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\begin{document}
 \parindent=0cm
  \section*{Frobenius theorem(Global part)}
\begin{thm}
  Let $\dc$ be a $k$-dimensional $\ccn$-involutive distribution on
  $M$.

  For $\forall p\in M,\exists! $ maximal connected integral
  manifold $C$ of $\dc$ through $p$.

   Furthermore, if $M$ is 2nd
  countable, so is $C$.
  \end{thm}

  \begin{proof}
Local part�κ��� �� $p\in M$�� ���� $\exists (U,x) $ coordinate
chart such that $x(U)=(-\eps,\eps)^n$ and each slice

$S_a^U=\{q\in
U\,\,|\,\,(x_{k+1}(q),\cdot\cdot\cdot,x_n(q))=a=(a_{k+1},\cdot\cdot\cdot,a_n)\},|a_{k+j}|<\eps$


is a integral manifold.\\

(1) $\{S_{\alp}^U\}$ defines a coherent topology(or foliation
topology) $\mathcal{F}$ on $M$.

$\hs{2em}$(Each $S_{\alp}^U$ has a topology of $x(S_{\alp}^U)$):\\

Check : Topology of $S_{a}^U\cap S_{b}^V$ induced from $S_{a}^U$
 is the same as that from $S_{b}^V$ and open in $S_{a}^U$(and in
 $S_{b}^V$).\\

 **�׸�24**\\

���� $S_{a}^U\cap S_{b}^V$�� topology�� $x(S_{a}^U)$�� topology��
������ ���̰� $y(S_{b}^V)$�ʵ� ���������̴�. �׷��� $M$���� chart
$x,y$�μ� transition map $h=y\circ x\inv$�� ���ǿ��� ġ��������
diffeomorphism�̹Ƿ� homeomorphism�̴�. ����  �̸�
$x(S_{a}^U\cap S_{b}^V)$�� restrict��Ų �� ����
homeomorphism�̹Ƿ� �̵鰣�� topology�� ��ġ�Ѵ�.



$S_{a}^U\cap S_{b}^V$��  $S_{a}^U$
 ,$S_{b}^V$ ���� open�̶�� �����?(Exercise)\\

 (2) ������ ���� $(M,\fc)$�� $M$�� k-dimensional submanifold�� �ǰ� $\dc$�� integral
 manifold�� �ȴ�  :  (1)������ ���������� $h|_{S_{a}^U\cap S_{b}^V}$�� $\ccn$�̹Ƿ�
 $(S_a^U,x|_{S_a^U})$�� submanifold�� �ǰ� �ϴ� chart�� �ǰ�, $(M,\fc)$�� �翬��
 $\dc$�� integral manifold�� �ȴ�.\\

 (3) $C:=$a connected component of $(M,\fc)$ containing
 $p$.

 $\fc$�� locally connected �̹Ƿ� component $C$�� open(in $\fc)$�̰�, (����
 component�� ���ǿ� ���� closed�̴�.) ���� $C$�� $\dc$�� integral
 manifold�� �ȴ�.\\

 (4) Show $C$ is maximal and unique :

 $(N,\vphi)$��  $p$�� ������ �Ǵٸ� $\dc$�� connected integral manifold��� ����. �� �� $\vphi:N\rightarrow(M,\fc)$��
 �������� ������.(����! $(M,\fc)\neq M$)



submanifold ���ǿ��� $\vphi:N\rightarrow M $�� $\ccn$�̹Ƿ�
continuous�̴�. ���� ���  cubic neighborhood $U_{\vphi(a)},a\in
N$ �� ���� $\exists W$,a connected neighborhood of $a$ such that
$\vphi(W)\subset U:=U_{\vphi(a)}$. �׷��� $W$�� connected integral
manifold�̹Ƿ� $\vphi(W)$�� single slice�� ���Եȴ�. ��
$\vphi:W\rightarrow U$�� �����̰� slice�� $U$�� subspace�̹Ƿ�
$\vphi:W\rightarrow slice$ �� �����̴�. ����
$\vphi:N\rightarrow(M,\fc)$�� continuous���� ������. (slice��
topology �� ($M,\fc$)�� subspace topology�� �����Ƿ�.)

���� $N$�� connected�̹Ƿ� $\vphi(N)$�� maximal connected $C$ in
$(M,\fc)$ �� ���Եȴ�. ���� $\vphi:N\rightarrow C$ �� �����̰�
submanifold ������ ������ ���� ������ ���� $\ccn$���� �ȴ�.\\

$(C,i)$ is a submanifold of M,
 $\psi=i\circ\vphi:\ccn\,\,\,\,and\,\,\vphi(N)\subset C\,\,.$ Then

$\hs{1em}N\overset{\psi}{\rightarrow} M$

 $\hs{1em}\vphi\searrow\hs{1em}\uparrow i\hs{4em}\,\,\vphi$ is
 continuous$\Rightarrow\vphi$ is $\ccn.$

 $\hs{3em}C$\\


�׸��� $(N,\vphi)$�� integral manifold �� $M$�� submanifold��
�ǹǷ� 1-1�� �ǰ� nonsingular(�� $\vphi_*:1-1$)�ϴ�. ���� $
\vphi$�� local diffeomorphism�� �ǰ�  $\vphi(N)\subset C$�̹Ƿ�
$(N,\vphi)$�� $C$�� open embedded submanifold�� �ȴ�. ���� $C$��
maximal, unique �ϴ�. (���� $N$�� maximal�ϴٸ� $\vphi$�� global
diffeomorphism��
�ȴ�.)\\

(5) $M$ is 2nd countable$\Rightarrow C$ is also 2nd countable:

���� $M$�� 2nd countable�̹Ƿ� $\exists$ countable cubic chart
which give slice solutions of $\dc$�̴�. �̵���
$\{U_0,U_1\cdot\cdot\cdot\}$ �̶� ����. $C$�� path connected�̹Ƿ�
������ $p\in U_0\cap C$�� ������ �� �� $q\in C$�� ���� $p$�� $q$��
�մ� path�� �����Ѵ�. �� �� path ���� �����鿡 ����  cubic����
����� covering�� ���� finite subcover�� �����ϹǷ� $\exists$ a
sequence $U_0=U_{i_1},\cdot\cdot\cdot,U_{i_p}=U_i$, $q\in U_i$ for
some $U_i$ such that $(C\cap U_{i_r})\cap(C\cap U_{i_{r+1}})\neq
\emptyset$ and $p\in U_o\cap C$ and $q\in U_i\cap C$. ����
������ ���̸� ����ϴ�.\\

1. $\exists$ only countably many possible choices of each
sequence,

$U_0=U_{i_1},\cdot\cdot\cdot,U_{i_p}=U_i$ : ���� $U_j$��
countable���̰� ���߿��� finite���� �̴� ���̹Ƿ� �翬�ϴ�.\\

2. A single slice $S$ of $U_j$ can intersect only countable many
slices of $U_k$ :

$S\cap U_k$�� $S$���� open�̰� $S$�� 2nd countable�̹Ƿ� $S\cap
U_k$ ���� 2nd countable�̴�. �� �� $S\cap U_k$�� component��
�����ϸ� �̴� locally connected�� $S$�� open set�̹Ƿ� open�� �ǰ�
 ���� $S\cap U_k$�� ���ƾ� countable ���� component�� ������.
 �̵�
 countable���� component���� $S$���� open�̹Ƿ� connected integral manifold�̴�. ���� $U_k$�� single slice�ȿ�
 ���ԵǾ�� �Ѵ�. �� $U_j$�� single slice $S$�� ���� �� �ִ�
 $U_k$�� slice�� ���ƾ� countable���̴�.
  \end{proof}



\end{document}
