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% 30/01/2023 11AM

\subsubsection*{Examples}

\begin{enumerate}[(1)]
    \item (See Example Sheet 1). The atlas of 2
        charts with stereographic projections
        gives $S^2$ the structure of an abstract
        smooth surface.
    \item The torus $T^2 = \RR^2 / \ZZ^2$. Recall
        that we obtained charts from (the inverses
        of) the projection restricted to small
        discs in $\RR^2$.
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/b40046a4a08f11ed.png}
        \end{center}
        The transition maps are
        \emph{translations} so $T^2$ inherits the
        structure of a smooth surface.
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/c8d2c71ea08f11ed.png}
        \end{center}
        $e(t, s) = (e^{2\pi i t}, e^{2\pi i s})$.
        Consider an atlas
        \[ \{(e(D_\eps(x, y)), e^{-1} \text{ on
        its image})\} \]
        $\eps < \frac{1}{3}$.
\end{enumerate}

\begin{definition*}
    Let $\Sigma$ be an abstract smooth surface and
    $f : \Sigma \to \RR^n$ a map. We say that $f$
    is smooth at $p \in \Sigma$ if whenever $(U,
    \varphi)$ is a chart at $p$, belonging to the
    smooth atlas for $\Sigma$, the map
    \[ f \circ \varphi^{-1} : \varphi(U) \to \RR^n \]
    is smooth at $\varphi(p) \in \RR^2$.
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/1fcb4de8a09011ed.png}
    \end{center}
\end{definition*}

\begin{note*}
    If it holds for one chart at $p$, it holds for
    \emph{all} charts at $p$:
    \[ f \circ \varphi^{-1} = f \circ
    \varphi_2^{-1} \circ \ub{(\varphi_2 \circ
    \varphi_1^{-1})}_{\text{diffeomorphism}} \]
    Chain rule!
\end{note*}

\begin{definition*}
    $\Sigma_1, \Sigma_2$ abstract smooth surfaces.
    A map $f : \Sigma_1 \to \Sigma_2$ is
    \emph{smooth} if it is smooth in local charts:
    there are charts $(U, \varphi)$ at $p$ and
    $(V, \psi)$ at $f(p)$ with $f(U) \subset V$
    such that $\psi \circ f \circ \varphi^{-1}$ is
    smooth at $\varphi(p)$.
    \begin{center}
        \includegraphics[width=0.8\linewidth]
        {images/99644d4ca09211ed.png}
    \end{center}
\end{definition*}

\begin{hiddenflashcard}[smooth-surface-function]
    When is $f : \Sigma_1 \to \Sigma_2$ smooth
    at $p$? \\
    \cloze{
    The map is smooth if it is smooth in local
    charts, i.e. there are charts $(U, \varphi)$
    at $p$ and $(V, \psi)$ at $f(p)$ with $f(U)
    \subset V$ such that $\psi \circ f \circ
    \varphi^{-1} : \RR^2 \to \RR^2$ is smooth
    at $\varphi(p)$.
    }
\end{hiddenflashcard}

\noindent
Again, if $f$ is smooth at $p$, then smoothness of
the local representation of $f$ at $p$ will hold
for all charts at $p$ and $f(p)$ on the given
atlas.

\begin{definition*}
    $\Sigma_1$ and $\Sigma_2$ are diffeomorphic if
    there exists $f : \Sigma_1 \to \Sigma_2$ that
    is smooth with smooth inverse.
\end{definition*}

\begin{definition*}
    If $Z \subset \RR^n$ is an arbitrary subset,
    we say that $f : Z \to \RR^n$ is smooth near
    $p \in Z$ if there exists open $B$, $p \in B
    \subset \RR^n$ and a \emph{smooth} $F : B \to
    \RR^n$ such that $F |_{B \cap Z} = f|_{B \cap
    Z}$, i.e. $f$ is locally the restriction of a
    smooth map defined on an open set.
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/2a62177aa09311ed.png}
    \end{center}
\end{definition*}

\begin{definition*}
    If $X \subset \RR^n$ and $Y \subset \RR^m$ are
    subsets, we say that $X$ and $Y$ are
    diffeomorphic if there exists $f : X \to Y$
    smooth with smooth inverse.
\end{definition*}

\begin{definition*}
    A \emph{smooth surface} in $\RR^3$ is a subset
    $\Sigma \subset \RR^3$ such that $\forall p
    \in \Sigma$ there exists an open set $p \in U
    \subset \Sigma$ such that $U$ is
    diffeomorphic to an open set in $\RR^2$. In
    other words, for all $p \in \Sigma$ there
    exists an open ball $B$ such that $p \in B
    \subset \RR^3$ and $F : B \to V \subset \RR^2$
    smooth such that $F|_{B \cap \Sigma} : B \cap
    \Sigma \to V$ is a homeomorphism with inverse
    $V \to B \cap \Sigma$ smooth.
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/9bafca9ea09311ed.png}
    \end{center}
    So we have 2 notions: one abstract and one
    taking advantage of the ambient space $\RR^3$.
\end{definition*}

\begin{hiddenflashcard}[smooth-surface-in-R3]
    \begin{definition*}[Smooth surface in $\RR^3$]
        A \emph{smooth surface} in $\RR^3$ is a
        subset $\Sigma \subset \RR^3$ \cloze{with the
        \fcemph{subspace topology} such that
        $\forall p \in \Sigma$ there exists
        an open set $p \in U \subset \Sigma$ such
        that $U$ is \fcemph{diffeomorphic} to an
        open set in $\RR^2$.}
    \end{definition*}
\end{hiddenflashcard}

\begin{theorem}
    For a subset $\Sigma \subset \RR^3$, the
    following are equivalent (TFAE):
    \begin{enumerate}[(a)]
        \item $\Sigma$ is a smooth surface in
            $\RR^3$
        \item $\Sigma$ is locally the graph of a
            smooth function over one of the
            coordinate planes, i.e. $\forall p \in
            \Sigma$ there exists open $p \in B
            \subset \RR^3$ and open $V \subset \RR^2$
            such that
            \[ \Sigma \cap B = \{(x, y, g(x, y)) :
            (g : V \to \RR), \text{$g$ smooth}\} \]
            (or the graph over the $xz$ or $yz$
            plane locally).
        \item $\Sigma$ is locally cut out by a
            smooth function with non-zero
            derivative, i.e. $\forall p \in
            \Sigma$, there exists $p \in B \subset
            \RR^n$ ($B$ open) and $f : B \to \RR$
            such that $\Sigma \cap B = f^{-1}(0)$
            and $Df|_x \neq 0$ for all $x \in B$.
        \item $\Sigma$ is locally the image of an
            \emph{allowable parametrisation} i.e.
            if $p \in \Sigma$, there exists open
            $p \in U \subset \Sigma$ and $\sigma :
            V \to U$ smooth such that $\sigma$ is
            a homeomorphism and $D\sigma|_x$ has
            rank 2 for all $x \in V$.
    \end{enumerate}
\end{theorem}

\begin{hiddenflashcard}[tfae-smooth-surface-in-R3]
\begin{theorem*}
    For a subset $\Sigma \subset \RR^3$, the
    following are equivalent:
    \begin{enumerate}[(a)]
        \item \cloze{$\Sigma$ is a \fcemph{smooth
            surface} in $\RR^3$.}
        \item \cloze{$\Sigma$ is \fcemph{locally}
            the graph of a
            \fcemph{smooth} function over one of the
            coordinate planes.}
        \item \cloze{$\Sigma$ is \fcemph{locally}
            cut out by a
            \fcemph{smooth} function with \fcemph{non-zero
            derivative}.}
        \item \cloze{$\Sigma$ is locally the image of an
            \fcemph{allowable parametrisation}.}
    \end{enumerate}
\end{theorem*}
\end{hiddenflashcard}

\begin{hiddenflashcard}[allowable-parametrisation-in-R3]
    What is an \emph{allowable} parametrisation?
    \\
    \cloze{
    An allowable parametrisation for a
    \fcemph{smooth surface in $\RR^3$} is a
    \fcemph{smooth} function $\sigma : V \to U$,
    $V$ open in
    $\RR^2$ and $p \in U \subset \Sigma$ open such
    that $\sigma$ is a \fcemph{homeomorphism} and
    $D\sigma|_x$ \fcemph{has rank 2} for all $x
    \in V$.
    }
\end{hiddenflashcard}

\begin{remark*}
    (b) says that if $\Sigma$ is a smooth surface
    in $\RR^3$, each $p \in \Sigma$ belongs to a
    chart $(U, \varphi)$ where $\varphi$ is the
    restriction of $\pi_{xy}$, $\pi_{yz}$,
    $\pi_{xz}$ from $\RR^3$ to $\RR^2$.
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/337b3ef4a0b611ed.png}
    \end{center}
    The transition map
    \[ (x, y) \mapsto (x, y, f(x, y)) \to (y, g(x,
    y)) \]
    has inverse
    \[ (y, z) \to (h(y, z), y, z) \to (h(y, z), y) \]
    are \emph{clearly smooth}. This gives $\Sigma$
    the structure of an abstract smooth surface.
\end{remark*}