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\begin{document}
 \parindent=0cm
  \section*{Definitions.}

\begin{defn}{\it
Let $\phi:M\rightarrow N$ be a $\ccn$ map.

 (1)$\phi$ is an immersion
if $d\phi_p$ is injective for $\forall p\in M$.

(2)$(M,\phi)$ is a submanifold of $N$ if $\phi$ is an injective
immersion.

(3)$\phi$ is an embedding if $\phi$ is an injective immersion
which is also a topological embedding.

(4) $\phi$ is a submersion if $d\phi_p$ is surjective for $\forall
p\in M$.}

\end{defn}


\begin{figure}[htb]
 \centerline{\includegraphics*[scale=0.5,clip=true]{grp15.eps}}

 \end{figure}

$\hspace{15em}$ �׸� 15\\


���� ��� projection map�� submersion�� �ȴ�.\\

{\bf Local picture}

\begin{lem}
$f:U(\subset\rb^m)\rightarrow\rb^n,\ccn$ and has constant rank $r$
on a neighborhood of $p\in U$. Then $\exists$a rectangular
coordinate charts $x$ about $p$ and $y$ about $f(p)$ such that

 $\hspace{3em}(y\circ
f\circ
x^{-1})(a_1,\cdot\cdot\cdot,a_m)=(\overset{n}{\overbrace{a_1,\cdot\cdot\cdot,a_r,0,\cdot\cdot\cdot,0}}).$

\end{lem}

\begin{proof}
May assume $det(\frac{\pa f_i}{\pa u_j})\neq 0$,
$i,j=1,\cdot\cdot,r$ on $U$ by rearranging coordinates $u_i$ on
$\rb^m$ and $\rb^n$ and by restricting $U$.

Define $x:U\rightarrow\rb^m$ by
$x(u)=(f_1(u),\cdot\cdot,f_r(u),u_{r+1},\cdot\cdot,u_m)$, then



\[ Dx= \left(
         \begin{array}{rr}
              (\frac{\pa f_i}{\pa u_j})_{r\times r} & * \\
              0            \,\,\,\,\,\,\,\,      &  I
          \end{array} \right)_{m\times m} :\,\,nonsingular \]

By the inverse function theorem, $x$ is a coordinate chart on a
neighborhood $"U"$ of $p$. Let $x(p)=(a,b)$ and $V_r,V_{m-r}$ as
in the picture so that $V_r\times V_{m-r}\subset
dom(x^{-1})$.\\

\vspace{3em}

\begin{figure}[htb]
 \centerline{\includegraphics*[scale=0.4,clip=true]{grp16.eps}}

 \end{figure}

$\hspace{15em}$ �׸� 16\\


$(a,b)\overset{x^{-1}}{\mapsto}(h(a,b),b)\overset{x}{\mapsto}(f_{1,\cdot\cdot\cdot,r}(h(a,b),b),b)=(a,b)$


$\Rightarrow f_{1,\cdot\cdot\cdot,r}(h(a,b),b)=a$ (independent of
$b$)

$\Rightarrow (f\circ
x^{-1})(a,b)=(a,f_{r+1,\cdot\cdot,n}(h(a,b),b))$

Consider

\[  D(f\circ x^{-1})= \left[
         \begin{array}{cc}
              I & 0 \\
              * & \,\,\,\,\, (\frac{\pa (f_i\circ x^{-1} )}{\pa
              u_j})
          \end{array} \right]_{i=r+1,\cdot\cdot,n\,\,and\,\,j=r+1,\cdot\cdot,m}
          \]


$rank\,\,\,D(f\circ x^{-1})=rank\,\,Df=r$ at $\forall p$    of
$U$.

$\therefore (\frac{\pa(f_i\circ x^{-1})}{\pa u_j})=0$ for
$i=r+1,\cdot\cdot,n$ and $j=r+1,\cdot\cdot,m$.(i.e.,
$\frac{\pa(f_{r+1\cdot\cdot, n}\circ x^{-1})}{\pa b}=0$)

$\therefore f_{r+1,\cdot\cdot,n}\circ
x^{-1}(a,b)=f_{r+1,\cdot\cdot,n}(h(a,b),b)=g(a)$ (i.e.,
 independent of $b$) for some $\ccn$ function $g$.



Define coordinate chart $y$ on $V_r\times \rb^{n-r}$ by
$y(a,c)=(a,c-g(a))$, then

 $(y\circ f\circ
x^{-1})(a,b)=y(a,g(a))=(a,0)\hspace{2em}$ and hence

$(y\circ f\circ
x^{-1})(a_1,\cdot\cdot\cdot,a_r,a_{r+1},\cdot\cdot,a_m)=(a_1,\cdot\cdot,a_r,0,\cdot\cdot,0)$
on $V_r\times V_{m-r}$.
\end{proof}\\

\begin{cor}
$\phi:M^m\rightarrow N^n\,\,is\,\,\ccn,$  $d\phi$ has constant
rank $\,r$ on neighborhood of $p\in M.$ Then $\exists$ rectangular
coordinate charts $x$ about $p$, y about $f(p)$ such that

$(y\circ \phi\circ
x^{-1})(a_1,\cdot\cdot\cdot,a_m)=(a_1,\cdot\cdot,a_r,0,\cdot\cdot,0)$.
\end{cor}

\begin{cor}
$\phi:M^m\rightarrow N^n$ ,  $\ccn$, is an immersion($m\leq n$).

$\Rightarrow \forall p\in M,\exists\,\,$rectangular coordinate
charts $x$ about $p$, y about $f(p)$ such that $(y\circ\phi\circ
x^{-1})(a_1,\cdot\cdot\cdot,a_m)=(a_1,\cdot\cdot,a_m,0,\cdot\cdot,0)$
\end{cor}

\begin{proof}
$d\phi$�� injective�ϹǷ� rank�� m�̰� ���� �ٷ� ���� ������
������ �ȴ�.
\end{proof}\\

\begin{cor}
$\phi:M^m\rightarrow N^n$,$\ccn$, is a submersion ($m\geq n) $
then $\forall p\in M$,

$\exists\,$ rectangular coordinate charts $x$ about $p$, y about
$f(p)$ such that

$(y\circ\phi\circ
x^{-1})(a_1,\cdot\cdot,a_n,\cdot\cdot,a_m)=(a_1,\cdot\cdot\cdot,a_n)$.\\
\end{cor}

{\bf Remark.} Hence an immersion is locally an inclusion and so
embedding.(the local topology is the same as that of  a slice)\\

\begin{cor}
(1) $i:M^m\hookrightarrow N^n$ is a submanifold

$\Leftrightarrow \forall p\in M,\exists$ rectangular coordinate
chart $(U,x)$ about $p$ such that $x(p)=0$, and $V=\{p\in
U\,\,|\,\,x_{m+1}(p)=\cdot\cdot\cdot=x_n(p)=0\}$ is a neighborhood
of $p$ in M and $(x_1|_{V},\cdot\cdot\cdot,x_m|_{V})$ is a
coordinate chart of $M$.\\

(2) $i:M^m\hookrightarrow N^n$ is an embedding

$\Leftrightarrow \forall p\in M,\exists$ rectangular coordinate
chart $(U,x)$ about $p$ such that $x(p)=0$, $V=\{p\in
U\,\,|\,\,x_{m+1}(p)=\cdot\cdot\cdot=x_n(p)=0\}$ is a neighborhood
of $p$ in M , $(x_1|_{V},\cdot\cdot\cdot,x_m|_{V})$ is a
coordinate chart of $M$ and $M\cap U=V$ .\\

\end{cor}


\begin{figure}[htb]
 \centerline{\includegraphics*[scale=0.5,clip=true]{grp17.eps}}

 \end{figure}

$\hspace{15em}$ �׸� 17\\


\begin{proof}
(1)�� $\Rightarrow$ �� �������� 3�� ���� ���� �� �ִ� coordinate
chart��� local inclusion map�� �̿��ϸ� �ǰ�, $\Leftarrow$��
$i$�� locally immersion�̶�� ��Ƿκ��� �翬�ϴ�.

(2)�� $\Rightarrow$�� ������. (1)�� ������ �����ϸ鼭 $M\cap U=V$
�� �����ϴ� $U$�� ������ ���̸� �ȴ�. embedding�� immersion�̱⵵
�ϹǷ� (1)�� ������ �����ϴ� $U$�� �����ϰ�, �̿� ���� $i(V)$��
�����غ���. $i$�� embedding�̹Ƿ� $i(V)$�� $i(M)$���� open�̴�.
���� $i(V)=i(M)\cap U'$ ( $U'$ is open in $N$) �� �� �� �ְ�,
�츮�� ���ϴ� $"U"$�� $"U"=U\cap U'$���� �θ� �ȴ�. (rectangular
coordinate�� �������� �ٽ� $"U"$ �ȿ��� �� ���� $p$�� rectangular
�ٹ��� ������ �ȴ�.)

(2)�� $\Leftarrow$�� ���̱� ���ؼ��� $i(open)=open$ in $i(M)=M$
���� ���̸� �ȴ�. �׷��� $i(V)=V=M\cap U$ ���� $U$�� N�� open
set�̹Ƿ� $M\cap U $�� $i(M)=M$���� open�� �ȴ�. basic open set��
���� $i(basic \,\,open)=open$�� �����ϹǷ� $i$�� open map(onto
$i(M)$ )���� �� �� �ִ�.

\end{proof}\\

{\bf Remark.} Corollary 3 shows {\it the uniqueness of smooth
structure on a submanifold }i.e., if a subset $M\subset N$ has a
topology, then there is at most one smooth structure on $M$ such
that $i:M\hookrightarrow N$ is a submanifold.\\

\begin{proof}
�� ���� $\mathcal{F}_1,\mathcal{F}_2$�� $i:M\hookrightarrow N$ ��
submanifold�� ����� smooth structure��� ����. �� structure ��
���� $i$�� submanifold�̹Ƿ� ��������5�� (1)�� ���� chart
$(U_1,\phi_1),V_1,(U_2,\phi_2),V_2$�� ���� �� �ִ�.

$(U_1,\phi_1)\in\mathcal{F}_1,(U_2,\phi_2)\in\mathcal{F}_2$ �� ��
$U_1\cap U_2$�� �����غ���.

���� $\phi_1,\phi_2\in \mathcal{F}(N)$�̹Ƿ� $\ccn$�� transition
map $\phi_2\circ \phi_1^{-1}$�� �����Ѵ�. �̰��� $\phi_1(V_1)$��
restriction��Ű�� �̰� ���� $\ccn$�̴�. �ֳ��ϸ� �츮�� ���ϴ�
$\phi_1(V_1)$���� $\phi_2(V_2)$�� ���� transition map
$\phi_2|_{V_2}\circ\phi_1|_{V_1}^{-1}$��
$\pi\circ\phi_2\circ\phi_1^{-1}\circ i$ �� ǥ���Ǳ� �����̴�.
���� $\phi_1|_{V_1}$, $\phi_2|_{V_2}$�� $\ccn$-related�Ǿ� �ְ�
$\mathcal{F}_1=\mathcal{F}_2$�̴�.
\end{proof}

\begin{prop}
Let $\varphi:M\rightarrow N$ be $\ccn$ , $P$ is a submanifold of
$N$ and $\vphi(M)\subset P$.

If $\vphi:M\rightarrow P$ is continuous, then $\vphi$ is $\ccn$.
\end{prop}

{\bf ���� 5.} �� ������ �����϶�.\\

�� �������� ���Ӽ��� �ʿ��� ������ ������ ���� �������� �� ��
�ִ�.\\


\begin{figure}[htb]
 \centerline{\includegraphics*[scale=0.5,clip=true]{grp18.eps}}

 \end{figure}

$\hspace{15em}$ �׸� 18\\


$N=\rb^2$,$M=(-1,1)$�̰� $P$�� figure eight���� ���� �� ��
$i:M\rightarrow N$�� $\ccn$ �̳� $M\rightarrow P$�� ���� map��
�ҿ����̴�.









  \end{document}
