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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 22.04.2013\\
Dr. Mihaela Pilca\\

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 2} $ $\\
\end{center}
{\bf Exercise 1}\\

Let $\Gamma$ be a discrete group acting smoothly on a differentiable manifold $M$.
\begin{itemize}
 \item[(a)] Show that the action is proper if and only if both of the following conditions are satisfied:
  \begin{itemize}
   \item[(i)] Each point $p\in M$ has a neighborhood $U$ such that $(\gamma \cdot U)\cap U =\emptyset$, for all but finitely many $\gamma\in \Gamma$.
   \item[(ii)] If $p,q\in M$ are not in the same $\Gamma$-orbit, there exist neighborhoods $U$ of $p$ and $V$ of $q$ such that $(\gamma\cdot U)\cap V =\emptyset$, for all $\gamma\in \Gamma$.
  \end{itemize}
 \item[(b)] If $\Gamma$ acts moreover freely, then show that the action is proper if and only if for each $p,q\in M$ there exist neighborhoods $U$ of $p$ and $V$ of $q$, such that for all $\gamma\in \Gamma$ with $q\neq \gamma\cdot p$ we have $(\gamma\cdot U)\cap  V=\emptyset$.
\end{itemize}

{\bf Exercise 2}\\
Let $X$ be a left-invariant vector field on a Lie group $G$ with unit element $e$ .
\begin{itemize}
 \item[i)] Show that there exists a curve $\gamma:\mathbb{R}\to G$ satisfying $\gamma(0)=e$ and $\dot{\gamma}(t)=X_{\gamma(t)}$, for all $t\in\mathbb{R}$.
 \item[ii)] Show that $\gamma(t+s)=\gamma(t)\cdot\gamma(s)$ and $\gamma(-t)=\gamma(t)^{-1}$, for all $s,t\in\mathbb{R}$.
\end{itemize}


{\bf Exercise 3}\\
Let $G$ and $H$ be two Lie groups and $e$ the unit element of $G$. If $f:G\to H$ is a smooth group homomorphism, then show that:
\begin{itemize}
 \item[i)] $d_e f:\mathfrak{g}\to \mathfrak{h}$ is surjective if and only if $f$ is a submersion.
 \item[ii)] $d_e f:\mathfrak{g}\to \mathfrak{h}$ is bijective if and only if $f$ is locally diffeomorphic.
 \item[iii)] If $H$ is connected and $d_e f:\mathfrak{g}\to \mathfrak{h}$ is surjective, then $f$ is surjective.
 (Hint: Show that $f(G)$ is open and closed. In order to prove that the image is closed one may cosider a sequence converging to any point in the closure of the imagine and translate it by left multiplication to the unit element of $H$.)
\end{itemize}

\pagebreak
{\bf Exercise 4}\\
For $\alpha\in\mathbb{R}\setminus\mathbb{Q}$, consider the following action of $\mathbb{R}$ on $M:=S^1\times S^1$: 
\[\mathbb{R}\times M\to M, \quad (t,p)\mapsto f_t(p) , \quad \text{where} \quad  f_t(x,y):=(e^{it}x, e^{i\alpha t}y).\]
\begin{enumerate}
\item[(a)]
Show that each orbit of this action is dense in $M$ and is neither closed nor a submanifold.
\item[(b)]
Is the map $\Theta:\mathbb{R}\times M\to M\times M$, 
$(t,p)\mapsto (f_t(p),p)$ closed? Is the action proper?
\item[(c)]Is $\mathbb{R}\backslash M$ (equipped with the quotient topology) a
Hausdorff space?
\end{enumerate}

\vskip1cm
{\sl Hand in the solutions on
\textbf{Monday, April 29, 2013} before the lecture}.\\
\end{document}



