lecture4.tex

\section{Differential Manifolds}
\begin{framed}
\textbf{Motivation}: So far we have dealt with topological manifolds which allow us to talk about continuity. But to talk about smoothness of curves on manifolds, or velocities along these curves, we need something like differentiability. Does the structure of topological manifold allow us to talk about differentiability? The answer is a resounding no.

So this lecture is about figuring out what structure we need to add on a topological manifold $M$ to start talking about differentiability of curves ($\mathbb{R} \to M$) on a manifold, or differentiability of functions ($M \to \mathbb{R}$) on a manifold, or differentiability of maps ($M \to N$) from one manifold $M$ to another manifold $N$. 
\end{framed}

\begin{tikzpicture}[decoration=snake]
  \matrix (m) [matrix of math nodes, row sep=2em, column sep=3em, minimum width=1em]
  {
\gamma : \mathbb{R} & U \\
& x(U) \subseteq \mathbb{R}^d \\ 
};
  \path[->]
  (m-1-1) edge node [above] {$$} (m-1-2)
          edge node [sloped, anchor=center, below] {$x \circ \gamma$} (m-2-2)
  (m-1-2) edge node [right] {$x$} (m-2-2);
\end{tikzpicture}

\underline{idea}. try to ``lift'' the undergraduate notion of differentiability of a curve on $\mathbb{R}^d$ to a notion of differentiability of a curve on $M$

\underline{Problem} Can this be well-defined under change of chart?

\begin{tikzpicture}[decoration=snake]
  \matrix (m) [matrix of math nodes, row sep=4em, column sep=6em, minimum width=2em]
  {
    & y(U\cap V) \subseteq \mathbb{R}^d \\ 
\gamma : \mathbb{R} & U \cap V \neq \emptyset \\
& x(U\cap V) \subseteq \mathbb{R}^d \\ 
};
  \path[->]
  
  (m-2-1) edge node [auto] {$$} (m-2-2)
          edge node [sloped, anchor=center, below] {$x \circ \gamma$} (m-3-2)
          edge node [sloped, anchor=center, above] {$y \circ \gamma$} (m-1-2)
  (m-2-2) edge node [auto] {$x$} (m-3-2)
          edge node [auto] {$y$} (m-1-2)
  (m-3-2) edge [bend right=40] node [right] {$y\circ x^{-1}$} (m-1-2);
\end{tikzpicture}

$x\circ \gamma$ undergraduate differentiable (``as a map $\mathbb{R} \to \mathbb{R}^d$'')

\[
\begin{gathered}
  \underbrace{y\circ \gamma}_{\text{maybe only continuous, but not undergraduate differentiable} } =  \underbrace{ ( \overbrace{ y\circ x^{-1}}^{\mathbb{R}^d \to \mathbb{R}^d }   )}_{\text{continuous}}  \circ \underbrace{ \overbrace{ (x\circ \gamma) }^{\mathbb{R}\to \mathbb{R}^d} }_{ \text{ undergrad differentiable } }  = y \circ (x^{-1} \circ x) \circ \gamma
\end{gathered}
\]

At first sight, strategy does not work out.  

\subsection{Compatible charts}

In section 1, we used any imaginable charts on the top. mfd. $(M,\mathcal{O})$.  

To emphasize this, we may say that we took $U$ and $V$ from the \emph{maximal atlas} $\mathcal{A}$ of $(M,\mathcal{O})$.  


\begin{definition}
Two charts $(U,x)$ and $(V,y)$ of a top. mfd. are called \ding{96}-compatible if 
either
\begin{enumerate}
  \item[(a)] $U \cap V = \emptyset$
or  \item[(b)] $U\cap V \neq \emptyset$
\end{enumerate}
chart transition maps have undergraduate \ding{96} property.

EY : 20151109 e.g. since $\mathbb{R}^d \to \mathbb{R}^d$, can use undergradate \ding{96} property such as continuity or differentiability.

\[
\begin{aligned}
  & y \circ x^{-1} : x(U \cap V) \subseteq \mathbb{R}^d  \to y(U\cap V) \subseteq \mathbb{R}^d  \\
  & x\circ y^{-1} : y(U\cap V) \subseteq \mathbb{R}^d   \to x(U\cap V) \subseteq \mathbb{R}^d
\end{aligned}
\]
\end{definition}

\underline{Philosophy}: 

\begin{definition}
  An atlas $\mathcal{A}_{\text{\ding{96}}}$ is a \ding{96}-compatible atlas if any two charts in $\mathcal{A}_{\text{\ding{96}}}$ are \ding{96}-compatible.

\end{definition}

\begin{definition}
  A \textbf{\ding{96}-manifold} is a triple $(\underbrace{ M,\mathcal{O} }_{\text{top. mfd.} }, \mathcal{A}_{\text{\ding{96}}})$ \quad \, $\mathcal{A}_{\text{\ding{96}}} \subseteq \mathcal{A}_{\text{maximal}} $
\end{definition}


\begin{tabular}{ l | c  l}
\ding{96} &  undergraduate  \ding{96} &  \\
\hline
$C^0$ & $C^0(\mathbb{R}^d \to \mathbb{R}^d) =$  &  continuous maps w.r.t. $\mathcal{O}$  \\
$C^1$ & $C^1(\mathbb{R}^d \to \mathbb{R}^d) = $  &  differentiable (once) and is continuous  \\
$C^k$ & & $k$-times continuously differentiable \\
$D^k$ & & $k$-times differentiable \\
$\vdots$ & & \\
$C^{\infty}$ & $C^{\infty}(\mathbb{R}^d \to \mathbb{R}^d)$ & \\
$\mathbin{\rotatebox[origin=c]{-90}{$\supseteq$}}$ & &  \\
$C^{\omega}$ & $\exists $  multi-dim. Taylor exp.  &  \\
$\mathbb{C}^{\infty}$ & satisfy Cauchy-Riemann equations, pair-wise & 
\end{tabular}


EY : 20151109 Schuller says: $C^k$ is easy to work with because you can judge $k$-times cont. differentiability from existence of all partial derivatives \textbf{and} their continuity.  There are examples of maps that partial derivatives exist but are not $D^k$, $k$-times differentiable.  

\begin{theorem}[Whitney]
%  Any $C^{k\geq 1}$-manifold $(M,\mathcal{O}, \mathcal{A}_{C^{k\geq 1}})$  
  Any $C^{k\geq 1}$-atlas, $\mathcal{A}_{C^{k\geq 1}}$ of a topological manifold \emph{contains} a $C^{\infty}$-atlas.  

Thus we may w.l.o.g. always consider $C^{\infty}$-manifolds, ``smooth manifolds'', unless we wish to define Taylor expandibility/complex differentiability \dots
\end{theorem}

EY : 20151109 Hassler Whitney \footnote{\url{http://mathoverflow.net/questions/8789/can-every-manifold-be-given-an-analytic-structure}}

\begin{definition}
  A smooth manifold $(\underbrace{ M,\mathcal{O} }_{\text{top. mfd. } }, \underbrace{ \mathcal{A}}_{C^{\infty}-\text{atlas}} )$ 
\end{definition}

\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=4em, column sep=6em, minimum width=2em]
  {
 \mathbb{R} & M   \\
&  \mathbb{R}^d \\ 
};
  \path[->]  
  (m-1-1) edge node [auto] {$\gamma$} (m-1-2)
          edge node [auto] {$x\circ \gamma$} (m-2-2)
  (m-1-2) edge node [auto] {$x$} (m-2-2);
\end{tikzpicture}
EY: 20151109 Schuller was explaining that the trajectory is real in $M$; the coordinate maps to obtain coordinates is $x\circ \gamma$

\subsection{Diffeomorphisms}

$M \xrightarrow{ \phi } N$

If $M,N$ are naked sets, the structure preserving maps are the bijections (invertible maps).  

e.g. $\lbrace 1,2,3 \rbrace \to \lbrace a,b \rbrace$

\begin{definition}
  $M \cong_{\text{set}} N$ (set-theoretically) isomorphic if $\exists \, $ bijection $\phi : M \to N$
\end{definition}

\underline{Examples}.  $\mathbb{N} \cong_{\text{set}} \mathbb{Z}$ \\
$\mathbb{N} \cong_{\text{set}} \mathbb{Q}$  (EY: 20151109 Schuller says from diagonal counting)\\
$\mathbb{N} \cancel{\cong_{\text{set}}} \mathbb{R}$

Now $(M, \mathcal{O}_M) \cong_{\text{top}} (N,\mathcal{O}_N)$ (topl.) isomorphic $=$ ``homeomorphic'' $\exists \, $ bijection $\phi : M \to N$  \\
\phantom{ \quad \quad \, } $\phi, \phi^{-1}$ are continuous.  

$(V,+,\cdot) \cong_{\text{vec}} ( W,+_w,\cdot_w)$ (EY: 20151109 vector space isomorphism) if \\
$\exists \, \text{ bijection } \phi : V \to W$ linearly

\underline{finally}

\begin{definition}
  Two $C^{\infty}$-manifolds \\
  $(M,\mathcal{O}_M, \mathcal{A}_M)$ and $(N,\mathcal{O}_N, \mathcal{A}_N)$ are said to be \textbf{diffeomorphic} if $\exists \, $ bijection $\phi : M \to N$ s.t. 
\[
\begin{aligned} & \phi : M \to N \\
  & \phi^{-1} : N \to M \end{aligned}
    \]
are both $C^{\infty}$-maps

\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=4em, column sep=6em, minimum width=2em]
  {
 \mathbb{R}^d & \mathbb{R}^e   \\
M \supseteq U &  V\subseteq N   \\ 
\mathbb{R}^d & \mathbb{R}^e \\
};
  \path[->]  
  (m-1-1) edge node [auto] {$\widetilde{y} \circ \phi \circ \widetilde{x}^{-1}$} (m-1-2)
  (m-2-1) edge node [auto] {$\widetilde{x}$} (m-1-1)
          edge node [auto] {$\phi$} (m-2-2)
          edge node [auto] {$x$} (m-3-1)
  (m-3-1) edge node [auto] {$ \substack{ y\circ \phi \circ x^{-1} \\ 
 \text{ undergraduate } C^{\infty} }$} (m-3-2)
          edge [bend left=50] node [auto] {$C^{\infty}$} (m-1-1)
  (m-2-2) edge node [auto] {$\widetilde{y}$} (m-1-2) 
          edge node [auto] {$y$} (m-3-2)
  (m-3-2) edge [bend right=50] node [auto] {$$} (m-1-2);
\end{tikzpicture}


\end{definition}

\begin{theorem}
  $\# = $ number of $C^{\infty}$-manifolds one can make out of a given $C^0$-manifolds (if any) - up to diffeomorphisms.  

\begin{tabular}{l | c r }
  $\text{dim}M$ &  $\#$ &  \\
  \hline
  1  & 1  & Morse-Radon theorems \\
 2  & 1  & Morse-Radon theorems \\
 3 & 1  & Morse-Radon theorems \\
4 & uncountably infinitely many & \\
5 &   finite  & surgery theory \\
6 &  finite & surgery theory \\
\vdots & finite & surgery theory
\end{tabular}

\end{theorem}

EY : 20151109 cf. \url{http://math.stackexchange.com/questions/833766/closed-4-manifolds-with-uncountably-many-differentiable-structures}  \\
\href{http://www.maths.ed.ac.uk/~aar/papers/scorpan.pdf}{The wild world of 4-manifolds}