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\begin{document}
\parindent=0cm
\section*{Orientation.}


\begin{defn}

$V$ : $n$-dimensional vector space.\\
Two ordered basis $b=\left( b_1, \cdots, b_n \right)$ and
$e=\left( e_1, \cdots, e_n \right)$ are equivalently oriented
$\left( b \sim
e \right)$ if $det A > 0,$ where $A=(a_{ij}), e_j=\sum_{i=1}^{n}a_{ij}b_i$.\\
Or equivalently, if $e_1 \we \cdots \we e_n=\alp b_1 \we \cdots
\we
b_n$ for some $\alp>$0.\\

\end{defn}


\begin{note}

$\sim$ is an equivalence relation and there exists two equivalent
classes in the set of all ordered basis. A choice of one of the
two equivalent classes is called an orientation.

\end{note}


\begin{note}

$f:(V^{n},\nu)\overset{\simeq} \to (W^{n},\mu)$ is orientation
preserving(or orientation reserving, resp.) , if $(v_1, \cdots ,
v_n) \in \nu \Rightarrow (fv_1 , \cdots , fv_n) \in \mu (\notin ,
resp.)$.(Well-definedness is clear)

\end{note}


\begin{defn}

$M$ : a $\ccn$-manifold (with boundary)\\
$\mu=\{\mu_p \arrowvert p \in M , \mu_p$ is an orientation of $T_p
M\}$ \\
$\mu$ is a (smooth) orientation of M if $\forall p \in M$,
$\exists$ a coordinate chart $(U,\varphi)$ of p s.t.
$(\varphi_*)_q$ is orientation-preserving $\forall q \in U$
(w.r.t. the standard orientation of $\rb ^n$)

\end{defn}


\begin{defn}

A connected M is orientable if M admits a (smooth) orientation.

\end{defn}


\begin{note}

M is orientable iff each component of M is orientable.\\

\end{note}


\begin{thm}

$M$ : connected $\ccn$-manifold of dimension n. TFAE.\\
(1) $M$ is orientable.\\
(2) $\exists$ a collection of coordinate charts $\Phi$=$\{(U,x)\}$
on $M$ s.t. $M$=$\bigcup_{U \in \Phi} U$ and det($\frac{\pa x}{\pa y}$)$>0$ on $U\cap V$ for $(U,x)$,$(V,y) \in \Phi$.\\
(3) $\bigwedge ^n(T^* M)\backslash 0-section$ \hspace{1.0em}has
exactly two components.\\
(4) $\exists$ non-vanishing n-form on $M$.(globally)\\
(5) $\forall$ loop in M, one can cover it by a finite positively
connected chain of coordinate charts.\\

\end{thm}


\begin{proof}

(1)$\Rightarrow$(4)$\Rightarrow$(3)$\Rightarrow$(2)$\Rightarrow$(1)\\

(1)$\Rightarrow$(4):\\
We can cover $M$ by a coordinate charts $(U_{\alp},x^{\alp})$ where $x^{\alp}_*$ is orientation-preserving.\\
Let $\{\varphi_{\alp}\}$ be a partition of unity subordinate to $\{U_{\alp}\}$.\\
Let $\ome (p):=\sum_{\alp} \varphi_{\alp}(p) dx_1^{\alp} \we
\cdots \we dx_n^{\alp}$. Then $\ome$ is clearly non-vanishing.\\

(4)$\Rightarrow$(3):\\
Let $\ome$ be a non-vanishing n-form.\\
Define $\bigwedge ^+ = \bigcup_{p \in M} \{c\ome_p \arrowvert
c>0\}$ and $\bigwedge ^- = \bigcup_{p \in M} \{c\ome_p \arrowvert
c<0\}$.\\
Then $\bigwedge ^+$ is connected, because $M$ is connected and
hence the graph of $\ome$ is connected.\\
Note also that $\bigwedge ^+$ and $\bigwedge ^-$ are disjoint open sets.\\

(3)$\Rightarrow$(2):\\
Suppose that $\bigwedge ^n \backslash 0$ has two components $C_1$ and $C_2$.\\
Let $\Phi =\{(U,x) \arrowvert dx^1 \we \cdots \we dx^n \in C_1
\}$. Then clearly $\Phi$ covers $M$.\\
($\because (U,x)=(x_1, \cdots, x_n)\notin \Phi \Rightarrow
(U,\tilde{x})=(-x_1,x_2, \cdots, x_n)\in \Phi$)\\
And for $(U,x),(V,y) \in \Phi$, \hspace{1.0em}$dx^1 \we \cdots \we
dx^n = det(\frac{\pa x}{\pa y})dy^1 \we \cdots \we dy^n
\Rightarrow det(\frac{\pa x}{\pa y})>0.$\\

(2)$\Rightarrow$(1):\\
Define $\mu_p=\{(\frac{\pa}{\pa x_1}(p), \cdots, \frac{\pa}{\pa
x_n}(p))\}$ for $(U,x) \in \Phi$. This is well-defined by (2).\\

(2)$\Leftrightarrow$(5): \bf exercise.\\

\end{proof}


{\bf ���� 21.} $f:M^n \to \rb ^{n+1}$ is an immersion.\\
$M$ is orientable $\Leftrightarrow \exists$ smooth non-vanishing
normal vector field along ($M$,$f$)\\

\begin{figure}[htb]

 \centerline{\includegraphics*[scale=0.6,clip=true]{hw1.eps}}

\end{figure}

\clearpage

{\bf Induced orientation on ${\pa M}$}\\

\begin{defn}

Let $M$ be orientable with an orientation $\mu.$\\
The induced orientation ${\pa \mu}$ on ${\pa M}$ is defined as
follows:\\
$\forall p \in \pa M$, let $(U,\varphi)$ be a coordinate chart and
let $out=\varphi_{*}^{-1}(\frac{\pa}{\pa u_1})$. \\
Then $(X_2, \cdots, X_n)\in {\pa \mu_p} \Leftrightarrow (out,X_2,
\cdots, X_n)\in \mu_p$\\

\end{defn}


\begin{figure}[htb]

\centerline{\includegraphics*[scale=0.6,clip=true]{induori.eps}}

\end{figure}


\end{document}
