10_05.tex

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\author{Professor Alejandro Uribe-Ahumada\\ \small\i{Transcribed by Thomas Cohn}}
\title{Math 591 Lecture 15}
\date{10/5/20} % Can also use \today

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

\thm{
	(Regular Value Theorem for Manifolds) Let $M$ and $N$ be manifolds, $F:M\to{}N$ $C^{\infty}$, and $q\in{}N$ a regular value of $F$. Then $F\inv(q)$ is a regular submanifold of $M$.\nn
	Proof: Let $p\in{}F\inv(q)$. We want to show there are coordinates of $M$ near $p$ which are adapted to the preimage of $F\inv(q)$. Because $q$ is a regular value, $F_{*,p}:T_{p}M\to{}T_{q}N$ is onto for any $p$. By the normal form for submersions, there are coordinates $(U,\phi=(x^{1},\ldots,x^{m}))$ near $p$ and $(V,\psi=(y^{1},\ldots,y^{n}))$ near $q$, with $U\subseteq{}F\inv(V)$, such that $\tilde{F}(r^{1},\ldots,r^{m})=(r^{1},\ldots,r^{n})$.
	\[
		\begin{tikzcd}[ampersand replacement=\&]
			U \arrow[r,"F"] \arrow[d,"\phi"] \& V \arrow[d,"\psi"]\\
			\phi(U) \arrow[r,"\tilde{F}"] \& \psi(V)
		\end{tikzcd}
	\]
	WOLOG assume $\psi(q)=0$. Split $\phi$, with $x'=(x^{1},\ldots,x^{n}):U\to\R^{n}$ and $x''=(x^{n+1},\ldots,x^{m}):U\to\R^{m-n}$. Then $F\inv(q)\cap{}U$ corresponds to $\tilde{F}\inv(0)$ by $\phi$, i.e., $F\inv(q)\cap{}U=\set{a\in{}U\mid{}x'(a)=0}$. Thus, $(x'',x')$ are adapted coordinates to $F\inv(q)\cap{}U$.\proven
}

\par\noindent
Observe: (Keeping the notation of the proof) $x'':F\inv(q)\cap{}U\to\R^{m-n}$ are coordinates on $F\inv(q)\cap{}U$. So $\dim{}F\inv(q)=m-n$. (Recall: $m\ge{}n$.)\n

\defn{
	The \u{codimension} of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.\n
}

\par\noindent
$\codim{}F\inv(q)=\dim{}M-\dim{}F\inv(q)=m-(m-n)=n$. This is the dimension of the target space.\n

\par\noindent
Observe: $\forall{}p\in{}F\inv(q)$ (if $q$ is a regular value), $T_{p}(F\inv(q))\subseteq{}T_{p}M$ as a subspace. In fact, $T_{p}(F\inv(q))$ is the kernel of $F_{*,p}$.\n

\subsection*{A General Observation on Tangent Spaces of Submanifolds}

\par\noindent
Let $S\subseteq{}M$ be a submanifold, and $p\in{}S$. Then $T_{p}S\subseteq{}T_{p}M$ by: $\forall\gamma:(-\varepsilon,\varepsilon)\to{}S$ with $\gamma(0)=p$, we have
\[
	\begin{tikzcd}
		\dot\gamma_{S}(0) \arrow[r,mapsto,"\iota_{*,p}"] & \dot\gamma_{M}(0)\\[-15pt]
		T_{p}S \arrow[u,symbol=\in] & T_{P}M \arrow[u,symbol=\in]
	\end{tikzcd}
\]
by using the differential of the inclusion $\iota:S\hookrightarrow{}M$. The inclusion in adapted coordinates is $x'\mapsto(x',0)$. If $[f]\in{}C^{\infty}_{p}(M)$, $\dot\gamma_{M}(0)[f]=\dot\gamma_{S}(0)[f\of\iota]$. Observe: $f\of\iota$ is the restriction of $f$ to $S$.\n

\par\noindent
Conclusion: Tangent spaces of submanifolds are subspaces of the tangent spaces of the original manifold.\n

\defn{
	A map $F:M\to{}N$ is a \u{submersion} iff $\forall{}p\in{}M$, $F_{*,p}$ is onto.\n
}

\cor{
	If $F$ is a submersion, then $\forall{}q\in{}N$, $q$ is a regular value, so $F\inv(q)$ (``the fiber of $f$ over $q$'') if either empty or a codimension $n$ submanifold of $M$.\n
}

\ex{
	Let $M=\R^{2}\cut{}S^{1}$, $N=\R$, $F:M\to{}N$ with $F(x,y)=x$. What are the fibers?
	\begin{itemize}[topsep=0pt, itemsep=0pt, leftmargin=3\parindent]
		\item For $q\in(-\infty,1)\cup(1,\infty)$, $F\inv(q)=\R$.
		\item For $q\in(-1,1)$, $F\inv(q)=(-\infty,-\sqrt{1-q^{2}})\cup(-\sqrt{1-q^{2}},\sqrt{1-q^{2}})\cup(\sqrt{1-q^{2}},\infty)$.
		\item For $q\in\set{-1,1}$, $F\inv(q)=\R\cut\set{0}$.
	\end{itemize}\up\n
	Note that in this example, some of the fibers are different topologically!\n
}

\ex{
	Let $M=S^{3}$, $F:S^{3}\to\CP^{1}\cong{}S^{2}$ (the Riemann Sphere). Then the fibers are all circles, and the map from $S^{3}$ to $S^{2}$ is called the Hopf fibration.\n
}

\defn{
	A $C^{\infty}$ map $F:M\to{}N$ is a \u{fibration} with fiber $\Phi$, where $\Phi$ is a manifold, iff there is an open covering $\set{U_{\alpha}}$ of $N$ (called the \u{base}) and diffeomorphic maps $\chi_{\alpha}:F\inv(U_{\alpha})\to{}U_{\alpha}\times\Phi$ (called \u{trivializations}) such that the diagram
	\[
		\begin{tikzcd}[ampersand replacement=\&]
			F\inv(U_{\alpha}) \arrow[rr, "\chi_{\alpha}"] \arrow[rd, "\restr{F}{F\inv(U_{\alpha})}", swap] \&[-10pt] \&[-10pt] U_{\alpha}\times\Phi \arrow[dl, "\pi\ptxt{ projection}"]\\
			\& U_{\alpha}
		\end{tikzcd}
	\]
	commutes (i.e. all paths are the same). We say that $F$ is a \u{fiber bundle} with \u{fiber} $\Phi$.\n
}

\par\noindent
Let's unpack what this means. Commutativity of the diagram means $\forall{}p\in{}F\inv(U_{\alpha})$, $\chi_{\alpha}(p)=(F(p),\star)$ where $\star\in\Phi$. So $\forall{}q\in{}U_{\alpha}$, $\chi_{\alpha}$ restricts to the fiber $F\inv(q)$, where $\chi_{\alpha}(p)\mapsto\star$.\n

\ex{
	The tangent bundle $TM$ is a fiber bundle, with fiber $\R^{m}$ (with $m=\dim{}M$).
	\[
		\begin{tikzcd}[ampersand replacement=\&]
			TM \arrow[d]\\
			M
		\end{tikzcd}
	\]
	Note: This has additional structure: $\Phi\cong\R^{m}$ is a vector space, the fibers are all vector spaces, and there exist trivializations that are linear on the fibers.\n
}

\end{document}