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\begin{document}
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\section*{Tensor fields and Forms}
���ݺ��� $T^r(TM)$�� $T^r(M)$���� $T^s(T^*M)$�� $T_s(M)$����
���������.

���� �̰͵��� �̿��ؼ� ������ �����Ѵ�.

$T_s^r(M):=T^r(M) \otimes T_s(M)$ ; tensor bundle of type (r,s)\\

\begin{defn}
A {\bf $\ccn$-section} $\sig$ of a vector bundle $E
\overset{\pi}{\longrightarrow} M$ over $U^{open} \subset M $ is a
$\ccn$-map $\sig : U \rightarrow E$ s.t. $\sig(p) \in E_p, \;
\forall p \in U $, i.e. $\pi \circ \sig = id$\\

$\ccn(M,E) := $ space of $\ccn$-sections.\\

A {\bf $\ccn$-frame} for $E$ over $U \subset M$ is a collection
$\{ \sig_1, \cdots , \sig_n \}$ of $\ccn$ sections over $U$ s.t.
$\{ \sig_1(p), \cdots , \sig_n(p) \}$ is a basis for $E_p$,
$\forall p \in U $.
\end{defn}

{\bf Note}: $U$�������� frame�� ������ �ִٴ� ���� $U$����
trivialization�������� �ִٴ� �Ͱ� ��ġ�̴�.\\

\begin{proof}
$\Leftarrow$) local chart �� �����Ѵٴ� ���� isomorphism
$\pi^{-1}U \overset{\varphi_U}{\longrightarrow}$ $U \times
\rb^n$�� �����Ѵٴ� ���̴�. ���� $\rb^n$�� ���Ա��� $e_1, \cdots,
e_n$����
$\varphi^{-1}$�� ������ frame�� �������.\\

$\Rightarrow$) frame�� �����ϹǷ� $\pi^{-1}U$�� ������ $\sum v^i
\sig_i (x)$�� ǥ���� �� �ִ�. ���� �� ���� $U \times \rb^n$���� ��
$(x, v^1, \cdots, v^n)$���� ������ map�� �����ϸ� �� map�� local
chart�� �ȴ�.
\end{proof}\\

������ ����鵵 ���� �� �� �ִ�.\\

1. $TM$�� section�� vector field�� �Ѵ�.

\h $T^*M$�� section�� 1-form�̶� �Ѵ�.

\h $\bigwedge^p(T^*M)$�� section�� p-form�̶� �Ѵ�.

\h $T_s^r(M)$�� section�� tensor field of type (r, s)�̶� �Ѵ�.\\

2. coordinate chart $(U, x)$�� ���Ͽ� bundle�� smooth structure
���Ƿ� ����

\h $\{\frac{\pa}{\pa x^1}, \cdots ,\frac{\pa}{\pa x^n} \}$ ��
$TM$�� $U$�������� $\ccn$-frame �� �ȴ�.

\h $\{dx^1, \cdots ,dx^n \}$ �� $T^*M$�� $U$�������� $\ccn$-frame
�� �ȴ�.\\

{\bf Remark}: ��������� tensor notation�� �� ���� covariant��
index�� �Ʒ�÷�ڷ�, contravariant�� index�� ��÷�ڷ� ǥ���ϸ�
�����ϱ� ������(���� ��� summation���� dummy index�� ����
$\sum$�� ���� ���� Einstein notation ���� �̿��ϱ⿡ ����)
$\frac{\pa}{\pa x_i}$, $dx_i$��ſ� $\frac{\pa}{\pa
x^i}$, $dx^i$�� ���⵵
�Ѵ�.\\

\h Exterior bundle, tensor bundle�� construction�� smooth
structure�� ���Ƿ� ���� $\{dx^I \}$ �� $\bigwedge^p(T^*M)$�� local
chart $U$�������� $\ccn$-frame �� �ǰ�,

(���⿡�� $I$�� multi-index�μ� �� $I = (i_1, i_2, \cdots, i_p)$ ,
$i_1 < i_2 < \cdots < i_p$�̰�  $dx^I = dx^{i_1}\wedge dx^{i_2}
\wedge \cdots \wedge dx^{i_p}$�̴�.)\\

\h $\{ \frac{\pa}{\pa x^{i_1}} \otimes \cdots \otimes
\frac{\pa}{\pa x^{i_r}} \otimes dx^{j_1} \otimes \cdots \otimes
dx^{j_s} \}$ �� $T_s^r(M)$�� $U$�������� $\ccn$-frame �� �ȴ�.\\


{\bf Note}: $U$���� $\ccn$-frame $\{\sig_1, \cdots, \sig_n\}$��
�־��� ���� ��, ������ section $\sig$�� $\dis{\sum_i f^i
\sig_i}$���� ��Ÿ�� �� �ִ�. �׷��� $\sig$�� $\ccn$�̶�� �Ͱ�
������ $i$�� ���Ͽ�
$f^i$�� $\ccn$($p$�� ���� �Լ��μ�)�̶�� ���� ��ġ�̴�.\\

\begin{proof}
) �� Note�� ���� �ڸ��ϴ�.
\end{proof}\\

��) $\alp$: p-form,\\
$\alp$ is $\ccn$ $\Leftrightarrow$ for some $(U,x)$, $\alp |_U =
\sum \alp_I dx^I $ where $\alp_I$ is $\ccn$.\\
\\

{\bf Notation}: For a smooth vector bundle $E \rightarrow M$,\\

\h$\Gamma(E) = \ccn(M,E)$ = the space of smooth global sections of E\\

\h$\ccn(M,TM)$ = $\mathcal{X}(M)$ = the space of $\ccn$ vector
fields \\


\h$\ccn(M,\rb)$ = $\mathcal{F}(M)$ = the space of $\ccn$ functions on M\\

\h$\ccn(M,\bigwedge^r(T^*M))$ = $\mathcal{E}^r(M)$ = the space of
r-forms \\

\h$\ccn(M,T_s^r(M)) = \mathcal{T}_s^r(M)$ = the space of tensor fields of type (r,s)\\

\newpage
3. View an r-form $\alpha \in \mathcal{E}^r(M)$ as an alternating
$\mathcal{F}$-linear map:\\

\h$\alpha \in \mathcal{E}^r(M)$�� �� $\alpha$�� alternating
$\mathcal{F}$-linear map $\alpha : \underbrace{
\mathcal{X}\times\cdots
\times\mathcal{X}}_{r}\longrightarrow\mathcal{F}$����

\h �� �� �ִ�. ���⼭\\

\h$\alpha(X_1,\cdots,X_r)(p) := \alpha_p(X_1(p),\cdots,X_r(p))$

\h for $\alpha_p \in
\bigwedge^r(T_p^*M)=(\bigwedge^r(T_pM))^*=A_r(T_pM)$
\footnote[1]{recall $\bigwedge^r(V^*)=(\bigwedge^rV)^*=A_r(V)$}(�Ʒ�����1)\\

\h locally $\alpha = \Sigma\alpha_I dx^I \Longrightarrow
\alpha(X_1,\cdots,X_r$) is clearly $\ccn$ .\\

\h ������ ���Ƿ� ���� $\alpha$��  $\mathcal{F}$ -linear, i.e.
$\alpha(X_1,\cdots,fX_i,\cdots,X_r) =
f\alpha(X_1,\cdots,X_i,\cdots,X_r)$

\h ���� �� �� �� �ִ�.\\
\\
Conversely $\forall \mathcal{F}$-linear alternating map is an r-form:\\

\h Given $\alpha : \mathcal{F}$-linear, alternating and $ X_1,
\cdots ,X_r \in T_pM$�� ����

\h $\alpha_p(X_1,\cdots,X_r) :=
\alpha(\widetilde{X_1},\cdots,\widetilde{X_r})(p)$ for
$\widetilde{X_i}:\ccn$-extension of $X_i$�� ������

\h ��. �׷��� $\alpha_p \in \bigwedge^r(T^*_pM)=A_r(T_pM)$�̴�. \\

\h ���� $\alpha(\widetilde{X_1},\cdots,\widetilde{X_r})(p)$��
extension�� ���ÿ� ������� �� ���ǵǴ� ����

\h ������.\\

\h $(*)$ $X_1, \cdots,X_r \in \mathcal{X}, X_1(p)=0$�� ��
$\alpha(X_1,\cdots,X_r)(p)=0$�� �Ǵ� ���� ��

\hh �̸� ����ϴ�. �ֳ��ϸ� $\widetilde{X}'$�� �Ǵٸ�
extension�̶�
�� ��\\

\hh $ X_1(p)=\widetilde{X_1}(p)=\widetilde{X_1}'(p)
\Longrightarrow(\widetilde{X_1}-\widetilde{X_1}')(p)=0$\\

\hh �׷��� $(*)$�� ����\\

\hh
$\alpha(\widetilde{X_1},\widetilde{X_2},\cdots,\widetilde{X_r})(p)
=
\alpha(\widetilde{X_1}',\widetilde{X_2},\cdots,\widetilde{X_r})(p)
=
\alpha(\widetilde{X_1}',\widetilde{X_2}',\cdots,\widetilde{X_r})(p)$

\hh = $\cdots =
\alpha(\widetilde{X_1}',\widetilde{X_2}',\cdots,\widetilde{X_r}')(p)$��
�����Ѵ�.\\
\\
\newpage

\h �׷��� $(*)$�� ������.\\

\h $X_1 = \Sigma a_i\frac{\pa}{\pa x^i}$ on U$\Longrightarrow
X_1(p) = \Sigma a_i(p)\frac{\pa}{\pa x^i}|_p = 0 \Longrightarrow
a_i(p) = 0 \h \forall i$\\

\h Let $\varphi$ be a bump function  s.t. $\varphi(x) = 1 ~for~x
\in $K (K compact neighborhood

\h $ \subset U$) and  $\varphi(x) = 0 ~for~x \in
U^c$\\

\h$X_1 = {\varphi}^2 X_1 + (1 - {\varphi}^2)X_1=\Sigma(\varphi
a_i)(\varphi \frac{\pa}{\pa x^i}) + (1 - {\varphi}^2)X_1$\\

\h$\alpha(X_1,\cdots,X_r)(p) = \Sigma (\varphi
a_i)(p)\cdot\alpha(\varphi \frac{\pa}{\pa x^i},X_2,\cdots,X_r)(p)
+
(1-{\varphi}^2)(p)\cdot\alpha(X_1,\cdots,X_r)(p)$\\

\hhhh\hhhh\h =0 \h ($\because a_i(p)=0$ and $(1-{\varphi}^2)(p) =0$) \\



\end{document}
