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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 29.04.2013\\
Dr. Mihaela Pilca\\

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 3} $ $\\
\end{center}

{\bf Exercise 1}\\

Let $\mathcal{H}_3:=\biggl\{\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1\end{pmatrix} |\, x,y,z \in \mathbb{R} \biggr\}$ and $\Gamma:=\biggl\{\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1\end{pmatrix} |\, x,y,z \in \mathbb{Z} \biggr\}$.
\begin{itemize}
 \item[i)] Show that $\mathcal{H}_3$ and $\Gamma$ are Lie groups. Does $\mathcal{H}_3$ admit a bi-invariant Riemannian metric?
 \item[ii)] Show that $\Gamma$ acts on $\mathcal{H}_3$ by left multiplication and this action is free and proper.
 \item[iii)] Consider the following action of $\mathbb{R}$ on $\mathcal{H}_3$:
 \[\mathbb{R}\times \mathcal{H}_3 \to \mathcal{H}_3,\quad \biggl(\tilde z, \begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1\end{pmatrix}\biggr) \mapsto \begin{pmatrix} 1 & x & z +\tilde z \\ 0 & 1 & y \\ 0 & 0 & 1\end{pmatrix}.\]
 Show that this action descends to an action of ${\mathbb{Z}}\backslash \mathbb{R}$ on the quotient $\Gamma \backslash \mathcal{H}_3$ and the quotient manifold obtained by this action is the $2$-dimensional torus.
\end{itemize}


{\bf Exercise 2}\\
Let $S^{4n+3}\subset \mathbb{H}^{n+1}$ be the unit sphere in the $(n+1)$-dimensional quaternionic vector space. 
\begin{itemize}
 \item[i)] Show that $S^3\subset \mathbb{H}$ acts smoothly, freely and properly on $S^{4n+3}$.
 \item[ii)] Give an atlas for the quotient manifold $\mathbb{H}\mathrm{P}^n:=S^3\backslash S^{4n+3}$. The manifold $\mathbb{H}\mathrm{P}^n$ is called the $n$-dimensional quaternionic projective space.\\
\end{itemize}


{\bf Exercise 3}
\begin{itemize}
 \item[i)] Determine the Lie bracket $[\cdot,\cdot]$ on $\mathfrak{gl}(n,\mathbb{R})$, the Lie algebra of the general linear group $GL(n,\mathbb{R})$.
 \item[ii)] For any Lie group $G$ with adjoint representation $\mathrm{Ad}: G \to \mathrm{Aut}(\mathfrak{g})$,
 let $\mathrm{ad}: \mathfrak{g} \to \mathrm{End}(\mathfrak{g})$ denote the differential of $\mathrm{Ad}$ at the unit element of $G$, $\mathrm{ad}:=\mathrm{d}_{\mathbb{1}}\mathrm{Ad}$.\\
 Show that for $GL(n,\mathbb{R})$, the map $\mathrm{ad}$ is given by $\mathrm{ad}(X)(Y)=[X,Y]$, for all $X,Y\in \mathfrak{gl}(n,\mathbb{R})$.
 \item[iii)] Let $X\in \mathfrak{gl}(n,\mathbb{R})$, $\widetilde X$ the corresponding left-invariant vector field on $GL(n,\mathbb{R})$ and $\gamma: \mathbb{R}\to GL(n,\mathbb{R})$ be a curve with $\gamma(0)=\mathbb{1}_n$, $\dot{\gamma}(t)=\widetilde{X}_{\gamma(t)}$. Show that $\gamma(t)=\sum_{n=0}^{\infty}\frac{1}{n!}(tX)^n$.
\end{itemize}

%{\bf Exercise 4}\\
%Let $G$ be a Lie group, $g$ a bi-invariant Riemannian metric on $G$ and \mbox{$\mathrm{ad}: \mathfrak{g} \to \mathrm{End}(\mathfrak{g})$} the map introduced in Exercise 3, ii). 

%\begin{itemize}
%\item[i)] Show that the map $\mathrm{ad}$ takes values into the skew-symmetric endomorphisms of $(\mathfrak{g}=T_{\mathbb{1}}G, g_{\mathbb{1}})$. Moreover, one can show that $\mathrm{ad}(X)(Y)=[X,Y]$, for all $X,Y\in\mathfrak{g}$ (we assume this result, it is not part of the exercise to prove it).
%\item[ii)] Use i) and the Koszul formula to show that the Levi-Civita connection of $g$ is given by $\nabla_X Y=\frac{1}{2}[X,Y]$, for all left-invariant vector fields $X, Y$. 
%\item[iii)] (Bonus points) Show that the sectional curvature of $g$ is nonnegative. (Hint: First compute the Riemannian curvature tensor using ii): $R(X,Y)Z=-\frac{1}{4}[[X,Y],Z]$, for all left-invariant vector fields $X,Y,Z$. Use also the Jacobi identity).
%\end{itemize}

%\vskip1cm
\bigskip
{\sl Hand in the solutions on
\textbf{Monday, May 6, 2013} before the lecture}.\\
\end{document}



