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\begin{document}
 \parindent=0cm
  \section*{Regular value and transversality.}
\begin{defn}{\it
$\vphi:M^m\rightarrow N^n$, $\ccn$, $q\in N$ is called a} {\bf
regular value} {\it of $\vphi$ if\\ $d\vphi_p$ is surjective for
$\forall p\in\vphi^{-1}(q)$}.\\

\end{defn}

{\bf Note.} $q\in N\backslash im\vphi$ is also a regular value.


$\hspace{3em}q\in N$ is a {\it critical value} if it is not regular.\\

\begin{thm}
$\vphi:M^m\rightarrow N^n,\ccn$. If $q\in N$ is a regular value of
$\vphi$, then $P=\vphi^{-1}(q)$ is  $\emptyset$ or an embedded
submanifold of M of dimension $m-n$. And $T_pP=ker \,\,d\vphi_p$.
\end{thm}

\begin{proof}
   $p\in P=\vphi^{-1}(q)$ ��� ����. $ d\vphi_p$�� $p$�� �ٹ濡��
   onto�̹Ƿ� ������ �������� 4�� ���� $p,q$������ chart $(U,x)$��
   $(V,y)$�� �����ؼ� $y\circ\vphi\circ
   x^{-1}(a_1,\cdot\cdot\cdot,a_m)=(a_1\cdot\cdot,a_n)$��
   �����Ѵ�.\\

   **�׸� 19**\\

   ����  $P$��
   $x_2|_{W}=(x_{n+1}|_{W},\cdot\cdot\cdot,x_m|_W)$�� $W=U\cap P$��
   coordinate chart�� ����. �׷��� �� �� $P$�� subspace topology�� ���� $W$�� open set
�̹Ƿ� �̷� chart�鿡 ����  topological manifold�� �ȴ�.
    �̷��� ���ǵ� chart���� $P$�� smooth structure�� �ִ� ���� ���̱� ���� �� chart
    $x_2|_W,x_2'|_W$�� ����
   $\ccn$-related��
   �� ������. ������ $x_2|_W\circ x_2'|_W^{-1}=\pi\circ x\circ x'^{-1}\circ
   i$�̰� �̴�  $\ccn$�̹Ƿ� $\ccn$-related���� �� �� �ִ�. ����  ������ �������� 5�� ���� $P$�� submanifold��
   �ȴ�.


   $M$�� embedded�ȴٴ� ���� �������� 5�� ���� $U\cap P=W$��
   �����ϴ� global chart $U$�� �����Ѵٴ� ��Ƿκ��� �� �� �ִ�.
  \end{proof}\\

  {\bf Remark.} �� �������� $M$�� second countable�ϴٰ� �����ϸ�
  $P$�� second countable�ϴ�.\\

  {\bf Examples.} $\vphi:\rb^m\rightarrow
  \rb^n=(\vphi_1,\cdot\cdot\cdot,\vphi_n)$.

  $d\vphi_p$ has rank $n$ $\Leftrightarrow$ $d\vphi_p:T_p\rb^m\rightarrow T_{\vphi(p)}\rb^n:\rb^m
  \rightarrow\rb^n$ is onto.



\[ (\,\,\,\,recall\,\,\,\,\,\,\,\,d\vphi_p= \left(
         \begin{array}{cc}
              \nabla\vphi_1   \\
               \cdot\\
                 \cdot\\
             \cdot\\
              \nabla\vphi_n

          \end{array} \right)\,\,\,\,\,\,\,\,\,\, ) \]



  $\Leftrightarrow\nabla\vphi_1(p),\cdot\cdot\cdot,\nabla\vphi_n(p)$
  are linearly independent.\\

  Ư�� $n=1$ �� ���� ������ $\nabla\vphi(p)\neq 0$ �̴�. ������
  �̹� $f(x)=|x|^2-1$�� ����($\nabla f\neq 0$) $M=f^{-1}(0)$�� smooth manifold�� ���� ����
  ���� �ִ�. �̴� ���� Ư���� ��쿡 ���Ѵ�. �̰��� �Ϲ�ȭ�� ����
  ������ �ٷ� $\nabla\vphi_1(p),\cdot\cdot\cdot,\nabla\vphi_n(p)$
  �� linearly independent�� �����̴�. \\

  \begin{defn}{\it
  $\vphi:M^m\rightarrow N^n$, $\ccn$ and $i:Q\hookrightarrow N$ is a
  sbmanifold. $\vphi$ is said to be {\bf transversal} to $Q$ ($\vphi\pitchfork
Q$) if
  $\,\,im(d\vphi_p)+T_{\vphi(p)}Q=T_{\vphi(p)}N$, $\forall p\in
  \vphi^{-1}(Q)$.}\\
  \end{defn}

���� ���Ƿ� ���� 1�� �� �Ϲ�ȭ�Ͽ� $\vphi^{-1}(Q)$�� submanifold�� ���� ������.\\

\begin{thm}
$\vphi:M^m\rightarrow N^n$, $\ccn$ and $i:Q\hookrightarrow N$, a
(embedded, respectively) submanifold, and let $\vphi\pitchfork Q$.
Then $P=\vphi^{-1}(Q)$ is a (embedded, respectively) submanifold
of $M$ with $codim \,\,P=codim\,\,Q$ and
$T_pP=d\vphi_p\inv(T_{\vphi(p)}Q)$.($P$ is second countable if $Q$
is second countable.)
\end{thm}

\begin{proof}

\begin{figure}[htb]
 \centerline{\includegraphics*[scale=0.75,clip=true]{grp20.eps}}

\end{figure}

�׸� 20\\

$Q$�� submanifold �̹Ƿ� $\forall q\in Q$�� ���� �� �׸��� ����
chart $(V,y)$ �� �����ؼ� local chart $W$�� �ش�. $W$�� slice
neighborhood��� �θ���.  �� �� �����Ǵ� $\rb^n,\rb^q$�� open
set�� ���� $"V","W"$ ��� ����.  ��  projection map

$p_2:\rb^n\rightarrow \rb^{n-q}$ �� �����ؼ� $p_2\circ y=\pi_2$
��� ����.

���� $\vphi\inv(V)=U$ ��� ���� $\vphi$�� $U$�� restriction��Ų
$\vphi_U$�� ���� $\pi_2\circ\vphi_U$�� �����غ���.
$\pi_2\circ\vphi_U$�� $U\rightarrow\rb^{n-q}$ �� ���� �Լ��̰�
$\forall p\in P\cap U$�� 0���� ������. �� �� 0��
$\pi_2\circ\vphi_U$�� regular value�� �ǹǷ�(transversality
$\vphi\pitchfork Q$�κ��� $d(\pi_2\circ\vphi_U)$�� onto�̱�
�����̴�.) ���� 1�� �� �� �ִ�. ����
$(\pi_2\circ\vphi_U)^{-1}(0)$�� $U$�� submanifold�� �ȴ�.
$(\pi_2\circ\vphi_U)^{-1}(0)=(\vphi_U^{-1}\circ\pi_2^{-1})(0)=\vphi_U^{-1}(W)=\vphi^{-1}(W)$
�� $U$�� embedded submanifold�� �ȴ�.\\

���� ���� locally�� �� ���ǵǾ��� ���� �� $\forall q\in Q$�� ����
���� ���� �����  submanifold $\vphi^{-1}(W)$���� �� �ٿ��� $P$��
�� �� �ִ�. $P$�� submanifold�� ���� ������.\\

{\bf 1. Topology of $P$.}

$\forall q\in Q$�� ���� �����ϴ� slice neighborhood $W$�� ����
���� �����ϰ�

$Q=\cup W_i$�� $W_i$���� ��� $\{W_i\},\{V_i\}$���� ��������.
(���� $Q$�� second countable�� ��쿡�� $\{W_i\}$ ��
conuntable�̶�� �� �� �ִ�.) �׷��� �� $\vphi\inv(W_i)$����
$U_i=\vphi\inv(V_i)$��, ���� $M$�� embedded submanifold�� �ȴ�.
�� �� $\vphi\inv(W_i)$ ���� $M$�� subspace topology�� �����Ƿ�
$P=\dis{\cup\vphi\inv(W_i)}$�� coherent topology�� ����. �̸� ����
���� �� ������ ���̸� ����ϴ�.

(check 1) $\vphi\inv(W_i)\cap\vphi\inv(W_j)$�� $\vphi\inv(W_i)$,
$\vphi\inv(W_j)$ ������ subspace topology�� ���� ��� $M$������
subspace topology�� ����. ���� �� ���� topology�� ����.

(check 2) $\vphi\inv(W_i)\cap \vphi\inv(W_j)$ ��
$\vphi\inv(W_i)$���� open���� ������.  ����


$\vphi|:\vphi\inv(W_i)\rightarrow W_i$�� �����Լ��̴�.  $W_i$��
topology�� $N$�� subspace topology�̰�, $\vphi\inv(W_i)$��
topology�� $M$�� subspace topology �̹Ƿ� $\vphi|$�� �����̴�.


$\vphi\inv(W_i)\cap\vphi\inv(W_j)=\vphi\inv(W_i\cap W_j)$ �̰�
$W_i\cap W_j $�� open in $W_i$ �̹Ƿ� �ᱹ $\vphi\inv(W_i)\cap
\vphi\inv(W_j)$ �� $\vphi\inv(W_i)$���� open�̴�.\\
���� $P$�� coherent topology�� �� �� �ִ�. �� ���� ������ ����
���� ���� �� coherent topology�� subspace topology���ٵ� finer
�ϴٴ� ���̴�.\\

{\bf ���� 6.} Note that coherent topology of $P\supset$ subspace
topology of $P$. i.e., $i:P\hookrightarrow M$ is continuous.\\



{\bf 2. Smooth structure of $P$.}

�̴� submanifold �� smooth structure�� ���� {\it uniqueness} ��
���� �� �ִ�. �� $\vphi\inv(W_i)\cap\vphi\inv(W_j)$�� smooth
structure�� $\vphi\inv(W_i)$�κ��� ���� ���� �ְ�,
$\vphi\inv(W_j)$�κ��͵� ���� �� �ִ�. �׷��� {\it uniqueness of
smooth structures of submanifold}�� ���� �� �� structure�� ���ƾ�
�Ѵ�. ���� $P=\dis{\cup \vphi\inv(W_i)}$�� ���� ��
$\vphi\inv(W_i)$�鿡 ���� smooth structure�� �ֵ� ��ġ��
�κп����� structure�� ���� �����Ƿ� ��ü������ $P$�� smooth
structure�� �ְ� �ǰ� $M$�� submanifold�� �ȴ�.\\

{\bf 3. $Q$ is embedded$\Rightarrow$ $P$ is embedded.}

$Q$�� embedded submanifold ��� �� $q\in Q$�� ���� $V\cap
Q=W(slice \,\,neighborhood)$

�� �Ǵ� �ٹ� $V$�� ���� �� �ִ�. $P$�� embedding���� ���̷���
$P$�� coherent topology�� subspace topology�� ���Ե��� ���̸�
����ϴ�. �ֳ��ϸ�

$i:P\,\,with\,\, subspace\,\,topology\hookrightarrow M$�� �翬��
embedding �̹Ƿ� ����


$coherent\,\,topology=subspace\,\,topology$�� ���̸� $P$��
(coherent topology�ν�) embedded submanifold�� �ȴ�.

$coherent\,\,topology\subset subspace\,\,topology$�� ������.


coherent topology������ open set�� $\vphi\inv(W)$���� open
subset�鿡 ���� generate�ǰ� $\vphi\inv(W)=\vphi\inv(V\cap
Q)=\vphi\inv(V)\cap\vphi\inv(Q)$�̴�.
$\vphi\inv(V)\cap\vphi\inv(Q)=U\cap P$�̰� �̸� $M$������ subspace
topology�� ���ڸ�  $U$�� $M$���� open�̹Ƿ� ����
$P\,\,with\,\,subspace\,\,topology$���� open�̴�.

�� $coherent\,\,topology\subset subspace\,\,topology$ �̰� ����
�� ���� 6�� ���� �� topology�� ����.
�׷��Ƿ� $P$ ���� embedded submanifold�� �ȴ�.\\

���������� $codim\,P=codim\,\vphi\inv(W)=n-q=codim\,Q$ �̰�,


\begin{eqnarray*}
T_pP&=&ker\,d(\pi_2\circ\vphi_U)_p\,\,\,\,\,\, (by \,\,thm\,1.)\\
&=&ker\,(d\pi_2\circ d\vphi_U)_p\\
&=&(d\pi_2\circ d \vphi_U)\inv(0)\\
&=&d\vphi_p\inv(T_{\vphi(p)}Q)\,\,\,\,\,\,\,(d\pi_2\inv(0)=T_{\vphi(p)}W=T_{\vphi(p)}Q)
\end{eqnarray*}







\end{proof}

\begin{defn}{\it
Let $P$ and $Q$ be submanifolds of $M$. $P$ and $Q$ are
transversal, $P\pitchfork Q$, if  $\,\,\,i:P\hookrightarrow M$ is
transversal to $Q$. i.e., if $\forall p\in P\cap Q$, $T_pP+
T_pQ=T_pM$ holds.}
\end{defn}

\begin{cor}
If two(embedded respectively) submanifold $P$ and $Q$ of $M$ are
transversal, then their intersection $P\cap Q$ is a (embedded
respectively) submanifold of $P$,$Q$ and hence of $M$.


\end{cor}

\begin{proof}
$P\cap Q=i_p\inv(Q)$($i_p:P\hookrightarrow M$ = inclusion map)�̰�
$Q$�� $M$�� submanifold �̹Ƿ� ���� 2�� ���� $i_p\inv(Q)$ ����
submanifold�� �ȴ�. $P$�� ���ؼ��� ���������̰� embedded
submanifold�� ��� ���� ���������̴�.
\end{proof}




\end{document}
