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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 13.05.2013\\
Dr. Mihaela Pilca\\

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 5} $ $\\
\end{center}



{\bf Exercise 1}\\
Let $S^3\subset \mathbb{H}$ be the unit sphere in the quaternion algebra. Consider the following map:
\[\theta: S^3\times S^3 \to \mathrm{Aut}(\mathbb{H})\]
\[(z,w)\mapsto (q\mapsto zq\overline{w}).\]
\begin{itemize}
 \item[i)] Show that $\theta$ defines a smooth action of $S^3\times S^3$ on $\mathbb{H}$, which preserves the standard norm on $\mathbb{H}\cong \mathbb{R}^4$. 
 \item[ii)] Compute the kernel of $\theta$. 
 \item[iii)] Show that the differential of $\theta$ at the identity element is bijective.
 \item[iv)] Conclude that $\theta$ is the universal covering of $\SO(4)$.\\
\end{itemize}



{\bf Exercise 2}\\
Let $\mathbb{Z}$ act on $\mathbb{R}^n$ by $k\cdot x:=2^k x$, for $k\in\mathbb{Z}$, $x\in\mathbb{R}^n$. 
\begin{itemize}
 \item[i)] Is this action proper on $M_1:=\mathbb{R}^n$, on $M_2:=\mathbb{R}^n\setminus\{0\}$, 
 on $M_3:=(0,\infty)\times (0,\infty)\times \mathbb{R}^{n-2}$?
 \item[ii)] Are the quotients $\mathbb{Z}\backslash M_i$ Hausdorff? Are they compact?\\
\end{itemize}



{\bf Exercise 3}\\
For $0<m<n$, let $G(m,n)$ be the set of all $m$-dimensional 
subspaces in $\mathbb{R}^n$.
Show that $\GL(n,\mathbb{R})$ and $\mathrm{O}(n,\mathbb{R})$ act transitively 
on $G(m,n)$. Determine the isotropy groups of $\mathbb{R}^m\times \{0\}$ 
for both actions, and write $G(m,n)$ as homogeneous space $G/H$ where
$G=\GL(n,\mathbb{R})$ or $G=\mathrm{O}(n,\mathbb{R})$. 

What is the interpretation of  
\begin{itemize}
 \item[i)] $\mathrm{O}(n,\mathbb{R})/(\mathrm{O}(m,\mathbb{R})\times \mathrm{O}(n-m,\mathbb{R}))$, 
 \item[ii)] $\mathrm{SO}(n,\mathbb{R})/(\mathrm{SO}(m,\mathbb{R})\times \mathrm{SO}(n-m,\mathbb{R}))$,
 \item[iii)]$\GL_+(n,\mathbb{R})/(\GL_+(m,\mathbb{R})\times \GL_+(n-m,\mathbb{R}))$, 
 \item[iv)] $\GL(n,\mathbb{R})/(\GL(m,\mathbb{R})\times \GL(n-m,\mathbb{R}))$. 
\end{itemize}
{\em Hint: Be cautious with the isotropy group of $\GL(n,\mathbb{R})$, and its relation to iii) and iv).}


\vskip1cm
{\sl Hand in the solutions on
\textbf{Monday, May 20, 2013} before the lecture}.\\
\end{document}



