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\newcommand{\dis}{\displaystyle}
\newcommand{\disi}{\displaystyle{\sum_{i=1}^n}}
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\newcommand{\Aff}{\mbox{\it Aff}}
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\newcommand{\bet}{\beta}
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\newcommand{\gam}{\gamma}
\newcommand{\vep}{\varepsilon}
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\newcommand{\kap}{\kappa}
\newcommand{\sig}{\sigma}
\newcommand{\ome}{\omega}
\newcommand{\Gam}{\Gamma}
\newcommand{\Ome}{\Omega}
\newcommand{\Sig}{\Sigma}
\newcommand{\Del}{\Delta}
\newcommand{\Lam}{\Lambda}


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\begin{document}
\parindent=0cm
\section*{Exterior derivative}


1. $d : \mathcal{F} = \mathcal{E}^0 (M) \rightarrow \mathcal{E}^1
(M)$ is defined by $d f_p (X_p ) = X_p f$, $\forall f \in
\mathcal{F}, X_p \in T_p M $

\hhhh\h$f$\hh$\mapsto$\h$df$\\
\\
{\bf Check} : $df$ �� $\ccn$ 1-form�̴�.\\

\hhhh Clearly $df_p \in T_p^*M$ \h ($\because df_p(X_p + cX'_p) =
df_p(X_p) + c\cdot df_p(X'_p))$

\hhhh $\Longrightarrow df = \dis{\sum} a_j dx^j $ locally on (U,x)

\hhhh $\Longrightarrow a_i =df(\frac{\pa}{\pa x^i}) = \frac{\pa
f}{\pa x^i} ~~: \ccn~~~(\because f ~~: \ccn)$

\hhhh $\Longrightarrow df =\dis{\sum_j} \frac{\pa f}{\pa
x^j}dx^j~~~ : \ccn$

\hhhh $\therefore df \in \mathcal{E}^1(M)$\\
\\
\\
{\bf Remark}\\
(1) $d (x^i) = d x^i$ :\\

\h $d(x^i)(\frac{\pa}{\pa x^j })= \frac{\pa}{\pa x^j} (x^i) =
\del_{ij} = d x^i (\frac{\pa}{\pa x^j})$ �Ǵ� $d(x^i)(X) = X(x^i) = dx^i(X)$\\
\\
(2) $f: M \rightarrow \mathbb{R}$, $\ccn$�� �� ���������� $df$��
$f_*$�� ����. ���� ��Ȯ�ϰ�\\

\h $df$ :TM $\overset{^{f_*}}{\rightarrow}
T\mathbb{R}\rightarrow\mathbb{R}$�� �ȴ�.

\hhh $X \mapsto f_* X $

\hhhh\hh $\shortparallel$

\hhhh\hh $\square\frac{d}{dt}\mapsto \square$(This map is a
canonical translation.)

\h $\square = f_*(X)(t) = X(t\cdot f)=Xf=df(X)$ where
$\mathbb{R} \overset{t} {\rightarrow} \mathbb{R}$ is the identity map.\\


(3) $d(f + g) = df + dg$

\h $d (fg) = g(df) + f(dg)$\\

\begin{math} \left( \begin{array}{l}
d(f + g)(X_p) = X_p (f + g ) = X_p f + X_p g\\
d(fg)(X_p) = X_p (fg) = g(X_p f) + f (X_p g) = (g(df) +
f(dg))(X_p) \end{array} \right)
\end{math}
\newpage

2. ���ݱ��� $d: \mathcal{E}^0 \rightarrow \mathcal{E}^1$�� ����
�Ͽ���. ���� �̸� �Ϲ�ȭ��Ų $d: \mathcal{E}^p \rightarrow
\mathcal{E}^{p+1}$�� ��������

\h ������.\\

\begin{thm}
$d: \mathcal{F} \rightarrow \mathcal{E}^1$ can be extended
uniquely to an anti-derivation of degree 1 on $\mathcal{E}$ s.t.

\h  $d^2 = 0$. i.e. $\exists ! \;\; d: \mathcal{E}^p
\rightarrow\mathcal{E}^{p+1}$, $\forall p = 0,1,2, \cdots $s.t.

\h (1) $d(\alp +\bet) = d \alp + d \bet$

\h (2) $d(\alp \wedge \bet) = d \alp \wedge \bet + (-1)^p \alp
     \wedge d \bet $\hh $p = deg(\alp)$

\h (3) $df$ are as in 1 for $f \in \mathcal{F}$

\h (4) $d^2 = 0$
\end{thm}

\begin{proof}

�켱 local�ϰ� d�� �����ϰ� ���� �� ���� property�� uniqueness��
check�� ���� global�ϰ� Ȯ������.\\

Local existence:\\

\h On $(U, x)$, let $\alp = \dis{\sum_I} f_I d x^I$ , where $d x^I
= d x^{i_1}\wedge\cdots\wedge d x^{i_p}$.

\h Define $ d \alp := \dis{\sum_I} d f_I \wedge d x^I $\\

\h (1)$\sim$(4) property�� �����ϴ� operator�� �ִٸ� �̷���
���ǵ� �� �ۿ� ����. �ֳ��ϸ�\\

\h $d\alpha = \dis{\sum_I} d(f_I d x^I) = \dis{\sum_I}(df_I\wedge
d x^I + f_I\wedge d(dx^I)) =  \dis{\sum_I} d f_I \wedge d x^I$

\hh $\smile$(by (1)) \hh $\smile$(by (2))\hhhh \hhhh \h
$\smile$(by �Ʒ�����\footnote {$d(dx^I)=d(d
x^{i_1}\wedge\cdots\wedge d x^{i_p})=\Sigma(-1)^{j+1}d
x^{i_1}\wedge\cdots
dd(x^{j+1})\cdots\wedge d x^{i_p}=0$ ($\because dd(x^{j+1})=0$ by (4)) }).\\
\\

\h ������ ���ǵ� d�� $d^2 f = 0$, $\forall f \in \mathcal{F}$��
�����Ѵ�.

\hh $\because d(d f) = d(\dis{\sum_i} \frac{\pa f}{\pa x^i} d
x^i)= \dis{\sum_i}d(\frac{\pa f}{\pa x^i}) \wedge d x^i$\hh (by
definition)

\hhhh\hhhh\hh $= \dis{\sum_{i,j}} \frac{\pa}{\pa x^j} (\frac{\pa
f}{\pa x^i}) d x^j\wedge d x^i$

\hhhh\hhhh\hh $= \dis{\sum_{i<j}} (\frac{\pa}{\pa x^j} (\frac{\pa
f}{\pa x^i})- \frac{\pa}{\pa x^i} (\frac{\pa f}{\pa x^j}))(d
x^j\wedge d x_i)$ ($\because d x^i \wedge d x^j = - d x^j \wedge d
x^i$)

\hhhh\hhhh\hh = 0\\
\\


\h ���� d on U�� (1)$\sim$(4)�� property�� ������Ű�� ����
check����.\\

\h (1) clear\h since $d(f +g) = df + dg $ and $ \alpha + \bet = \dis{\sum} (f_I +g_I)dx^I$

\hhhh\hh for  $\alp = \sum f_I d x^I $, $\bet = \sum g_J d x^J$ \\

\newpage

\h (2) $d(\alp \wedge \bet) = d (\dis{\sum_{I,J} f_I g_J}
dx^I\wedge dx^J )=\dis{\sum_{I,J}} d(f_I g_J)\wedge dx^I\wedge
dx^J$

\hhhh\hh =$\dis{\sum} \{(df_I) g_J \wedge dx^I \wedge d x^J\} +
\dis{\sum} \{f_I(d g_J) \wedge dx^I \wedge d x^J\}$

\hhhh\hh $= d \alp \wedge \bet + (-1)^p \alp \wedge d \bet$\\

\h (3) ���� ���� ���ʿ�.\\

\h (4) $d^2 \alp = d ( \dis{\sum} d f_I \wedge d x^I) =
\dis{\sum_I}\{d(d f_I) \wedge d x^I - d f_I \wedge d(dx^I)\} = 0$\\
\\
\\
Local uniqueness:\\

\h $d'$ �� ���� ������ ��� �����ϴ� �Ǵٸ� local operator�� ����.
�׷��� (3)�� ���ؼ�

\h $d'x_i = d x_i$�� �����Ѵ�. �̸� �̿��Ͽ� ������ ���� local uniqueness�� ������ �� �ִ�. \\

\h $d' \alp = \dis{\sum} d'(f_I dx^I) = \dis{\sum} (d' f_I \wedge
dx^I + f_I d'(dx^I))$

\hhh $\smile$ (by (1)) \hh $\smile$ (by (2)) \\

\hhh $=\dis{\sum} (d' f_I) \wedge d x^I$ \h (by �Ʒ�����
\footnote{ (3)�� (4)�� ����

\h $d'(dx^{i_1}\wedge\cdots dx^{i_p}) = d'(dx^{i_1})\wedge
dx^{i_2}\cdots dx^{i_p} - dx^{i_1}\wedge d'(dx^{i_2})\cdots
dx^{i_p} \cdots +(-1)^p dx^{i_1}\wedge dx^{i_2}\cdots
d'(dx^{i_p})$

\hhhh\hhhh\h $= d'^2(x^{i_1})\wedge dx^{i_2}\cdots dx^{i_p} -
dx^{i_1}\wedge d'^2(x^{i_2})\cdots dx^{i_p} \cdots + (-1)^p
dx^{i_1}\wedge dx^{i_2}\cdots d'^2(x^{i_p})$

\hhhh\hhhh\h $= 0 $})


\hhh $=\dis{\sum} (d f_I) \wedge d x^I $ \h (by (3))

\hhh $= d \alpha$\\
\\
\\
Global existence\\

\h �־��� $\alpha \in \mathcal{E}^p $�� ���� ������ local�� $d$��
����� ���ϼ��� �����Ͽ���. ���� $d$�� ������

\h ���� global�ϰ� Ȯ���Ű��.\\

\h $d \alp |_U := d(\alp |_U )$\\

\h �� ��������. �׷���  $d\alpha$�� local uniqueness�� ���� �� ���ǵȴ�. �׸��� �� $d$�� (1)$\sim$(4)

\h ������ �������� ������ ���� �� �� �ִ�.\\

\h (1), (2)�� $+, \wedge$�� pointwise operation�̹Ƿ� �� �����Ѵ�.
������\\

\hh$d(\alpha +\bet)|_U = d((\alpha +\bet)|_U) = d(\alpha|_U +
\bet|_U) = d(\alpha|_U )+d(\bet|_U ) = d\alpha|_U + d\bet|_U =
(d\alpha +d\bet)|_U$\\

\h (3) ���� ���� ���ʿ�.\\

\h (4) $d^2\alpha = 0$\h $\because d^2 \alpha|_U = d(d\alpha|_U )
= d d(\alpha|_U ) = d^2 (\alpha|_U ) = 0$

\newpage

Global uniqueness\\

\h $d'$�� (1)$\sim$(4)�� ������Ű�� �� �ٸ� global operator��
����. �׷��� ���� $d'$�� (1)$\sim$(4)��

\h �����ϴ� U�󿡼��� operator $d' : \mathcal{E}^p (U)\rightarrow
\mathcal{E}^{p+1}(U)$�� induce�ϴ� ���� ������. �̰���

\h  ���̱� ���� $d'$�� "local operator"���� ������. ��, \\

\h $(*) ~ \alpha|_U \equiv 0 \Longrightarrow d\alpha|_U \equiv 0 ~
for ~ \alpha \in \mathcal{E}^p(U)$\\

\h �� �������� ������. �� ������ U�󿡼��� $d\alpha$����
$\alpha$�� U�󿡼��� ������ �����Ѵٴ�

\h ���̴�. $(*)$�� ���̷��� $\forall p \in U ,~ d\alpha (p) =0$���� ���̸� �ȴ� :\\

\hh U�� ���� p�� ���� neighborhood V�� ���� ,

\hh $\exists \varphi$ a bump function s.t. 1 on V and 0 on $U^c$

\hh $d(0)(p) = d(\varphi\cdot\alpha)(p) =
d\varphi(p)\wedge\alpha(p) + \varphi(p)d\alpha(p) = d\alpha(p)$

\hh $\therefore d\alpha(p) =0$ \h (by �Ʒ�����\footnote{(1)�� ����
$d(0) + d(0) = d(0) \Longrightarrow d(0) =0$})\\

\h ���� $\alp \in \mathcal{E}^p(U)$�� �־����ٰ� ����. ��
$\alpha$�� ���� $ d'\alp(p) := d'(\varphi\alp)(p) , ~~\forall p
\in U$��� ����

\h �ϸ� �� $(*)$�� ���� bump function $\varphi$�� ���ÿ�
������� �� ���ǵȴ�.\\

\h �̷��� ���ǵ� $d' :
\mathcal{E}^p(U)\rightarrow\mathcal{E}^{p+1}(U)$�� (1)$\sim$(4)��
������Ű�� ���� ���� check�� ��

\h �ִ�. ������, $\widetilde{\alpha}$�� $\alpha$�� global
extension�̶�� $d'\alpha = d'\widetilde{\alpha}|_U$�� �������� ��
�� �ִ�.

\h $(\because \forall p \in U , d'\alpha(p) := d'(\varphi
\alpha)(p) = d' \widetilde{\alpha}(p)$ by $(*)$)\\

\h ���� local uniqueness�� ���� $d=d'$ on U�̰�

\h $(d'\alpha)|_U = d'(\alpha|_U) = d(\alpha|_U ) =
(d\alpha)|_U$�� ��� U�� ���� �����ϹǷ�\\

\h$\therefore $ globally $d' = d$

\end{proof}

\newpage
3. Invariant definition of $d\ome$:\\

Recall $r-form \Longleftrightarrow \mathcal{F}$-linear alternating
map $\underbrace{ \mathcal{X}\times\cdots
\times\mathcal{X}}_{r}\longrightarrow\mathcal{F}$\\

I. $d\ome(X_0, \cdots, X_r) = \dis{\sum_{i=0}^{r}}(-1)^i X_i
(w(X_0, \cdots, \widehat{X_i}, \cdots, X_r))$

\hspace{9em} + $\dis{\sum_{0 \leq i<j \leq r}}(-1)^{i+j}\ome([X_i,
X_j], X_0, \cdot, \widehat{X_i}, \cdots, \widehat{X_j}, \cdots,
X_r)$

I\!I. $d\ome(X_0, \cdots, X_r) = \frac{1}
{r+1}\{\dis{\sum_{i=0}^{r}}(-1)^i X_i (w(X_0, \cdots,
\widehat{X_i}, \cdots, X_r))$

\hspace{9em} + $\dis{\sum_{0 \leq i<j \leq r}}(-1)^{i+j}\ome([X_i,
X_j], X_0, \cdot, \widehat{X_i}, \cdots, \widehat{X_j}, \cdots,
X_r)\}$ (�Ʒ����� \footnote{$\alpha^1 \wedge \cdots \wedge
\alpha^r (v_1,\cdots,v_r) = det (\alpha^i(v_j))~ \cdots$  I.

\hhhh\hhhh\hhhh$ =\frac{1}{r!}det(\alpha^i (v_j))\cdots $  I\!I.\h ���� ���� pairing�� �����Ͽ��� ��})\\
\\
\\
\\
$\ome$�� 1-form�� �� �� ���� ������ ����. $\ome$�� 1-form�� �� ��
���� �캯�� ������ ���� ����.\\

\h $"d\ome" (X_0 , X_1) \overset{^{I.}}{:=} X_0\ome(X_1) -
X_1\ome(X_0) - \ome([X_0 , X_1])$\\

\hhhh\hhh $\overset{^{I\!I.}}{:=} \frac{1}{2} (X_0\ome(X_1) -
X_1\ome(X_0) - \ome([X_0 ,
X_1]))$ \\

�켱 $"d\ome"$�� 2-form���� ���̱� ���� 2-form��
$\mathcal{F}$-linear alternating map���� �̿�����. �̸� ���̷��� ���� �� ������ check�ϸ� �ȴ�.\\

\h $"d\ome"(X_0 +X_0 ',X_1) = "d\ome"(X_0,X_1) + "d\ome"(X_0
',X_1)$

\h $"d\ome"(fX_0,X_1) = f"d\ome"(X_0,X_1)$

\h $"d\ome"(X_0,X_1) = - "d\ome"(X_1,X_0)$\\

�׷��� �� �������� ���Ƿ� ���� �� �������� ���� �� �� �ִ�.\\
\\
\\
���� $"d\ome" = d\ome$���� ������. Locally check �ϸ� ����ѵ�\\

\h local chart U������ $\ome = \dis{\sum} a_j dx^j \Longrightarrow
d\ome=\dis{\sum_j} da_j \wedge dx^j = \dis{\sum_{i,j}} \frac{\pa
a_j}{\pa x^i} dx^i \wedge dx^j$.

\h Let $"d\ome" = \dis{\sum_{i<j}} ~\square~ dx^i \wedge dx^j$

\h $\Longrightarrow \square = "d\ome"(\frac{\pa}{\pa
x^i},\frac{\pa}{\pa x^j}) = \frac{\pa}{\pa x^i}a_j - \frac{\pa}{\pa x^j}a_i$�̹Ƿ� $"d\ome" = d\ome$�� �����Ѵ�.\\
\\
\\
(����) �� ���� �Ϲ����� p-form�� ���ؼ� �����Ͽ���. (���� Spivak p
289)


\end{document}
