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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 29.10.2012\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 3} $ $\\
\end{center}
{\bf Exercise 1}\\
Let $M^m$ be an $m$-dimensional submanifold of $\mR^k$ an $p\in M^m$ be a point.
Prove that the tangent space of the manifold $M^m$ at $p$ as defined in the lecture can be identified with the tangent space of the submanifold $M^m$ at $p$ you already know from previous lectures.\\

{\bf Exercise 2}\\
Show that any topological manifold carries an atlas with countably many charts.\\

{\bf Exercise 3}\\
Let $M$ be a set and $n\in \mathbb{N}$. Further, we assume that
a family of bijective maps $(\phi_\alpha:U_\alpha\to V_\alpha)_{\alpha \in A}$ 
is given, where $U_\alpha$ is a subset of $M$ and where $V_\alpha$ is an open 
subset of $\mathbb{R}^n$. This family is supposed to satisfy:
\begin{enumerate}[{\rm (i)}]
\item $M=\bigcup_{\alpha\in A} U_\alpha$,
\item $\phi_\alpha(U_\alpha\cap U_\beta)$ is open in $\mathbb{R}^n$ for all 
$\alpha,\beta\in \mathbb{R}^n$
\item $\phi_\beta \circ \phi_\alpha^{-1}:\phi_\alpha(U_\alpha\cap U_\beta)\to \phi_\beta(U_\alpha\cap U_\beta)$ is continuous for all $\alpha,\beta\in A$.
\end{enumerate}
Show that
\begin{enumerate}[{\rm (a)}]
\item There is a unique topology on $M$ such that all 
$U_\alpha$ are open and 
such that all $\phi_\alpha$ are homeomorphisms.
\item If $A_1\subset A$ satisfies $M=\bigcup_{\alpha\in A_1} U_\alpha$,
then $(\phi_\alpha:U_\alpha\to V_\alpha)_{\alpha \in A}$ and 
$(\phi_\alpha:U_\alpha\to V_\alpha)_{\alpha \in A_1}$ induce the same topology on $M$.
\item
The topology on $M$ is second countable if $A$ is countable.
\item
Suppose that for any $p,q\in M$ we have:
\begin{enumerate}[{\rm (i)}]
\item there is $\alpha\in A$ with $p,q\in U_\alpha$, {\bf or}
\item there are $\alpha,\beta\in A$ with $p\in U_\alpha$, $q\in U_\beta$, 
$U_\al\cap U_\beta=\emptyset$.
\end{enumerate}
Then the topology on $M$ is Hausdorff.
\end{enumerate}
Are the sufficient 
conditions in (c) resp.\ (d) for second countability resp. Hausdorff property 
also necessary? 
\newpage


{\bf Exercise 4}\\
Let $\mCP^n:=\rquot{\mC^{n+1}\setminus\{0\}}{\sim}$ denote the complex projective space where, by definition, $x\sim y$ $\iff$ $x\in\mC\cdot y$.
We note by $[x^1,\ldots,x^{n+1}]$ the equivalence class of $(x^1,\ldots,x^{n+1})\in\mC^{n+1}\setminus\{0\}$.
For $\alpha\in\{1,\ldots,n+1\}$ we let $U_\alpha$ be the subset of all $[x^1,\ldots,x^{n+1}]\in\mCP^n$ with $x^\alpha\neq0$ and define the map
\begin{eqnarray*}
\phi_\alpha:U_\alpha&\lra&\mC^n\\
{} [x^1,\ldots,x^{n+1}]&\lmt&\left(\frac{x_1}{x_\alpha},\ldots,\wih{x_\alpha},\ldots,\frac{x_{n+1}}{x_\alpha}\right),
\end{eqnarray*}
where, as usual, $\wih{x_\alpha}$ means that the $\alpha^{\rm th}$ coordinate is omitted.
\begin{enumerate}
 \item Show that $U_\alpha$ and $\phi_\alpha$ are well-defined and that $\phi_\alpha$ is bijective, for all $\alpha\in\{1,\ldots,n+1\}$.
\item Show with the help of Exercise 3 that $\mathcal{A}:=\left\{(U_\alpha,\phi_\alpha)\,,1\leq \alpha\leq n+1\right\}$ defines a structure of $C^\infty$ manifold on $\mCP^n$.
\item Show that the underlying topology coincides with the quotient topology, where $\mC^{n+1}$ carries the standard topology.\\
\end{enumerate}


\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 5.11.2012} vor der Vorlesung}.\\
\end{document}



