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\begin{document}
 \parindent=0cm
  \section*{Examples and review of implicit function theorem.}

{\bf 1. $\rb^n$.}

$\{(\rb^n,id)\}$�� �����ϸ�, �̴� atlas�� �ǰ� �� �� �̰���
standard structure��� �Ѵ�. ������, n=4�̿��� ��� $\rb^n$��
���ؼ��� standard structure �̿��� structure(exotic structure)��
�������� ������ �˷��� �ִ�. n=4�� ��� ������ ���� exotic
structure�� ������ �ִ�.\\

{\bf 2.  $\mathbb{S}^n$} :
�� ���� �������  ���캸��.\\


(1)
\begin{figure}[htb]
    \centerline{\includegraphics*[scale=0.6,clip=true]{grp6.eps}}

    \end{figure}

$\hspace{15em}$ �׸� 6\\


$\sbb^n$����  �ϱ� $p_{+}$�� ������ ������ $U_+$�� $p_{-}$��
������ ������ $U_-$  �ΰ���  open set�� �����غ���. �׸��� ������
open set���� stereographic projection�� ���غ���. ����
$\mathbb{S}^n \backslash \{p_{+}\}$ ������ stereographic
projection $\phi_{+}$ �� ���غ���

$\phi_+ : \sbb^n \backslash \{p_+\} \rightarrow
\rb^n\hspace{0.5em},\hspace{0.5em}$ $(x_1,\cdot\cdot\cdot
,x_{n+1})\mapsto y=\frac{1}{1-x_{n+1}}(x_1,\cdot\cdot\cdot
,x_n)$�̰� ����

$\phi_- : \sbb^n \backslash \{p_-\} \rightarrow
\rb^n\hspace{0.5em},\hspace{0.5em}$ $(x_1,\cdot\cdot\cdot
,x_{n+1})\mapsto y=\frac{1}{1+x_{n+1}}(x_1,\cdot\cdot\cdot ,x_n)$
�̴�.


�� ��, �� ���� chart�� $\cc^k$-related ���� ���̱�����
$\phi_{+}^{-1}$�� ���غ���

 $\phi_+^{-1} : \rb^n \backslash \{0\} \rightarrow
\sbb^n\backslash \{p_+\}\hspace{0.5em},\hspace{0.5em}$
$(y_1,\cdot\cdot\cdot ,y_{n})\mapsto
y=\frac{2}{1+|y|^2}(y_1,\cdot\cdot\cdot ,y_n,\frac{|y|^2-1}{2})$

�̹Ƿ� ���� $(\phi_- \circ \phi_+^{-1}) (y)=\frac{y}{|y|^2}$
�̴�. $\phi_- \circ \phi_+^{-1} $�� ���ǿ����� 0�� �����Ƿ� �̴�
smooth map�̰� ���� $\cc^k$-related ���� ������.(�����
conformal, real analytic ������
�����Ѵ�)\\

$\vspace{2em}$

(2)
\begin{figure}[htb]
    \centerline{\includegraphics*[scale=0.4,clip=true]{grp7.eps}}

    \end{figure}

$\hspace{15em}$ �׸� 7\\




$U_1,U_2$�� $\sbb^1$�� ���κа� �Ʒ��κ����� �ΰ� $U_3,U_4$��
����, ������ �κ����� �� ���� ���� projection map�� �����ؼ� �̸�
$\phi_i,i=1,2,3,4$�� ����. �׷��� �� ��
$\{(U_i,\phi_i)\,|\,i=1,2,3,4\}$�� atlas�� ���� ���� �� �ִ�.\\

{\bf ����1.} (1)�� ���� generated�� maximal atlas $\mathcal{F}$��
(2)�� ����
generated�� maximal atlas $\mathcal{F}$�� ������ ������. (�ٽ� ���� (1)��  (2)�� $\sbb^n$��
 ���� smooth structure�� �ش�.)\\

(3) $\sbb^n=\{x\in \rb^{n+1}\,|\,f(x)=|x|^2-1=0\}$ ���� ǥ���� ��
�ְ� �̴� smooth structure�� ������ ���� ������ ���� ���� ��
�ִ�.\\

\begin{prop}
Let $f:\rb^n\rightarrow \rb$ be a $\cc^\infty$-map,
$M=f^{-1}(0)=\{p\in \rb^n\,|\,f(p)=0\}$(�̷� ���� hypersurface��
�θ���.) Assume $M\neq\emptyset,\nabla f=grad\,\,f$ is nonsingular
on $M$. Then M is a $\cc^\infty$-manifold.

\end{prop}

�� �� ������ �����ٸ� $f(x)=|x|^2-1=x_1^2+x_2^2+\cdot\cdot
+x_{n+1}^2-1$, $\frac{\partial f}{\partial x_i}(x)=2x_i$ �̹Ƿ�
$\nabla f=2(x_1,\cdot\cdot\cdot,x_{n+1})\neq 0$ �� �Ǿ� M�� smooth
structure�� �����ٴ� ���� �� �� �ִ�.

�� ������ ������ ���ؼ� Inverse function theorem, Implicit
function
theorem�� �ٽ� �� �� ���캸��.\\

\begin{thm}
(Inverse function theorem)

Let $U\subset\rb^n$ be an open subset, $f:U\rightarrow \rb^n$ be a
$\cc^k$-map, $1\leq k \leq \omega$. If the Jacobian matrix
$\frac{\partial f}{\partial x}=(\frac{\partial f_i}{\partial
x_j})$ is nonsingular at $p\in U$, then $f$ is a local
diffeomorphism at $p$, i.e.,

$\exists\,V$ , a neighborhood of $p$ such that $f|_V:V\rightarrow
f(V) $ is a $\cc^k$-homeomorphism and $f|_V^{-1}$ is also $\cc^k$.


\end{thm}
(����) Munkres : Analysis on Manifold ����.\\
�� �����κ��� Implicit function theorem�� �̲��� �� �� �ִ�.\\
\begin{thm}
(Implicit function theorem)

Let $U\subset\rb^n \times \rb^m$ be open, $f:U\rightarrow \rb^m$
be a $\cc^k$-map ($1\leq k \leq \omega$), $p=(a,b)\in U$. If
$f(p)=0$ and the $m\times m$ matrix $\frac{\partial f}{\partial
y}(p)$ is nonsingular, then $f(x,y)=0$ can be solved locally for
$y=g(x)$, i.e., $\exists V\subset\rb^n$,a neighborhood of $a$ and
$W\subset\rb^m$ of b with $V\times W\subset U$ and $\exists
\cc^k$-map $g:V\rightarrow W$, such that $f(x,y)=0\Leftrightarrow
y=g(x),\forall (x,y)\in V\times W.$
\end{thm}
(sketch of proof)


\begin{figure}[htb]
    \centerline{\includegraphics*[scale=0.5,clip=true]{grp8.eps}}

    \end{figure}

$\hspace{15em}$ �׸� 8\\





 $(x,y)\in\rb^{n+m}$ �� ����
 $F:U\subset\rb^{n+m}\rightarrow\rb^{n+m}$, $(x,y)\mapsto(x,f(x,y))
$�� ��������. �� �� F�� Jacobian matrix�� ���캸��




\[ DF= \left[
         \begin{array}{rr}
                              I &  0 \\
\frac{\partial f}{\partial x} &\frac{\partial f}{\partial y}
          \end{array} \right], \]

�̰� ���� det DF$=det\frac{\partial f}{\partial y}\neq 0$ �̹Ƿ�
Inverse function theorem�� ������ �����Ѵ�. �� $ G=F^{-1}$ ��
�����ؼ� $G(x,z)=(x,h(x,z))$ �� ǥ���� �� �ְ� �� ��
$g(x):=h(x,0)$ �� �ٷ� ���ϴ� explicit
function�� �ȴ�. $\hspace{7em}\Box$\\

���� �̷κ��� ����1�� �����غ���.

(���� 1�� ����) $\forall\,p\in M, \nabla f(p)\neq 0$ �̹Ƿ�
$\frac{\partial f}{\partial x_n}(p)\neq 0$ �̶�� �����ص�
�Ϲݼ��� ���� �ʴ´�. Implicit function theorem�� ����
$f(x_1,\cdot\cdot\cdot ,x_n)=0$�� ������ $p$�� �ٹ濡��
$x_n=g(x_1,\cdot\cdot\cdot,x_{n-1})$, $g\in \cc^{\infty}$ ���� Ǯ
�� �ִ�.

�� �� ������ ���� �Լ� $G$�� ��������.


$\hspace{2em}G:V\rightarrow\rb^n$ , $G(q)=(q,g(q))\in M$.

�׷��� �̴� �翬�� $\cc^{\infty}$-map�� �ǰ� �̰��� �ٷ� graph
map�� �ȴ�. ���� $M$�� coordinate chart�� �ֱ� ���� ���� �Լ���
��������.

$\hspace{2em}\phi:=G^{-1}:U=G(V)\rightarrow V(\subset \rb^{n-1})$.

 �׷��� ��
$(U,\phi)$�� $p$�ٹ濡���� $M$�� coordinate chart �� �ȴ�.
$\phi$�� graph map(�� embedding�̴�) $G$�� ���Լ��̹Ƿ�
homeomorphism�� �ǰ� ���� �̷� ������ ���ǵ� $(U,\phi)$�鰣��
$\cc^k$-related ���θ� Ȯ���ϸ� �ȴ�. ����
$(\widetilde{U},\widetilde{\phi})$�� ���� ������� ������� ��
�ٸ� chart��� �ϸ� $g\in\cc^{\infty}$�̰� $\widetilde{\phi}$��
projection�̹Ƿ�
$\widetilde{\phi}\circ\phi^{-1}=\widetilde{\phi}\circ G$��
$\cc^{\infty}$ �̴�. $\phi\circ\widetilde{\phi}^{-1}$�� ���ؼ���
���������̴�. ���� $\{(U_p,\phi_p)\,|\,p\in M\}$ �� $M$�� chart
�� �ȴ�. $\hspace{1em}\Box$\\

�� section���� $\sbb^n$�� chart�� (2) �� ���� ��������� atlas��
(3)�� atlas�� ���� ����. �ֳ��ϸ� ��� (2)�� $\phi$ map ��ü��
projection map�̹Ƿ� �̰��� �ٷ� (3)���� ���� explicit map�̱�
�����̴�.\\

{\bf ��.} $f(x,y,z)=(r-2)^2+z^2-1$ where $r=\sqrt{x^2+y^2}$.

�� �Լ��� $\nabla f\neq 0$�� �����Ѵ�. ���� ���� 1�� ���� ��
�Լ��� 0-locus $M$�� $\cc^{\infty}$ structure�� ������ ��
�� �ִ�. ��� $M$��  torus�� �ȴ�.\\

\begin{figure}[htb]
    \centerline{\includegraphics*[scale=0.6,clip=true]{grp9.eps}}

    \end{figure}

$\hspace{15em}$ �׸� 9\\




{\bf Exercise.} Generalize the above proposition to
$f:\rb^n\rightarrow\rb^k$.










  \end{document}
