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

\begin{defn}{\it
Let M,N be smooth manifolds.

A continuous map $f:M\rightarrow N$ is $\cc^{\infty}
(\cc^k,\,\,respectively)$ if $\psi\circ f\circ\phi^{-1}$ is
$\cc^{\infty}(\cc^k,\,\,respectively)$ for all coordinate maps
$\phi$ on $M$ and $\psi$ on $N$.

(�Ǵ� �Ϲ������� differentiable of class
$\cc^{\infty}(\cc^k,\,\,respectively)$����� �Ѵ�.)}

\end{defn}
$\vspace{3em}$

\begin{figure}[htb]
\centerline{\includegraphics*[scale=0.5,clip=true]{grp12.eps}}

\end{figure}

$\hspace{15em}$ �׸� 12\\



$\ccn(M,N)=$the set of $\ccn$-maps from $M$ to $N$.\\

{\bf Note.}

(1) $f$�� differentiability�� check�ϱ� ���ؼ� ��� coordinate
map�� ���� Ȯ���� �� �ʿ�� ����. $M$�� $N$�� atlas�� ���ؼ���
�غ��� �ȴ�.

(2)Differentiability�� ���Ӽ��� ���������� local concept �̴�. ��

{\it $\hspace{3em}f:M\rightarrow N$ is $\ccn$ if and only if

$\hspace{5em}\forall\,p\in M, \exists U, $a neighborhood of p such
that $f|_U$ is $\ccn $.} ({\bf Exercise.})

(3) A composite of $\ccn$ maps is also $\ccn$.

(4) $f:M\rightarrow N$ is $\ccn\Longleftrightarrow g\circ f\in\ccn
(M,\rb) $ for $\forall g\in \ccn(N,\rb)$. ({\bf ����3(1)})\\

\begin{defn}
A diffeomorphism $f:M\rightarrow N$ is a homeomorphism such that
$f$ and $f^{-1}$ are smooth.\\
\end{defn}

{\bf ���� 3(2)} manifold $\rb$�� ���� $\,\phi(t)=t,\phi(t)=t^3$��
���� generated�� structure�� ����
$\mathcal{F}_1,\mathcal{F}_2$��� �� �� $(\rb,\mathcal{F}_1)$��
$(\rb,\mathcal{F}_2)$�� diffeomorphic����
������.\\

{\bf ��.}

1. $M,N$�� manifold�̰�, $(U,\phi)$�� $M$�� coordinate chart�� ��

$\phi:U\rightarrow \rb^n$�� $\ccn$, $\phi:U\rightarrow \phi(U)$��
diffeomorphism�̴�.\\



2. $U$ is open in $M$ and $U$ has the induced smooth structure
$\Rightarrow i:U\hookrightarrow M$ is
$\ccn$.\\

3. $M\times N$�� �տ��� ����� product smooth structure�� ���� ��


the projection map $p:M\times N\rightarrow M$ is $\ccn$.\\

4. $p:\widetilde{M}\rightarrow M$ is the covering with pull back
structure$\Rightarrow p$ is $\ccn$.


\end{document}
