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\author{Professor Alejandro Uribe-Ahumada\\ \small\i{Transcribed by Thomas Cohn}}
\title{Math 591 Lecture 35}
\date{11/30/20} % Can also use \today

\begin{document}
\maketitle
\setlength\RaggedRightParindent{\parindent}
\RaggedRight

\par\noindent
Final remarks on orientation:\n

\par\noindent
Recall: If $M$ is oriented, $\exists\set{(U_{\alpha},\varphi_{\alpha})}$, a positive atlas of $M$. This means all transition functions $\varphi_{\beta}\of\varphi_{\alpha}\inv$ have the determinant of their Jacobian positive at every point, and the coordinate frames are positive.\n

\par\noindent
Conversely, if $M$ is an atlas satisfying he above property, then one can define an orientation of $M$ by requiring the coordinate frames are positive.\n

\par\noindent
In general, to show a manifold is orientable, exhibit such an atlas.\n

\ex{
	Check: The atlas of $\CP^{n}$ used in homework has this condition, so it is orientable.\n
}

\par\noindent
Observe: If $S\subset{}M$ is a $\codim$-$1$ submanifold, and $M$ is oriented, and there exists a continuous field of normal vectors on $S$,
\[
	S\ni{}p\mapsto\vec{n}_{p}\in{}T_{p}M\quad\ptxt{\st{}}\quad{}T_{p}M=T_{p}S\oplus\R\vec{n}_{p}
\]
then $S$ is orientable, and the convention for its orientation is: a basis $\set{b_{1},\ldots,b{n-1}}$ of $T_{p}S$ is positive iff $\set{\vec{n}_{p},b_{1},\ldots,b_{n-1}}$ is positive w.r.t. $M$.\n
\i{TL;DR, put the normal vector first.}\n

\ex{
	We can embed the Klein bottle in a $\dim$-$3$ manifold $M$ \st{} there exists a continuous $\vec{n}$, but $M$ is non-orientable.\n
}

\subsection*{Partitions of Unity}

\par\noindent
This is a very technical, but very useful tool. We begin with point-set topology.\n

\defn{
	An indexed covering (not necessarily open) $\set{S_{\alpha}}_{\alpha\in{}A}$ of (a manifold) $M$ (doesn't have to be a manifold) (with $S_{\alpha}\subset{}M$) is said to be \u{locally finite} iff every $p\in{}M$ has a neighborhood $U$ \st{} $\set{\alpha\in{}A\mid{}S_{\alpha}\cap{}U\ne\emptyset}$ is finite. That is, every $p$ is in only finitely-many $S_{\alpha}$.\n
}

\thm{
	(Thm 1.15 in Lee) Any topological manifold is \u{paracompact}: every open cover $\set{U_{\alpha}}_{\alpha\in{}A}$ has a countable, locally-finite refinement $\set{V_{i}}_{i\in\N}$. That is, $\forall{}i\in\N$, $V_{i}$ is open, and $\exists\alpha\in{}A$ \st{} $V_{i}\subset{}U_{\alpha}$ and $M$ is covered by $\set{V_{i}}_{i\in\N}$.\nn
	Proof: This proof is long and complex, but it only uses point-set topology. This is the first time we're using the fact that $M$ is second-countable!\n
}

\par\noindent
Observe: If $\mathscr{B}$ is any basis of $M$, the $V_{i}$ can be chosen to be in $\mathscr{B}$.\n

\defn{
	Let $M$ be a smooth manifold. A \u{partition of unity} on $M$ is an indexed family $\set{\chi_{\alpha}}_{\alpha\in{}A}$ of $C^{\infty}$ functions on $M$ \st{}
	\begin{enumerate}[leftmargin=4\parindent, itemsep=0pt, label=(\arabic*)]
		\item $\set{\supp(\chi_{\alpha})}_{\alpha\in{}A}$ is a locally finite cover of $M$.
		\item $\forall{}p\in{}M$, $\sum_{\alpha\in{}A}\chi_{\alpha}(p)=1\in\R$. (Note that this is a finite sum by (1).)
	\end{enumerate}
}

\thm{
	Let $\set{U_{\alpha}}_{\alpha\in{}A}$ be an open cover of $M$. Then $\exists\set{\chi_{\alpha}}_{\alpha\in{}A}$, a partition of unity, that is \u{subordinate} to $\set{U_{\alpha}}_{\alpha\in{}A}$, i.e., $\forall\alpha\in{}A$, $\supp\chi_{\alpha}\subseteq{}U_{\alpha}$.\nn
	Proof: We'll use paracompactness. (It may be easier to start by just thinking of a compact manifold.)\n
	Let $\mathscr{B}$ be the set of normal coordinate balls in $M$; we define $B\subset{}M$ to be a \u{normal coordinate ball} in $M$ iff there's a chart $(U,\phi)$ \st{} $\closure{B}\subset{}U$ and $\phi(B)=B_{r}(0)\subset\R^{n}$, the ball of radius $r$ centered at $0$ in $\R^{n}$, and also $\exists{}r'>r$ \st{} $\closure{B_{r}(0)}\subset{}B_{r'}(0)\subset\phi(U)$.\nn
	We claim that $\mathscr{B}$ is a basis of the topology of $M$. Let $\set{U_{\alpha}}_{\alpha\in{}A}$ be any open cover. Use the theorem on paracompactness: $\exists\set{B_{i}}_{i\in\N}$, a locally-finite refinement, and $\forall{}i\in\N$, $B_{i}$ is a normal coordinate ball. $\forall{}i\in\N$, let
	\[
		\phi_{i}(B_{i})=B_{r_{i}}(0)\subset\closure{B_{r_{i}}(0)}\subset{}B_{r_{i}'}(0)
	\]
	and $H_{i}$ be a function on $\Im(\phi_{i})$ such that $H_{i}:\Im(\phi_{i})\to\R$ is smooth, with
	\begin{itemize}[leftmargin=4\parindent, label=-]
		\item $H_{i}>0$ on $B_{r_{i}}(0)$
		\item $H_{i}=0$ on $B_{r_{i}}(0)\comp$
	\end{itemize}\up\n
	Thus, $\supp{}H_{i}=\closure{B_{r_{i}}(0)}$.\nn
	Define $\psi_{i}\in{}C^{\infty}(M)$ by $\psi_{i}=H_{i}\of\phi_{i}$ on $\dom\phi_{i}$, and $0$ everywhere else. Then $\supp\psi_{i}=\closure{B_{i}}\subset{}M$. We claim that $\set{\closure{B_{i}}}_{i\in\N}$. Observe that $\forall{}p$, $\sum_{i\in\N}\psi_{i}(p)>0$, because $\set{B_{i}}_{i\in\N}$ forms a cover of $M$, and $\restr{\psi_{i}}{B_{i}}>0$.\nn
	Now, define
	\[
		f_{i}=\frac{1}{\displaystyle\sum_{j\in\N}\psi_{j}}\psi_{i}
	\]
	so that $\set{\supp{}f_{i}}_{i\in\N}=\set{\closure{B_{i}}}$ is locally finite, and $\sum_{i\in\N}f_{i}=1$, $\forall{}p\in{}M$. Then, we just have to fix it so that he indexing sets are the same as $\set{U_{\alpha}}_{\alpha\in{}A}$ by $\forall{}i\in\N$, pick $\alpha(i)\in{}A$ such that $B_{i}\subset{}U_{\alpha(i)}$ and $\forall\alpha\in{}A$, let
	\[
		\chi_{\alpha}=\sum_{\substack{i\ptxt{ \st{}}\\ \alpha(i)=\alpha}}f_{i}
	\]
	(Note that $\chi_{\alpha}=0$ if the sum is empty.)\nn
	We claim that $\set{\supp\chi_{\alpha(i)}}_{i\in\N}$ is still locally finite. This follows from $\set{\supp{}f_{i}}_{i\in\N}$ being locally finite.\proven
}

\par\noindent
There are many applications of partitions of unity!

\ex{
	Existence of $C^{\infty}$ sections of any vector bundle.\nn
	Say $E\to{}M$ is a vector bundle of rank $r$. Then there exist $\set{(U_{\alpha},f_{\alpha})}$ local trivializations:
	\[
		\begin{tikzcd}[ampersand replacement=\&]
			\pi\inv(U_{\alpha}) \arrow[rr, "f_{\alpha}"] \arrow[dr] \&[-10pt] \&[-10pt] U_{\alpha}\times\R^{n} \arrow[dl]\\
			\& U_{\alpha} \arrow[ul, bend left=45, "s_{\alpha}"] \arrow[ur, bend right=45, "\ptxt{pick any section $s$}", swap]
		\end{tikzcd}
	\]
	Let $\set{\chi_{\alpha}}$ be a partition of unity on $M$ subordinate to $\set{U_{\alpha}}$. Then let $s=\sum_{\alpha\in{}A}\chi_{\alpha}\cdot{}s_{\alpha}$ (we interpret $\chi_{\alpha}\cdot{}s_{\alpha}$ as a $C^{\infty}$ section on $M$).\n
}

\par\noindent
The main application of partitions of unity is integrating forms.\n

\defn{
	Let $M$ be an oriented $n$-dimensional manifold. Let $\mu\in\Omega^{n}_{0}(M)$ be a top degree form with compact support. Let $\set{\phi_{\alpha}}$ be a positive atlas, and $\set{\chi_{\alpha}}$ a subordinate partition of unity (i.e. $\supp\chi_{\alpha}\subseteq\supp\phi_{\alpha}$, $\forall\alpha$). Then we define
	\[
		\int_{M}\mu=\sum_{\alpha}\int(\phi_{\alpha}\inv)^{*}(\chi_{\alpha}\mu)
	\]
}

\par\noindent
We have to check that the right hand side is independent of choice of coordinates. We'll do this next time...

\end{document}