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\begin{document}
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\section*{Tangent bundle.}

$M$�� n���� $\ccn$-manifold �� �� ������ ���� $TM$�� �����Ѵ�.

$TM=\dis{\bigcup_{p\in M}}T_p M=\{X_p=(p,X)\,|\,X_p\in
T_pM\,,\,p\in M\}$

$\pi:TM\rightarrow M$, $\pi(X_p)=p$.

{\it Then $TM$ is a $\ccn$-manifold of dimension 2n, called {\bf
tangent bundle} of $M$, such that $p$ is $\ccn$. }\\

������ $TM$�� manifold�� ���� ������.\\

(1) Coordinate chart for $TM$ :

$(U,\phi)$�� $M$���� coordinate chart�� �ΰ� $x_i=u_i\circ\phi$��
coordinate function

���� �θ� $(\frac{\pa }{\pa x_1},\cdot\cdot\cdot,\frac{\pa }{\pa
x_n})$�� $T_pM$ �� basis�� �̷��. �̰��� dual basis��


$(dx_1\cdot\cdot\cdot,dx_n)$�̶� ����. ��

$dx_i:T_pM\rightarrow \rb$, linear functional such that
$dx_i(\frac{\pa }{\pa x_j})=\del_{ij}$.

�׷��� $p$������ tangent vector $X_p$��
$X_p=\disj(Xx_j)\frac{\pa}{\pa x_j}$�� �����ϹǷ�

$dx_i(X_p)=\disj(Xx_j)dx_i(\frac{\pa}{\pa x_j})=Xx_i$ �� �ȴ�.

������ �� �� �ֵ��� $dx_i$�κ��� $X_p$�� �� ������� �˾Ƴ� ��
�����Ƿ� �̸� �̿��ؼ� ������ ���� $TM$�� chart
$(\widehat{U},\widehat{\phi})$�� �ش�.

$\widehat{U}=\pi^{-1}(U)$�� �ְ�,
$\widehat{\phi}:\widehat{U}\rightarrow\phi(U)\times\rb^n\subset\rb^{2n}$
��


$\hspace{5em}\widehat{\phi}(X_p)=(\underbrace{x_1(p),\cdot\cdot\cdot,x_n(p}),\underbrace{
dx_1(X_p),\cdot\cdot\cdot,dx_n(X_p}))$

$\hspace{12em}\phi(p)\hspace{2em}$coefficients of $X_p \,\,w.r.t
(\frac{\pa}{\pa x_1},\cdot\cdot,\frac{\pa}{\pa x_n})$

�׷��� $\widehat{\phi}$�� bijection�� �ְ�(set���μ�) ���� $TM$��
topology�� ����.\\

(2) Topology of $TM$ : weak topology.

�� $\{\widehat{U}\,|\,(U,\phi):coordinate\,\,chart \,\,for\,\,M\}$
�� �����ϴ� coherent topology�� �ش�. (�տ��� �̹� �ߴ� coherent
topology�� ������ ���� �� ���� ������ �����ϹǷ� topolgy�� �� �� �ִ�.)\\

(3) ���������� chart���� $\ccn$-related ���� ������ :


�� chart
$(\widehat{U},\widehat{\phi}),(\widehat{V},\widehat{\psi})$
�� ���� $U\cap V=W$�� ����.\\

\vspace{2em}


\begin{figure}[htb]
 \centerline{\includegraphics*[scale=0.5,clip=true]{grp14.eps}}

 \end{figure}

$\hspace{15em}$ �׸� 14\\



$W$�� $\widehat{\phi}$�� ���� image�� $\phi(W)\times \rb^n$�� �ǰ�
$\widehat{\psi}$�� ���� image�� $\psi(W)\times \rb^n$�� �ȴ�. ��
���̿� transition map $\,\widehat{\psi}\circ\widehat{\phi}^{-1}$��
����ϱ� ���Ͽ� $\phi,\psi$������ transition map
$f=\psi\circ\phi^{-1}$�� �̿�����.

$X=\disi a_i\paxi=\disj b_j\payj\,\,$�̶� �θ� $(p,X)\in
\pi^{-1}(W)$�� ����
\\



$\hspace{7em}\pi^{-1}(W)$

$\hspace{5em}\widehat{\phi}\swarrow\hspace{2em}\searrow\widehat{\psi}$

$\hspace{2.5em}\phi(W)\times\rb^n\rightarrow\psi(W)\times\rb^n\,\,\,\,\,$,
$\,\,\,\,\,\phi(p)=x,\psi(p)=y$

$\hspace{4em}(x,a)\hspace{1em}\mapsto\hspace{1.5em}(y,b)$\\

���� �� �� �ְ� $y=f(x)$ , $b=df_x(a)$ where $df_x=\frac{\pa
y}{\pa x}(x)$ �� �־����ٴ� ���� ���� �� �ִ�. ��ǻ� $x$����
$y$�� ���� map�� $f$�� ���� �� ���ǵǰ� �������� ����� �����
$\dis{\frac{\pa}{\pa x_i}=\disj\frac{\pa y_j}{\pa
x_i}\frac{\pa}{\pa y_j }}$�� $X=\disi a_i\frac{\pa}{\pa x_i}$��
��������. �׷���

$X_p=\disi a_i\disj \frac{\pa y_j}{\pa x_i}\frac{\pa}{\pa y_j}=
\disj (\disi a_i \frac{\pa y_j}{\pa x_i})\frac{\pa}{\pa y_j}$

�� �Ǿ� $b_i=\disj \frac{\pa y_i}{\pa x_j}a_j$ �� ���� �� �ִ�.
���� $b=df_x(a)$�� �ǰ� �� $\frac{\pa y_i}{\pa x_j}$�� $x$��
���� smooth function�̹Ƿ�
$\widehat{\psi}\circ\widehat{\phi}^{-1}$�� smooth function �� �ȴ�.\\

{\bf Remark.} $X\in T_pM$�� $x,y$ coordinate �� ���� ��Ÿ����
$X=\disi a_i\paxi=\disi b_i\payi$�� �� �� �ִµ� ������
$dy_i(X)=b_i=\disj\frac{\pa y_i}{\pa x_j}a_j=\disj\frac{\pa
y_i}{\pa x_j}dx_j(X)$ �̹Ƿ� $dy_i=\disj\frac{\pa y_i}{\pa
x_j}dx_j$ �� �������� �� �� �ִ�. �̰��� ��ķ� ǥ���ϸ�
$dy=\frac{\pa y}{\pa x}dx$�̰� ���⼭ $dx,dy$�� �����ͷ� ����. ��
����� �տ��� tangent vector�� $x,y$ coordinate chart���� ��
����� $\frac{dy}{dt}=\frac{\pa y}{\pa x}\frac{dx}{dt}$�� formally
��ġ���� �� �� �ִ�. ���� �츮�� ��� ���� �̺� $dx$�� $dy$����
coordinate change���� tangent vector�� dual vector�� ���� ��ȭ����
�� �� �ִ�. �̷� �ǹ̿��� �츮�� ���������� ���Ѽ��� ��ȭ��
�����ϰ� �ִ� �̺��� ���г��������δ� tangent vector��
dual(cotangent vector)�� ������ �� �ִ�.\\


���� $dx_i$�� $\paxi$�� dual�� �ǰ� �� $dx_i$���� �����ϴ� vector
space���� ������ �� �ִ�. �̴� �� �� $p$���� $T_pM$�� dual vector
space $(T_pM)^*$��
������ �� �ְ�, �̵��� ��ü�� cotangent bundle $T^*M$�̶�� �Ѵ�.\\


{\bf ���� 4.} $T^*M=\dis{\bigcup_{p\in M}}(T_pM)^*$ : cotangent
bundle. Do the same for $T^*M$ as we did for $TM$.









\end{document}
