09_18.tex

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\author{Professor Alejandro Uribe-Ahumada\\ \small\i{Transcribed by Thomas Cohn}}
\title{Math 591 Lecture 8}
\date{9/18/20} % Can also use \today

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

\defn{
	Let $M,N$ be $C^{\infty}$ manifolds, and $F:M\to{}N$ a continuous map. Let $p\in{}M$. Then we say $F$ is \u{smooth} at $p$ iff there exist charts $(U,\phi)$ of $M$ and $(V,\psi)$ of $N$ \st{} $p\in{}U$, $F(p)\in{}V$, and
	\[
		\psi\of{}F\of\phi\inv:\phi(F\inv(V)\cap{}U)\to\R^{n}
	\]
	is $C^{\infty}$.\n
}

\par\noindent
Observe: Since $F$ is continuous, $F\inv(V)$ is open, so $F\inv(V)\cap{}U$ is an open neighborhood of $p$. Thus, $\phi(F\inv(V)\cap{}U)$ is open in $\R^{m}$.\n

\defn{
	Let $M,N$ be $C^{\infty}$ manifolds, $F:M\to{}N$ continuous. Then $F$ is \u{smooth} iff $\forall{}p\in{}M$, $F$ is smooth at $p$.\n
}

\lemma{
	Let $M,N$ be $C^{\infty}$ manifolds, $F:M\to{}N$ continuous. Then $F$ is smooth iff there are atlases $\set{(U_{\alpha},\phi_{\alpha})}$ of $M$ and $\set{(V_{\beta},\psi_{\beta})}$ of $N$ \st{} $\forall\alpha,\beta$, $\psi_{\beta}\of{}F\of\phi_{\alpha}\inv:\phi_{\alpha}(F\inv(V_{\beta})\cap{}U_{\alpha})\to\R^{n}$ is smooth. This, in turn, is true iff for any pair of atlases $\set{(U_{\alpha},\phi_{\alpha})}$ and $\set{(V_{\beta},\psi_{\beta})}$, the previous condition holds.\n
	Proof: (exercise)\n
}

\par\noindent
The key outcome is that if a function is smooth according to one atlas, it's smooth according to all atlases.\n

\ex{
	\[
		\map{\GL(n,\R)\times\GL(n,\R)}{\GL(n,\R)}{(g_{1},g_{2})}{g_{1}g_{2}}\qquad\ptxt{and}\qquad\map{\GL(n,\R)}{\GL(n,\R)}{g}{g\inv}\qquad\ptxt{are smooth.}
	\]
	\[
		\map{O(n)\times{}O(n)}{O(n)}{(g_{1},g_{2})}{g_{1}g_{2}}\qquad\ptxt{and}\qquad\map{O(n)}{O(n)}{g}{g\inv}\qquad\ptxt{are smooth.}
	\]
	\n
}

\defn{
	A \u{Lie group} $G$ is a group which also has a $C^{\infty}$ structure \st{}
	\[
		\map{G\times{}G}{G}{(g_{1},g_{2})}{g_{1}g_{2}}\qquad\ptxt{and}\qquad\map{G}{G}{g}{g\inv}
	\]
	are smooth.
}

\subsection*{Tangent and Cotangent Spaces}

\par\noindent
We want to construct tangent vectors without requiring an ambient space!\n
Idea: Vectors in $\R^{n}$ define ``directional'' derivatives.\n

\par\noindent
Pick $p\in{}U\overset{\ptxt{open}}{\subseteq}\R^{n}$, and $v\in\R^{n}$. Then if $f:U\to\R$ is $C^{\infty}$, we can define $D_{v}f(p)=\del{}f(p)\cdot{}v$.\n

\par\noindent
Remark: We can regard $D_{v}$ as an operator $C^{\infty}\ni{}f\mapsto{}D_{v}f(p)\in\R$. It has the following properties:
\begin{enumerate}[topsep=0pt, itemsep=0pt, label=\arabic*)]
	\item Linear over $\R$: $D_{v}(f+cg)(p)=D_{v}f(p)+cD_{v}g(p)$.
	\item Leibniz' rule: $D_{v}(fg)(p)=f(p)D_{v}g(p)+D_{v}f(p)g(p)$.
\end{enumerate}
This was all motivation. Now, for the formalization.\n

\newpage
\defn{
	Let $M$ be a smooth manifold, $p\in{}M$. Then the \u{space of germs of functions of $M$ at $p$} is
	\[
		C^{\infty}_{p}(M)=\set{(f:U\to\R,U)\mid{}U\subseteq{}M\ptxt{ open },p\in{}U,f\in{}C^{\infty}}/\sim
	\]
	where $(f,U)\sim(g,V)\Leftrightarrow\exists{}W\subseteq{}U\cap{}V$ \st{} $p\in{}W$ and $\restr{f}{W}=\restr{g}{W}$.\n
	A \u{germ} at $p$ is an equivalence class $[f]=[(f,U)]$.\n
}

\par\noindent
Notation: Given $(f,U)$ as above and $p\in{}U$, $[f]$ is the calss of $(f,U)\in{}C^{\infty}_{p}(M)$.\n

\lemma{
	$C^{\infty}_{p}(M)$ is an $\R$-vector space and a ring.
	\begin{enumerate}[label=\alph*), topsep=0pt, itemsep=0pt, leftmargin=5\parindent]
		\item $[f]+c[g]\eqdef{}[f+cg]$
		\item $[f]\cdot{}[g]\eqdef{}[fg]$ (defined by $\restr{fg}{U\cap{}V}:U\cap{}V\to\R$)
	\end{enumerate}\up\n
	EXER: Show the remaining properties.\n
}

\defn{
	A \u{derivation} on $M$ at $p$ is an $\R$-linear map $D:C^{\infty}_{p}\to\R$ \st{} $\forall{}[f],[g]\in{}C^{\infty}_{p}(M)$, $D([f]g])=f(p)D[g]+g(p)D[f]$.\n
}

\par\noindent
Observe: $f(p)=[f](p)\in\R$ is well defined by $[f]$.\n

\defn{
	The \u{tangent space} to $M$ at $p$ is $T_{p}M=\set{\ptxt{all derivations of $M$ at $p$}}$.\n
}

\end{document}