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\usepackage{longtable}
\newcommand{\pauo}{\frac{\partial}{\partial u_1}}
\newcommand{\paui}{\frac{\partial}{\partial u_i}}
\newcommand{\pauj}{\frac{\partial}{\partial u_j}}
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\newcommand{\ml}{\mathcal{L}}
\newcommand{\hs}{\hspace}
\newcommand{\inv}{^{-1}}
\newcommand{\vphi}{\varphi}
\newcommand{\pax}{\frac{\partial}{\partial x}}
\newcommand{\pay}{\frac{\partial}{\partial y}}
\newcommand{\paz}{\frac{\partial}{\partial z}}
\newcommand{\paxi}{\frac{\partial}{\partial x_i}}
\newcommand{\payj}{\frac{\partial}{\partial y_j}}
\newcommand{\paxj}{\frac{\partial}{\partial x_j}}
\newcommand{\payi}{\frac{\partial}{\partial y_i}}
\newcommand{\li}{[X,Y]}
\newcommand{\dis}{\displaystyle}
\newcommand{\disi}{\displaystyle{\sum_{i=1}^n}}
\newcommand{\disj}{\displaystyle{\sum_{j=1}^n}}
\newcommand{\pa}{\partial}
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\newcommand{\dc}{\mathcal{D}}
\newcommand{\ccn}{\mathcal{C}^{\infty}}
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\begin{document}
 \parindent=0cm
  \section*{Distribution and Frobenius theorem(Local part)}

  \begin{defn}
A $k-$dimensional {\bf distribution} $\dc$ on $M^n$ is a choice of
a $k-$dimensional subspace $\dc(p)\subset T_pM$ for each $p\in M.$
 $\dc$ is $\ccn$ if $\forall p\in M,\,\,\exists U$ a neighborhood
 of $p$ and  $X_1,\cdot\cdot\cdot,X_k$, $\ccn$ vector fields on $U$
 such that $\dc(p)=span\{X_1(p),\cdot\cdot\cdot,X_k(p)\}$.

 For a vector field $X$, $X\in\dc$ if $X(p)\in\dc(p)\,,\,\,\forall p\in
 M$.

 We consider only $\ccn$ distributions. $\dc$ is {\bf involutive}(or
 completely integrable) if $\li\in\dc$ for $\forall\ccn$ vector fields $ X,Y\in\dc.$

 \end{defn}

\begin{defn}
A submanifold $(N,\vphi)$ of $M$ is an {\bf integral manifold} of
$\dc$ on $M$ if $\vphi_*(T_pN)=\dc(\vphi(p))\,\,\,\,\forall p\in
N.$\\
\end{defn}

{\bf ���� 12-1.} $X_1,\cdot\cdot\cdot,X_k:\ccn$ vector fields
spanning $\dc$. Then

$ \dc$ is involutive if and only if
$[X_i,X_j]\in\dc\,\,,\,\,\forall i,j.$

\begin{prop}
Let $\dc$ be a distribution on $M$.

 $\forall p\in M,\exists$
an integral manifold of $\dc$ through $p\Rightarrow\dc$ is
involutive.
\end{prop}

\begin{proof}
$X,Y\in\dc$ �� ���� $\li\in\dc$�� ���̱� ���� �� $p\in M$�� ����
$(N,\vphi)$�� $p$�� ������ $\dc$�� integral manifold�� ����.
integral manifold�� ���ǿ� ����
$\vphi_*(\overline{X})=X,$$\vphi_*(\overline{Y})=Y$ �� �����ϴ�
vector fields $\overline{X},\overline{Y}$ �� $N$�� �����Ѵ�.  ��
�� $\overline{X},\overline{Y}$�� $N$���� $\ccn$ vector field��
�ȴ�.(�� �׷��� ������.)

$\li_{\vphi(p)}=[\vphi_*\overline{X},\vphi_*\overline{Y}]_{\vphi(p)}=\vphi_*[\overline{X},\overline{Y}]_p
\in\vphi_*(T_pN)\in\dc $.

���� $X,Y\in\dc\Rightarrow\li\in\dc$���� ������.
\end{proof}

\begin{thm}(Frobenius theorem : Local part)


Let $\dc$ be a $k-dimensional\,\,\,\ccn$ distribution on $M^n$.

$\dc$ is involutive $\Rightarrow\forall\,p\,\in M,\,\exists\,$ an
integral manifold of $\dc$ through $p$.

In fact , $\exists$a coordinate chart $(U,x)$ with $x(p)=0$ and
$x(U)=(-\eps,\eps)^n$ such that
$\forall(a_{k+1},\cdot\cdot,a_n)\in\rb^{n-k}$ with $|a_i|<\eps,$
the slice

$\{q\in U,\,|\,\,x_{k+1}(q)=a_{k+1},\cdot\cdot,x_n(q)=a_n\}$ is an
integral manifold of $\dc$.

And if $(N,\vphi)$ is a connected integral manifold of $\dc$ such
that $\vphi(N)\subset U$,

then $\vphi(N)$ lies in one of these slices.
\end{thm}

**�׸� 23**\\


\begin{proof}
local problem�̹Ƿ� $p=0\in\rb^n$���� �ΰ�
$\dc(0)=span\{\pauo(0),\cdot\cdot,\frac{\pa}{\pa u_k}(0)\}$

�̶� ����. (linear coordinates change�� ���� �̷��� �ξ
�������.) canonical projection $\pi:\rb^n\rightarrow\rb^k$��
�����ϸ�

$\pi_*:\dc(0)\rightarrow T_0\rb^k$�� ��ǻ� identity�̹Ƿ�
isomorphism�� �ȴ�.

(0���� $\dc(0)=span\{\pauo(0),\cdot\cdot,\frac{\pa}{\pa u_k}(0)\}$
�̰� �̴� $T_0\rb^k$�� ����.)

�׷��� continuity���������� $0$�ٹ��� $p$�� ���ؼ���
$\pi_*:\dc(p)\rightarrow T_{\pi(0)}\rb^k$�� isomorphism�� �ȴ�.
����

$\exists! X_1(q),\cdot\cdot,X_k(q)\in\dc(q)$ such that
$\pi_*X_i(q)=\paui(\pi(q))\,\,,\,\,i=1,\cdot\cdot,k$.

Note that each $X_i $ is $\ccn$({\bf ���� 12-2.})  and
$\pi-$related to $\paui.$

$\therefore\pi_*[X_i,X_j]_q=[\pi_*X_i,\pi_*X_j]_{\pi(q)}=[\paui,\pauj]_{\pi(q)}=0$

$\rb^n$������ $\paui,\pauj$�� ���� commute�ϹǷ� ������ 0�� �ȴ�.
�׷��� involutive�ϴٴ�  ���ǿ��� $[X_i,X_j]_q\in\dc(q)$�̰�,
$\pi_*$�� $\dc(q)$��  isomorphism�� �ֹǷ� $[X_i,X_j]_q=0$ �̾��
�Ѵ�. ���� ������ ����6�� ���� $\exists$ coordinate chart
$(U,x)$ such that $X_i=\paxi$ on $U$ �̰� ���� �� slice��
$\dc$�� integral manifold�� �ȴ�.\\

$(N,\vphi)$�� connected integral manifold with $\vphi(N)\subset U
$ �� ���\\

$\hs{2em}N\overset{\vphi}{\rightarrow}U\overset{x}{\rightarrow}x(U)\overset{p_2}{\rightarrow}\rb^{n-k}$\\

���� $N$�� integral manifold��� ��� ������ $(p_2\circ
x\circ\vphi)_*=0$ �� �ǰ� ���� $(p_2\circ
x\circ\vphi)=locally\,\,\,constant$ �̴�. �׷��� $N$�� connected
�̹Ƿ� $(p_2\circ x\circ\vphi)=constant$ �̴�. ���� $\vphi(N)$��
single slice�� ���ԵǾ�� �Ѵ�.
\end{proof}\\

$<${\bf Frobenius theorem as P.D.E. problem}$>$\\


{\bf ��.} $M=\rb^3$, $\dc(p)=span\{X=\pax+f(p)\frac{\pa}{\pa
z},Y=\pay+g(p)\frac{\pa}{\pa z}\}$.

$\dc$�� involutive������ ���� ���ؼ��� $X,Y$�� ���ؼ���
$\li\in\dc$ ������ üũ�ϸ� �ȴ�. ���� ����� �غ���
$\li=(\frac{\pa g}{\pa x}-\frac{\pa f}{\pa y}+f\frac{\pa g }{\pa z
}-g\frac{\pa f}{\pa z})\paz$ �̰� �� ���� $\dc$�ȿ� ��������

$\hs{3em}\frac{\pa g}{\pa x}-\frac{\pa f}{\pa y}+f\frac{\pa g
}{\pa z }-g\frac{\pa f}{\pa z}=0\hs{2em}(*)$

�� �Ǿ�� �Ѵ�. �� $\dc$�� involutive���� Ȯ���ϴ� ���� $(*)$��
���̴� �Ͱ� ����.  ���� $\dc$�� $(*)$�� �����Ѵٸ� Frobenius
������ ����, Ȥ�� ���������� ������ ���� integral manifold��
�������� ���� �� �ִ�.\\


Show integrability directly using $(*)$ :

We want $\alp:\rb^2\rightarrow\rb$ such that

\[  (*)'\left\{
\begin{array}{cc}
\frac{\pa\alp}{\pa x}=&f(x,y,\alp(x,y))  \\
              \frac{\pa\alp}{\pa y}=&g(x,y,\alp(x,y))
\end{array} \right. \]\\


���� $(*)'$�� �����ϴ� $\alp$�� �����Ѵٸ� $\frac{\pa^2\alp}{\pa
x\pa y }=\frac{\pa^2\alp}{\pa y\pa x }$ �������� $(*)$�� ��������
�� �� �ִ�.  ������ $\alp$�� ���缺�� $(*)$�� ������ ����ϴٴ�
���� ������.

(�̴� Frobenius ������ �����ϴ� �ٸ� ����̱⵵ �ϴ�. )\\

(1st step) $y=b$�� ������Ű�� $\frac{\pa\alp}{\pa
x}=f(x,b,\alp(x,b))$ �� Ǯ�� �̴� O.D.E �̹Ƿ� �ذ� �����ϰ�
�����Ѵ�.\\

(2nd step) �� ������ $x$�� ����  ���� ���� ��������.

$  \frac{\pa\alp}{\pa y}=g(x,y,\alp(x,y))$ with $\alp(x,b)=$given
by 1st step.

�̰� ���� O.D.E�̹Ƿ� ������ �ظ� ã�� �� �ְ�, local flow
theorem�� ����($\ccn$ dependence on initial data)  $\alp(x,y)$��
$x,y$�� ���� ��� $\ccn$ �̴�.\\

(3rd step) ������ ã�� $\alp$�� ������ $(*)'$�� �����ϴ�����
Ȯ������. $(*)'$�� �Ʒ��Ŀ��� $\alp$�� ������Ƿ� ���� ������
�����ϴ����� ���캸�� �ȴ�.

Let $h(y)=\frac{\pa\alp}{\pa x}-f(x,y,\alp(x,y)) $. Then $h(b)=0$
and

$h'(y)=\pay\frac{\pa\alp}{\pa x}-(\frac{\pa f}{\pa y}+\frac{\pa f
}{\pa z}\frac{\pa \alp }{\pa y})\hs{3em} where\,\,\,z=\alp(x,y)$.

$\hs{2.5em}=\frac{\pa g}{\pa x}+\frac{\pa g}{\pa
z}\frac{\pa\alp}{\pa x}-(\frac{\pa f}{\pa y}+\frac{\pa f }{\pa
z}g)$

$\hs{2.5em}=\frac{\pa g}{\pa x}+\frac{\pa g}{\pa
z}(h+f)-(\frac{\pa f}{\pa y}+\frac{\pa f }{\pa z}g)$

$\hs{2.5em}=\frac{\pa g}{\pa z}h\hs{2em} by \,\,\,(*)$

$\therefore h'(y)=\frac{\pa g}{\pa z}h$     ;     O.D.E

�׷��� $h(y)\equiv 0$�� �� �̺й������� �ذ� �ǹǷ�, uniqueness��
���� $h(y)\equiv 0$�̴�. ���� 2nd step���� ã�� $\alp$�� �ٷ� �츮�� ã�� �ذ� �ȴ�.\\

�� �Ϲ������� ������ ���� $\ccn$ distribution�� �����غ���.

$\dc(p)=\dis{<\pauo+\sum_{i=1}^{l} f_1^i\frac{\pa}{\pa
u_{k+i}},\cdot\cdot\cdot,\frac{\pa}{\pa u_k}+\sum_{i=1}^{l}
f_k^i\frac{\pa}{\pa u_{k+i}}>}\equiv<X_1,\cdot\cdot\cdot,X_k>$

integrability condition : $[X_i,X_j]=[\paui+\sum_{p}^{l}
f_i^p\frac{\pa}{\pa u_{k+p}}\,\,\,,\,\,\pauj+\sum_{q}^{l}
f_j^q\frac{\pa}{\pa
u_{k+q}}]\in \dc$\\

$\Rightarrow \sum_p(*)\frac{\pa}{\pa u_{k+p}}\in\dc$ $\therefore
(*)=0\,\,,\,\forall i,j.$

$(*)$�� vector notation $f_j=(f_j^1,\cdot\cdot\cdot,f_j^l)$��
�̿��� �ٽ� ���� ������ ���� ǥ���� �� �ִ�.

$\frac{\pa f_j}{\pa u_i}-\frac{\pa f_i}{\pa u_j}+\sum_q
f_i^q\frac{\pa f_j}{\pa x_q}-\sum_qf_j^q\frac{\pa f_i}{\pa
x_q}=0$, $x_q=u_{k+q}$  ($q=1,\cdot\cdot\cdot,l$).\\\\

\begin{thm}
Let $U\times V\subset\rb^k\times\rb^l=\{(u,x)\}$, with open $U$
and $V$, $k+l=n$ and $f_i:U\times V\rightarrow\rb^l$, $\ccn$
functions, $i=1,\cdot\cdot,k$. Then for each $(a,b)\in U\times V$,
$\exists W\subset U$, a neighborhood of $a$ and $\exists !
\alp:W\rightarrow V$ such that

\[  \left\{
\begin{array}{ll}
\alp(a)=&b  \\
              \frac{\pa\alp}{\pa u_i}(u)=&f_i(u,\alp(u))\,\,\,,\,\, u\in W.
\end{array} \right. \]\\

if and only if $(*)$ holds on $U\times V$.


\end{thm}

{\bf(����)} �տ��� �̹� ���� $(k,l)=(2,1)$�� ���� ����.


 Reference : Spivak p.254











  \end{document}
