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\begin{document}
 \parindent=0cm
  \section*{Vector Bundle.}
\begin{ex}

Typical example : Tangent bundle.\\
$TM$ = $\underset{p}\bigcup T_p M$ : tangent bundle.\\
$\downarrow \pi$\\
$M$\\
For each p$\in M$, $T_p M$ is an n-dim'l vector space and these
are
pieced together smoothly to give a manifold $TM$.\\
i.e. $\forall p \in M, \exists U$ a neighborhood of $p$ and a
diffeomorphism\\

\hspace{4.0em} $\varphi_U : \pi\inv(U) \longrightarrow U\times\rb^n $ \\
\hspace*{4.0em} $\hspace*{3.0em} X_p \mapsto (p,a^1,\cdots, a^n)$\\
\hspace*{5.0em} where $X_p=\sum_{i=1}^{n}a^i\paxi |_p$ and
$a^i=dx^i(X_p)=X_p(x^i)$\\
s.t. $\varphi_U|_p : \pi\inv(p) = T_p M \to \{p\}\times\rb^n$ is
a vector space isomorphism.\\

\end{ex}

\begin{figure}[htb]

\centerline{\includegraphics*[scale=0.4,clip=true]{vectorbun1.eps}}

\end{figure}


\begin{defn}

A n-dim'l vector space over $M$ is a triple ($E$,$\pi$,$M$), where
$\pi : E \to M$ is a $\ccn$-map between $\ccn$-manifolds toghther
with a vector space (over $\rb$ or $\mathbb{C}$) structure on each
fiber $\pi\inv(p)$ s.t. the following local
triviality condition is satisfied:\\
$\forall p \in M $,$ \exists U$ a neighborhood of $ p $ and a
diffeomorphism\\

\hspace{4.0em} $\varphi_U: \pi^{-1}(U) = E_U \longrightarrow U \times \rb^n$\\
\hspace*{4.0em} $\hspace*{6.0em}\pi \searrow \hspace{2.5em} \swarrow p$\\
\hspace*{4.0em} $\hspace*{9.0em}U$\\

s.t. $\varphi_U |_p : \pi\inv (p)\longrightarrow \{p\} \times
\rb^n $ is a vector space isomorphism $\forall p \in U$.

\end{defn}

\clearpage

{\bf transition map}\\
Given two local trivilizations $\varphi_U $,$ \varphi_V $, the map
$g_{UV}(p)$ : $U\cap V \longrightarrow GL(n,\rb) $ given by
$g_{UV}(p)$ = $\varphi_{U}\circ\varphi_{V}{\inv}|_{p\times\rb^n}$
is called a transition function relative to $\varphi_U$ and
$\varphi_V$.\\


\begin{figure}[htb]

\centerline{\includegraphics*[scale=0.4,clip=true]{vectorbun2.eps}}

\end{figure}


{\bf TM case :}\\
$X_p=\sum a^{i}\paxi = \sum b^{j}\payj$\\
$\hspace*{4.0em} \downarrow \hspace{4.0em} \searrow \\
(p,a^1,\cdots,a^n)\overset{g_{UV}(p)}\longleftarrow
(p,b^1,\cdots,b^n)$\\

¿¡¼­ $g_{UV}(p)(b)=a $, ¿©±â¼­ $a$¿Í $b$ÀÇ °ü°è¸¦ ±¸Çϸé
$a^i=dx^i(X_p)=\sum\frac{\pa x_i}{\pa y_j}dy^j(X_p) = \sum\frac{\pa x_i}{\pa y_j}b^j$ÀÌ´Ù.\\
$\therefore  g_{UV}(p)=\frac{\pa x}{\pa y}(p) \leftarrow n\times
n$ \hspace{0.5em} Jacobian matrix.\\


\begin{cl}
Given $\varphi_U$ and $ \varphi_V$ without condition $\ccn$.\\
Show that $g_{UV}$ is $\ccn$ as a map from $U\cap V$ to $GL(n,\rb)
\Leftrightarrow \varphi_U\circ\varphi_V\inv$ is $\ccn$.{\bf
(¼÷Á¦\#18)}
\end{cl}

\end{document}