\documentclass[11pt]{article}

  \usepackage{latexsym}
  \usepackage{comment}
  %\usepackage{showlabels}
  \usepackage{amsmath,amsthm,amsfonts,amssymb,graphicx,epsfig,latexsym,color,amssymb,mathabx}
  %\usepackage{showkeys}
  \usepackage{epsf,enumerate,bbm}
  \usepackage[shortlabels]{enumitem}
  \usepackage{tikz}
  \def\rank{\operatorname{rank}}

  \newtheorem*{thm}{Theorem}
  \newtheorem*{lemma}{Lemma}
  
  \newcommand{\mb}[1]{\marginpar{\tiny #1 --mb}}

  \bibliographystyle{plain}   %plain

  \def\R{{\mathbb R}}
  \def\Q{{\mathbb Q}}

  \def\Z{{\mathbb Z}}
  \def\N{{\mathbb N}}
  \def\H{{\mathcal H}}  
  \def\G{{\Gamma}}
  \def\S{{\mathcal S}}
  \def\s{{\sigma}}
  \def\t{{\tau}}
  \def\e{{\varepsilon}}
  \def\F{{\mathcal F}}
  \def\X{{\mathcal X}}
  \def\P{{\mathcal P}}
  \def\pa{pseudo-Anosov\ }
  \def\diam{\operatorname{diam}}
  \def\spec{\operatorname{spec}}
  \def\stab{\operatorname{Stab}}
  \def\diag{\operatorname{diag}}
  \def\SL{\operatorname{SL}}
  \def\GL{\operatorname{GL}}
  \def\Sym{\operatorname{Sym}}
  \def\FF{{\mathbb F}}
  \def\A{{\mathcal A}}
  \def\l{{\lambda}}
  \def\C{{\mathbb C}}

  \newcommand{\<}{\langle}
  \renewcommand{\>}{\rangle}

  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Definitions and examples of manifolds}}

  

  \subsection*{Guillemin-Pollack}
In this part we use the Guillemin-Pollack definitions, in particular
of a smooth function defined on a subset $X\subset \R^n$: $f:X\to\R^m$
is smooth if for every $x\in X$ there is a neighborhood $U$ of $x$ in
$\R^n$ and a smooth function $F_U:U\to\R^m$ that agrees with $f$ on
$U\cap X$.
  
  
\begin{enumerate}[1.]
\item For $k<n$ view $\R^k$ as a subset of $\R^n$ via
  $(x_1,\cdots,x_k)\mapsto (x_1,\cdots,x_k,0,\cdots,0)$. Show that a
  function $f:\R^k\to\R$ is smooth (using the G-P subset definition)
  if and only if it is smooth in the usual sense.

  \item Let $X\subset \R^m,Y\subset \R^k,Z\subset \R^n$. If $f:X\to Y$
    and $g:Y\to Z$ are smooth (as maps to the ambient Euclidean
    spaces) then so is the composition $gf:X\to Z$. If $f,g$ are
    diffeomorphisms so is $gf$.

    \item Let $X\subset \R^n, Y\subset \R^m, X'\subset\R^k,
      Y'\subset\R^l$. If $f:X\to X'$ and $g:Y\to Y'$ are smooth, so is
      $f\times g:X\times Y\to X'\times Y'$. Here we view $X\times Y$
      as a subset of $\R^{n+m}$ and $X'\times Y'$ as a subset of
      $\R^k\times \R^l$.

    \item Let $X\subset \R^n$ and let $\Delta=\{(x,x)\in\R^{2n}\mid
      x\in X\}$ be the diagonal. Show that $\Delta$ is diffeomorphic
      to $X$. More generally, if $f:X\to Y$ is smooth, then the graph
      of $f$
      $$graph(f)=\{(x,f(x))\mid x\in X\}$$ is diffeomorphic to $X$. I
      am suppressing various ambient Euclidean spaces.

\item Show that the map $a:S^n\to S^n$ defined by $a(x)=-x$ is a
  diffeomorphism. 
  
\item Show that the letters L and I are homeomorphic but not
  diffeomorphic. Here $I=\{0\}\times \R$ and $L=\{0\}\times
  [0,\infty)\cup [0,\infty)\times\{0\}$. Hint: View $I$ as a subset of $\R$ and
  a diffeomorphism $I\to L$ as a local parametrization around the
  corner point, the derivative should be injective.

  

  \end{enumerate}

  \subsection*{Chart definition}
\begin{enumerate}[resume]
\item Show that $X=\{(x,y)\in\R^2\mid xy=0\}$ is not a topological
  manifold. Hint: How many components does $X\smallsetminus\{(0,0)\}$
  have?

  \item Let $X$ be a smooth manifold. Show that the set $C^\infty(X)$
    of all smooth functions $X\to\R$ is an algebra, i.e. if $f,g\in
    C^\infty(X)$ then so are $af+bg$ and $fg$ for any $a,b\in\R$. You
    can use the fact that this is so when $X$ is an open set in
    $\R^n$.

  \item The group $GL_n(\R)$ of $n\times n$ matrices is naturally an
    open set in the space $M(n)$ of all
    real $n\times n$ matrices, which is naturally identified with
    $\R^{n^2}$. Show that the maps $GL_n(\R)\times GL_n(\R)\to
    GL_n(\R)$ defined by $(A,B)\mapsto AB$ and $GL_n(\R)\to GL_n(\R)$,
    $A\mapsto A^{-1}$ are smooth. This shows that $GL_n(\R)$ is a Lie
    group.

%% \item Show that the group $\Z/2\cong \{id,a\}=G$ satisfies (F) and (PD),
%%   where $a:S^n\to S^n$ is the antipodal map $a(x)=-x$. Show that
%%   $S^n/G$ is diffeomorphic to $\R P^n$ defined via charts. Also show
%%   that $\R P^1$ is diffeomorphic to $S^1$.

\end{enumerate}

\subsection*{The Regular Value Theorem, Lie groups}

\begin{enumerate}[resume]
\item Let $f:\R^3\to\R$ be given by $f(x,y,z)=xyz+x^3+y^3+z^3$ and let
  $M_a=f^{-1}(a)$. Show that $M_a$ is a manifold for $a\neq 0$, and
  that $M_0\smallsetminus\{(0,0,0)\}$ is a manifold.

\item The {\it unitary group} $U(n)$ is the group of complex $n\times
  n$ matrices $M$ such that $MM^*=I$, where $M^*$ is transpose
  followed by conjugation of all entries. Show that $U(n)$ is a
  compact Lie
  group.

\item The (real) symplectic group $Sp(2n,\R)$ is the group of real
  $2n\times 2n$ matrices $M$ that satisfy $M\Omega M^t=\Omega$, where
  $\Omega$ is the block matrix
  $$\Omega=\begin{pmatrix}
  0 & I_n\\
  -I_n & 0\end{pmatrix}$$
  with blocks of size $n\times n$ and $I_n$ the identity matrix.
  In other words, if you think of $\Omega$ as defining a skew-symmetric
  bilinear form $(v,w)\mapsto v^t\Omega w$ on $\R^{2n}$, then the condition is that $M$ preserves this
  form. Show that $Sp(2n,\R)$ is a Lie group. Also show that
  $Sp(2,\R)=SL_2(\R)$ (the form $(v,w)\mapsto v^t\Omega w$ is the
  signed area of the parallelogram spanned by $v$ and $w$).
  Hint: Use the argument for $O(n)$ as a template. This time the range
  of the function $F$ is the space of {\it skew-symmetric} matrices
  $S$, those with $S^t=-S$.

\end{enumerate}
  

\end{document}
