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\begin{document}
\section*{Orientation}
\begin{defn}
For each point p in a plane, there are two different choices of
rotations, namely clockwise and counter-clockwise. The choice of
one rotation is called an \textit{orientation at p}.

 (Or equivalently, choice of ordered basis $(e_1,e_2)$ at p, such
 that the rotation from $e_1$ to $e_2$ matches up the given
 orientation. In this case $(e_1,e_2)$ and $(b_1,b_2)$ give the
 same orientation iff det A $>$ 0 where $(e_1,e_2)A=(b_1,b_2)$.)
\end{defn}

�� basis�� �̿��� ����� ���� �Ϲ����� �������� orientation��
������ �� �ִ� ����� �ش�.

\begin{defn}
Let M be a surface (with or without boundary). Since M is locally
Euclidean(i.e., plane), each point of M has a two choices of
orientation. M is called \textit{orientable} if we can select a
"continuous"(i.e locally constant) choice of orientation at each
point.
\end{defn}
{\bf ��.} $\textbf{R}^2$ (or any open subset of $\textbf{R}^2$)
is orientable.\\\\
{\bf Surface�� $\,\,\,$orientation.}\\
1. $S^2$ : $\textbf{R}^3$ ��ü��  orientable�̹Ƿ� ��
orientation(������ �����)�� �������� ����. �׷��� normal vector
N�� $S^2$�� orientation�� ���Ͽ� $\textbf{R}^3$�� orientation��
�����ϰ� ������ �� �����Ƿ� normal vector�� ���������� ���� ��
������ $S^2$�� orientation�� ���������� ������ �� �ִ�. ����
$\textbf{R}^3$�� embedded�� surface�� ���ؼ��� orientable�̳�
�ƴϳ� �ϴ� ����  normal vector�� ���������� ������ �� �ִ���
�ƴϳĿ� ����. $S^2$�� ��� normal vector�� ������ �ٱ� ��������
������ �����̹Ƿ� orientble�̴�.\\
2. $T^2$ : ���� ���������̹Ƿ� ���� orientable�̴�.\\
3. M\"{o}bius band : normal vector�� �������� ������ �Ұ����ϹǷ�
non-orientable. \\\
{\bf Exercise.} Continuous choice of orientation �� �� coordinate
chart�� ���ÿ� �����ϴ�. (�� ������ Annulus Thm �� ����ϸ� ���������� ������ �����ϴ� ������ �����Ѵ�. Let
$$
A = \{x /in \mathbb{R}^2 \,|\,1\leq||x||\leq2\}\textrm{ and } S_r=\{x\in\mathbb{R}^2\,|\,||x||=r\}.
$$
Suppose an orientation on each of $S_r$, $r=1,2$ is given. Then a homeomorphism of $A$ preserves or changes the orientations of both of $S_r$ simultaneously.)
 \\
{\bf Note.} \textit{Orientability is a topological invariant.}\\\\
{\bf Orientation through triangulation.}\\
{\bf 1.} Suppose M has a triangulation. Then M is orientable if
and only if the two "induced orientation" on each edge from two
neighboring triangles are opposite.

��, �� ������ �ﰢ�������� orientation�� ���� edge���� ���ǰ�
�־����ٸ� �� �� �ﰢ�������� orientation�� ���� ����. �ٽø���
���������� �� orientation�� ����ȴٴ� ���� ���Ѵ�. ���� ���
�ﰢ���� ���Ͽ� orientation�� �ֵ� ���� edge���� ���� ���� ��
�ְ� �� �� �ִٸ� M�� orientatable�ϰ� �ȴ�.\\

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$\hspace{-2em}\,\,\,\,$ {\bf 2}. Suppose M is obtained from
polygon identifying each pair of edges. The same idea applies.

�� ���� identify�Ǵ� �ٱ� edge�鿡 ���ؼ��� induced orientation�� cancel �Ǵ����� ���캸�� �ȴ�.\\
(�׸�����)
\begin{prop} M is non orientable if and only if M contains a
M\"{o}bius band.
\end{prop}
\begin{proof} $\Leftarrow$) Clear. M�� triangulation�� �� �ﰢ���� orientation�� ��� �ִ���
M\"{o}bius band�� �߽ɼ��� ���� triangulation������ �� �ﰢ����
orientation�� ���� edge���� ���� cancel�ǰ� �� ��
����.\\$\Rightarrow$) Let $M = D / \sim$. ( D is a polygon.) We
extend an orientation at a point in D toward the boundary in a
unique way. �տ��� �� ��ó�� D�� boundary�� identify�ϴ� ������
���� �ð�������� ������ $\cdot\cdot\cdot a \cdot\cdot\cdot a'
\cdot\cdot\cdot$ ó�� ������ �� �ִ�. ���⼭ ���
identify�ϴ��Ŀ� ���� $a'=a^{\pm1}$�� �ȴ�. ���� M��
non-orientable �ϸ� �̵� �������� induced orientation�� ����
������ �ʴ� edge�� �־�� �ϹǷ� � edge $a$ �� ����
$\cdot\cdot\cdot a \cdot\cdot\cdot a \cdot\cdot\cdot$ �� ����
�Ǿ�� �ϰ� �� $a$�� �� ������ �ϴ� �簢���� �������� $a$��
identify �� ���� M\"{o}bius band�� �ȴ�.
\end{proof}

\begin{cor} M is a closed(i.e.,compact without boundary)
surface.
\\$\Rightarrow$ M = $ T^2 \sharp T^2 \cdot\cdot\cdot \sharp T^2 $ if M is orientable, and

$\hspace{1em} = P^2 \sharp P^2 \cdot\cdot\cdot \sharp P^2 $ if M is non-orientable.
\end{cor}

\end{document}
