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\begin{document}
\parindent=0cm
\section*{Construction of new vector bundles from given ones}

{\bf General rule}: A functorial construction of a vector space
from given ones
gives rise to the corresponding vector bundle construction.\\

��: ���� �������� ���ǵ� $*, \oplus , \otimes , \wedge , S ,
\cdots
$���� functorial construction�̰� ���� �̰͵��� vector bundle�� ���ε� �״�� Ȯ���� �� �ִ�.\\

���� ������ construction�� ���캸�ڴ�.\\

���� vector bundle\\

\hhhh \hhhh \hhhh$E = \dis{\bigcup_{p \in M} E_p}$ with $\{
\varphi_{U} \}$ and $\{
g_{UV}\} $\\

\hhhh \hhhh \hhhh\h$\downarrow \pi$\\

\hhhh \hhhh \hhhh\h$M$\\

�� �־��� �ִٰ� ����.\\

1. Dual bundle\\

a. total space: \\

�켱 ������ $E_p$�� ���Ͱ����̹Ƿ� �ִ����(dual) $E_p^*$��
�����ϴ°��� �ڸ��ϰ� �����Ǵ� $\pi^*$�� ���������� ���ǰ� �ȴ�.
���� �켱 set���μ��� $E^*$�� ������ �� �ִ�. \\

\hhhh \hhhh \hhhh$E^* = \dis{\bigcup_{p \in M} E_p^*}$ \\

\hhhh \hhhh \hhhh \h$\downarrow \pi^*$\\

\hhhh \hhhh \hhhh \h$M$\\
\newpage
b. local chart(local trivialization): \\

�������� $E^*$�� local trivialization $\varphi_U^*$�� $E$�� local
trivialization $\varphi_U : \pi^{-1} U \rightarrow U \times
\rb^n$���κ��� �� point $p \in M$�� ���� ${\varphi_U}|_p : E_p
\rightarrow {p} \times \rb^n$�� ��´�. �̷κ��� dual map
$({\varphi_U}|_p)^* : {p} \times {\rb^n}^* \rightarrow {E_p}^*$��
induce�ǰ� �̷κ��� $E^*$�� local trivialization\\

${\varphi_U^*}^{-1} : {\pi^*}^{-1} U \rightarrow U \times
{\rb^n}^*$\\

�� ���ǵȴ�.\\

c. transition:\\

���������� �̷��� ����� local trivialization�� ������
transition�� ���캸��.\\

${\varphi_U^*}^{-1} \cdot \varphi_V^* = (\varphi_V \cdot
\varphi_U^{-1})^* = ({(\varphi_U \cdot \varphi_V^{-1})^{-1}})^*
$\\

$g_{UV}^* (p) = {\varphi_U^*}^{-1} \cdot \varphi_V^* |_{p \times
{\rb^n}^*}$ and  $g_{UV} (p) = \varphi_U \cdot \varphi_V^{-1} |_{p \times \rb^n}$\\

$\therefore g_{UV}^* = ^t(g_{UV}^{-1})$ as matrices\\

�����ð��� ���������κ��� $g_{UV}$�� $\ccn$���� �� �� �ְ�
�̰���(inverse�� ���ϰ� transpose�� ��Ű�� ����)�� algebraic
process�̹Ƿ� $\ccn$�� �����ǹǷ� $g_{UV}^*$�� $\ccn$�̴�. ����
���� ���������� ���� ${\varphi_U^*}^{-1} \cdot \varphi_V^*$��
$\ccn$�̰�  $\{{\varphi_U^*}^{-1}\}$ �� $E^*$�� $\ccn$ ������
�شٴ� ���� �� �� �ִ�.(�̰�� topology�� ${\varphi_U^*}^{-1}$��
������ coherent topology) �� �̷��� ������ dual bundle�� ������ $\ccn$ vector bundle�� �ȴٴ� ���̴�.\\
\newpage
2. Direct sum\\

���� base�� ������ �� bundle $E \rightarrow M$(with
$\{\varphi_U\}, \{g_{UV}\} $), $F \rightarrow M$(with $\{\psi_U\},
\{h_{UV}\} $
�� �־��� �ִٰ� ����.\\

a. total space:\\
������ fibre $E_p$, $F_p$�� ���� �����̹Ƿ� direct sum $E_p \oplus
F_p $�� ������ �� �ִ�. ���� \\

\hhhh \hhhh \hhhh$E \oplus F := \dis{\bigcup_p (E_p \oplus F_p)}$\\

\hhhh \hhhh \hhhh \h $\downarrow \pi $\\

\hhhh \hhhh \hhhh \h$M$\\

�� �� ���ǵȴ�.\\

b. local trivialization:\\

$E$�� local trivialization $\varphi_U : \pi^{-1} U \rightarrow U
\times \rb^n$, $F$�� local trivialization $\psi_U : \pi^{-1}
U\rightarrow U\times \rb^m$ ���κ��� �տ���ó�� ���� fiberwise
���ǵ� map \\
$\varphi_U |_p \oplus \psi_U |_p : E_p \oplus F_p \rightarrow p
\times ( \rb^n \oplus \rb^m)$�� $\varphi_U |_p$�� $\psi_U |_p$�κ��� ���, �̷κ��� \\
$\varphi_U \oplus \psi_U : \pi^{-1} U\rightarrow U\times(\rb^n
\oplus \rb^m )$ �� ���� �� �ִ�. \\

c. transition:\\

\hhhh\hhhh\hhhh\hhh$E_p \oplus F_p $\\

\hhhh\hhhh\hhhh$\overset{\varphi_U \oplus \psi_U} {\swarrow} \hhhh
\overset{\varphi_V \oplus \psi_V} {\searrow} $ \\

\hhhh\hhhh$p\times (\rb^n\oplus\rb^m)\ \overset{^{g_{UV}(p) \oplus h_{UV}(p)}}{\longleftarrow} p\times (\rb^n\oplus\rb^m)$\\

���⼭ $g_{UV} \oplus h_{UV}$�� ������ ���� ��ķ� �־����Ƿ�
$g_{UV}$�� $h_{UV}$�� $\ccn$�̸� $g_{UV} \oplus h_{UV}$��
$\ccn$�̴�.\\
\begin{displaymath} \left(\begin{array}{c|c}
 g_{UV}(p) & 0\\
\hline 0 & h_{UV}(p)
\end{array}\right)
\end{displaymath}
���� �տ��� ó�� $\{\varphi_U \oplus \psi_U \}$�� $\ccn$ vector
bundle structure�� �� �����Ѵ�.\\

3. Tensor product\\

a. total space:\\

���� base�� ������ �� bundle $E \rightarrow M$(with
$\{\varphi_U\}, \{g_{UV}\} $), $F \rightarrow M$(with $\{\psi_U\},
\{h_{UV}\} $)�� ���� direct sum�� ���������� ������ fibre $E_p$,
$F_p$�� ���� �����̹Ƿ� tensor product $E_p \otimes F_p $�� ������
�� �ִ�. ����\\

\hhhh\hhhh\hhhh$E \otimes F := \dis{\bigcup_{p \in M} E_p \otimes F_p}$\\

\hhhh\hhhh\hhhh\h$\downarrow \pi$\\

\hhhh\hhhh\hhhh\h$M$ \\

�� �� ���ǵȴ�.\\


b. local chart:\\

\hhhh\hhhh $\varphi_U \otimes \psi_U : \pi^{-1}(U) \longrightarrow
U \times
(\rb^n \otimes \rb^m)$\\

�̰� ���� �տ��� ó�� ������ �� p���� pointwise�ϰ� ���ǵ�
$\varphi_U|_p \otimes \psi_U|_p : E_p \otimes F_p \longrightarrow
{p}\times (\rb^n \otimes \rb^m )$���κ��� ������� map�̴�.\\

c. transition:\\

\hhhh\hhhh\hhhh\hhh$E_p \otimes F_p $\\

\hhhh\hhhh\hhhh$\overset{\varphi_U \otimes \psi_U} {\swarrow}
\hhhh
\overset{\varphi_V \otimes \psi_V} {\searrow} $ \\

\hhhh\hhhh$p\times (\rb^n\otimes\rb^m)\ \overset{^{g_{UV}(p) \otimes h_{UV}(p)}}{\longleftarrow} p\times (\rb^n\otimes\rb^m)$\\

$g_{UV}$�� $(g_{ij})$��, $h_{UV}$�� H�� ��Ÿ����. �׷��� $g_{UV}
\otimes h_{UV}$�� ������ ����
$nm \times nm$��ķ� �־����Ƿ�\\

\begin{displaymath}
\left(\begin{array}{ccc}
g_{11}H & g_{12}H & \ldots\\
g_{21}H & g_{22}H & \ldots\\
\vdots & \vdots & \ddots
\end{array} \right)
\end{displaymath}
$g_{UV}$�� $h_{UV}$�� $\ccn$�̸� $g_{UV} \otimes h_{UV}$��
$\ccn$�̴�.\\
\\
\\
4. Exterior product\\

a. total space:\\

�־��� bundle $E \rightarrow M$(with $\{\varphi_U\}, \{g_{UV}\}
$)�� ���� $E_p$�� vector space�̹Ƿ� $\bigwedge^r (E_p)$�� ������
��
�ִ�. ���� \\

\hhhh\hhhh\hhhh\ $\bigwedge^r (E) := \dis{\bigcup_{p \in M}} {\bigwedge}^r (E_p)$\\

\hhhh\hhhh\hhhh\h $\downarrow \pi$ \\

\hhhh\hhhh\hhhh\h $M$\\

�� �� ���ǵȴ�.\\

�� �Ϲ�������  $\bigwedge(E) = \oplus \bigwedge^r (E)$��
$\bigwedge^r (E)$�� �˸� direct sum�� ����
�� �� �ִ�.\\

\hhhh\hhhh\hhhh\ $\bigwedge(E) = \oplus \bigwedge^r (E)$\\

\hhhh\hhhh\hhhh\h $\downarrow \pi$\\

\hhhh\hhhh\hhhh\h $M$ \\

b. local chart:\\

������ �� p���� pointwise�ϰ� $\bigwedge^r ( \varphi_U|_p ) :
\bigwedge^r ( E_p ) \longrightarrow {p}\times \bigwedge^r ( \rb^n
)$�� �������, �̷κ��� local trivialization\\

\hhhh\hhhh$\bigwedge^r ( \varphi_U ) : \pi^{-1}(U) \longrightarrow
U \times
\bigwedge^r  (\rb^n )$ \\

�� �� ���ǵȴ�.\\

c. transition:\\

\hhhh\hhhh\hhhh\hhh$\bigwedge^r (E_p)$\\

\hhhh\hhhh\hhh$\overset{\bigwedge^r (\varphi_U |_p)} {\swarrow}
\hhhh
\overset{\bigwedge^r (\varphi_V |_p)} {\searrow} $ \\

\hhhh\hhhh$p\times (\bigwedge^r (\rb^n)) \overset{^{\bigwedge^r
(g_{UV} (p))}}{\longleftarrow}
 p\times (\bigwedge^r (\rb^n))$\\

�׸��� $\bigwedge^r (g_{UV})$�� $g_{UV}$�� p�� ����
$\ccn$map�̹Ƿ� $\ccn$�̴�.\\
\\
\\
5. Pullback bundle \\

a. total space:\\

�־��� vector bundle $E \rightarrow M$(with $\{\varphi_U\}$)��
 $\ccn$ map $f:N \rightarrow M $�� ���� pullback bundle�� ������ $p
\in N$�� ���� $f(p)\in M$������ vector space $E_{f(p)}$��
������Ų��. �� pullback bundle�� ������ ���� ���ǵȴ�. \\

\hhhh\hhhh\hhhh $f^* E := \dis{\bigcup_{p \in N} E_{f(p)}}$\\

\hhhh\hhhh\hhhh\h $\downarrow \pi$\\

\hhhh\hhhh\hhhh\h $N$\\

b. local chart:\\

$f(p)$�� �ٹ� U�� ���� \\

\hhhh\hhhh $f^*\varphi_U : f^*E|_{f^{-1} (U)} \longrightarrow
f^{-1}(U)
\times \rb^n$ \\

�� ������ �� $p \in N$���� pointwise�ϰ� ���ǵ� map\\

\hhhh\hhhh $f^* E|_p = E_{f(p)} \overset{\varphi_U|_{f(p)}}
{\longrightarrow} f(p) \times \rb^n = p \times \rb^n$\\

���κ��� ���ǵȴ�.\\

c. transition:\\

\hhhh\hhhh\hhhh\h$f^* E|_p := E_{f(p)}$\\

\hhhh\hhhh\hhhh$\overset{\varphi_U|_{f(p)}} {\swarrow} \hhhh
\overset{\varphi_V|_{f(p)}}  {\searrow} $ \\

\hhhh\hh$p\times \rb^n = f(p)\times \rb^n
\overset{^{g_{UV}(f(p))}} {\longleftarrow}
f(p)\times \rb^n = p\times \rb^n$\\

���⼭ $g_{UV}$�� $f$�� ���� $\ccn$�̹Ƿ� $f^* g_{UV} := g_{UV}
\circ f$�� $\ccn$�̴�.\\



\end{document}
