure I understand this part).
\end{proof}
\begin{proof}[Proof of Lemma 2]
We have that $w_1(p)\neq 0$, so (as we have shown earlier) we can assume WLOG that $w=\frac{\partial}{\partial x_1}$. For any vector $v=\sum a_i\frac{\partial}{\partial x_i}$, we have that
\[[w,v]=\sum\frac{\partial a_i}{\partial x_1}\cdot\frac{\partial}{\partial x_i}.\]
By assumption, this is zero, so the $a_i$'s are (in some neighborhood) functions of $(x_2,\ldots,x_n)$. Because the statement is local, we can assume that we are working on $(-\epsilon,\epsilon)\times\Omega$ for an open subset $\Omega\subset\R^{n-1}$. Let $c\in(-\frac{\epsilon}{2},\frac{\epsilon}{2})$. Let $h_c:(-\frac{\epsilon}{2},\frac{\epsilon}{2})\to(-\epsilon,\epsilon)\times\Omega$ be defined by
\[h_c(x_1,x_2,\ldots)=(x_1+c,x_2,\ldots).\]
Then $v$ and $v|_{(-\frac{\epsilon}{2},\frac{\epsilon}{2})}$ are $h_c$-related.

Therefore, if $\delta$ is an integral curve of $v$, then $h_c\circ\delta$ is also an integral curve. Let $\phi_t$ denote the one-parameter group associated to $v$. Then we have that
\[h_c\circ \phi_t=\phi_t\circ h_c.\]
But $h_c=\psi_c$ where $\psi_c$ is the one-parameter group associated to $w$.
\end{proof}

\begin{theorem}[Thom's ambient isotopy lemma]
Let $I=[0,1]$, let $A$ and $B$ be $\Cinfty$ manifolds where $A$ is compact, and let $F:A\times I\to B$ be a $\Cinfty$ map. Let $f_t:A\to B$ be defined by $f_t(a)=F(a,t)$ for all $a\in A$ and $t\in [0,1]$. If $f_t$ is an embedding for all $t\in I$, then there is a $\Cinfty$ map $G:B\times I\to B$ such that $g_t$ is a diffeomorphism for all $t\in I$, and $f_t=g_t\circ f_0$ for all $t\in I$, where $g_t(b)=G(b,t)$.
\end{theorem}

Recall that if $A$ is an arbitrary subset of a $\Cinfty$ manifold $M$, then given a map $f:A\to\R$, we say that it is $\Cinfty$ map when there exist open sets $U_\lambda\subset M$  for all $\lambda\in \Lambda$ such that $f|_{A\cap U_\lambda}=f_\lambda|_{A\cap U_\lambda}$ and $W:=\bigcup U_\lambda$ contains $A$. Then $\{U_\lambda\}_{\lambda\in\Lambda}$ is an open cover of $W$, so there is a partition of unity subordinate to this cover. Let $\varphi_\lambda:W\to\R$ be subordinate to $U_\lambda$.

Consider $\varphi_\lambda|_{U_\lambda}\circ f_\lambda:U_\lambda\to\R$, which has support contained in $U_\lambda$, and extends by zero to a $\Cinfty$ function on $W$ denoted by $\varphi_\lambda f_\lambda$. If we then define $\widetilde{f}=\sum_{\lambda\in\Lambda}\varphi_\lambda f_\lambda$, then $\widetilde{f}$ is a $\Cinfty$ function defined on $W$ that extends $f$. More generally, if we have a $\Cinfty$ bundle
\begin{center}
\begin{tikzcd}
{}& V \ar{d}{p} \\ A\ar{ru}{s} \ar[hook]{r} & M
\end{tikzcd}
\end{center}
where $A$ is arbitrary, then what we've shown is that it extends to a $\Cinfty$ section on an open $W\supset A$.

A variant of this result is that if $A$ is a closed set, then note that $\{U_\lambda\mid\lambda\in\Lambda\}\cup\{M-A\}$ is also an open cover, so we can create a partition of unity $\{\varphi_\lambda\mid\lambda\in\Lambda\}\cup\{\varphi_0\}$. If we define $f_0:(M-A)\to\R$ to be zero, then let
\[\widetilde{f}=\sum_\lambda \varphi_\lambda f_\lambda+\varphi_0f_0.\]
Once again, $\widetilde{f}:M\to\R$ and $\widetilde{f}|_A=f$. Finally, if $A$ is compact, then we see that $\widetilde{f}$ can be chosen to have compact support.


\begin{proof}[Proof of Thom's lemma]
WLOG, we can assume that $B\subset\R^N$, so that $F:A\times I\to B$ can be extended to a $\Cinfty$ map $F:A\times(-\epsilon,1+\epsilon)\to B$. This is because we can extend to a map $A\times\R\to\R^N$, and letting $U$ be a tubular neighborhood around $B$ in $\R^N$, we can find an open neighborhood $V$ around $A\times I$ in $A\times\R$ that maps into $U$, and because $A$ is compact we can take $V$ to be of the form $A\times(-\epsilon,1+\epsilon)$, and then we can use the retraction from $U$ to $B$ to map everything into $B$.
\begin{center}
\begin{tikzcd}
A\times I \ar{d}\ar[white]{r}[description]{\text{\normalsize\color{black}{$\subset$}}} & V \ar{d} \ar[white]{r}[description]{\text{\normalsize\color{black}{$\subset$}}} & A\times\R \ar{d}\\
B \ar[white]{r}[description]{\text{\normalsize\color{black}{$\subset$}}} & U \ar[white]{r}[description]{\text{\normalsize\color{black}{$\subset$}}}  & \R^N
\end{tikzcd}
\end{center}
Because $A$ is compact, we can assume that $f_t$ is an embedding for all $t\in (-\epsilon,1+\epsilon)$. Define $\widetilde{F}:A\times(-\epsilon,1+\epsilon)\to B\times(-\epsilon,1+\epsilon)$ to be the map sending $(a,t)\mapsto(F(a,t),t)$. Then $\widetilde{F}$ sends $(0,\frac{d}{dt})$ to a vector field $(w,\frac{d}{dt})$. Let $C=\widetilde{F}(A\times(-\epsilon,1+\epsilon))$.

We have that $C\hookrightarrow B\times (-\epsilon,1+\epsilon)$ is closed and a section $w$ of $p_1^*TB|C$, where $p_1:B\times(-\epsilon,1+\epsilon)\to B$. There exists a global $\Cinfty$ section $\widetilde{w}$ that extends $w$. Consider $v=(\widetilde{w},\frac{d}{dt})$, which is a vector field on $B\times(-\epsilon,1+\epsilon)$. Let $\phi_t$ be the flow associated to $v$.

\textbf{Fact 1:} We know that for all $a\in A$, the map $t\mapsto (f_t(a),t)$ is an integral curve.

\textbf{Fact 2:} We may assume that $\supp(\widetilde{w})\xrightarrow{\;p_2\;}(-\epsilon,1+\epsilon)$ is proper. This implies that for all $z\in (-\epsilon,1+\epsilon)$, the flow $\phi_t(B\times z)$ is defined for all $t$ with $|t|<\delta$, say. In particular, $\phi_t(B\times z)$ is defined for all $z\in I$ and fro all $t$ with $|t|<\delta$.

\textbf{Fact 3:} We have that $\phi_t(B\times z)\subset B\times\{z+t\}$, from which it follows that for all $0\leq z\leq 1$, $\phi_t$ is defined on $B\times z$ for all $-z\leq t\leq 1-z$.

From these facts, we have that $\phi_t|_{B\times 0}\xrightarrow{\;\cong\;}B\times t$ is a diffeomorphism for all $0\leq t\leq 1$. Now define $g_t=\phi_t$ and we are done.
\end{proof}