\documentclass[12pt ]{article}
\setlength{\textwidth}{14 true cm} \setlength{\textheight}{20 true
cm}
\usepackage{graphicx}
\usepackage{hangul}
\usepackage{amscd,amsmath}
\usepackage{amsfonts}
\usepackage{amssymb,theorem}
\usepackage{longtable}
\newcommand{\Aff}{\mbox{\it Aff}}
\newcommand{\aff}{\mbox{\it aff}}
\newcommand{\cc}{\mathcal{C}}
\newcommand{\ccn}{\mathcal{C}^{\infty}}
\newcommand{\sbb}{\mathbb{S}}
\newcommand{\rb}{\mathbb{R}}
\newcommand{\rc}{\mathcal{R}}
\newcommand{\alp}{\alpha}
\newcommand{\bet}{\beta}
\newcommand{\del}{\delta}
\newcommand{\gam}{\gamma}
\newcommand{\vep}{\varepsilon}
\newcommand{\eps}{\epsilon}
\newcommand{\lam}{\lambda}
\newcommand{\kap}{\kappa}
\newcommand{\sig}{\sigma}
\newcommand{\ome}{\omega}
\newcommand{\Gam}{\Gamma}
\newcommand{\Ome}{\Omega}
\newcommand{\Sig}{\Sigma}
\newcommand{\Del}{\Delta}
\newcommand{\Lam}{\Lambda}


\newtheorem{thm}{Á¤¸®}
\newtheorem{cor}[thm]{µû¸§Á¤¸®}
\newtheorem{lem}[thm]{º¸Á¶Á¤¸®}
\newtheorem{prop}[thm]{¸íÁ¦}
\newtheorem{cl}{Claim}

{\theorembodyfont{\rm}
\newtheorem{ex}{¿¹}
\newtheorem{que}{Áú¹®}
\newtheorem{notation}{Notation}[section]
\newtheorem{defn}{Á¤ÀÇ}
\newtheorem{rem}{ÁÖ}
\newtheorem{note}{Note}
}
\renewcommand{\thenote}{}
\renewcommand{\therem}{}

\newenvironment{proof}{{\bf Áõ¸í}}{\hfill\framebox[2mm]{}}
\newenvironment{proof1}{{\bf Á¤¸®Áõ¸í}}{\hfill\framebox[2mm]{}}

\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 $N$.

(¶Ç´Â ÀϹÝÀûÀ¸·Î differentiable of class
$\cc^{\infty}(\cc^k,\,\,respectively)$¶ó°íµµ ÇÑ´Ù.)}

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

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

\end{figure}

$\hspace{15em}$ ±×¸² 12\\



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

{\bf Note.}

(1) $f$ÀÇ differentiability¸¦ checkÇϱâ À§Çؼ­ ¸ðµç coordinate
map¿¡ ´ëÇØ È®ÀÎÇØ º¼ ÇÊ¿ä´Â ¾ø´Ù. $M$°ú $N$ÀÇ atlas¿¡ ´ëÇؼ­¸¸
Çغ¸¸é µÈ´Ù.

(2)Differentiability´Â ¿¬¼Ó¼º°ú ¸¶Âù°¡Áö·Î local concept ÀÌ´Ù. Áï

{\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 ¼÷Á¦3(1)})\\

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

{\bf ¼÷Á¦ 3(2)} manifold $\rb$¿¡ ´ëÇØ $\,\phi(t)=t,\phi(t)=t^3$¿¡
ÀÇÇØ generatedµÈ structure¸¦ °¢°¢
$\mathcal{F}_1,\mathcal{F}_2$¶ó°í ÇÒ ¶§ $(\rb,\mathcal{F}_1)$°ú
$(\rb,\mathcal{F}_2)$°¡ diffeomorphicÇÔÀ»
º¸¿©¶ó.\\

{\bf ¿¹.}

1. $M,N$Àº manifoldÀÌ°í, $(U,\phi)$´Â $M$ÀÇ coordinate chartÀÏ ¶§

$\phi:U\rightarrow \rb^n$Àº $\ccn$, $\phi:U\rightarrow \phi(U)$´Â
diffeomorphismÀÌ´Ù.\\



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

3. $M\times N$ÀÌ ¾Õ¿¡¼­ ¾ð±ÞÇÑ product smooth structure¸¦ °¡Áú ¶§


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}