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\begin{document}
\pagestyle{empty}
Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 26.11.2012\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 7} $ $\\
\end{center}

{\bf Exercise 1}\\
Let $(M,g)$ be a smooth compact Riemannian manifold.
Show that every geodesic of $(M,g)$ is defined in $\mR$.\\

{\bf Exercise 2}
\begin{enumerate}
 \item Let $M_1$ and $M_2$ be two smooth surfaces in $\mR^3$, and assume that 
we have a smooth curve $c:I\to\mR^3$ with $c(I)\subset M_1\cap M_2$. 
Further we assume that $T_{c(t)}M_1=T_{c(t)}M_2$ for all $t\in I$.
Show that the parallel transports along $c$ in $M_1$ and in $M_2$ coincide.
\item Given $\theta\in\mo]0,2\pi\mc[$ let $C:=\left\{(r\cos \varphi,r\sin\varphi)\,,\,r\in\mo]0,\infty\mc[,\,\varphi\in \mo]0,\theta\mc[ \right\}\subset\mR^2$.
Determine the parallel transport along the curve $c_r:]0,\theta[\to C$, $t\mapsto (r\cos t,r\sin t)$, where $r>0$.
\item Deduce an explicit formula for the parallel transport along a circle of latitude $t\mapsto(\cos t \cos \varphi ,\sin t \cos \varphi,\sin\varphi)$ in $S^2$, where $\varphi\in\mo]-\frac{\pi}{2},\frac{\pi}{2}\mc[$.
\end{enumerate}


{\bf Exercise 3}{\it (Geodesics in hyperbolic space)}
\begin{enumerate}
\item Let $\phi:M\to M$ be an isometry of $(M,g)$. For any $X\in TM$ let 
$\gamma_X:I\to M$ be a geodesic with $\dot\gamma_X(0)=X$.\\
Show: $\phi(\gamma_X(t))=\gamma_X(t)$ for all $t\in I$ if and only if
$d\phi(X)=X$. If $\gamma:I\to M$ is an arbitrary curve in $M$ with 
$\phi(\gamma(t))=\gamma(t)$ for all $t\in I$, then 
$d\phi(\frac{\nabla}{dt}\dot\ga(t))=\frac{\nabla}{dt}\dot\ga(t)$ for all 
$t\in I$.
\item Let $\mathbb{H}^n:=\left\{x\in\mR^{n+1}\,,\,\lala x,x\rara=-1\textrm{ and }x_0>0\right\}$ denote the $n$-dimensional hyperbolic space (see Exercise no. 4 in Sheet 5). We identify $T_x\mathbb{H}^n$ with 
$x^\perp:=\left\{V\in\mR^{n+1}\,,\,\lala V,x\rara=0\right\}.$
For $V$ in $x^\perp$ we define $\|V\|:= \sqrt{\lala V,V\rara}$.
For $V\in x^\perp\setminus\{0\}$ show that 
  $$\gamma_{x,V}(t):= \cosh(\|V\|t)x + \sinh(\|V\|t)\frac{V}{\|V\|}$$ 
is a curve in  $\mathbb{H}^n$.
\item Determine a linear map $\Phi:\mR^{n+1}\to \mR^{n+1}$ with 
$\Phi^*\lala\,\cdot\,,\,\cdot\,\rara=\lala\,\cdot\,,\,\cdot\,\rara$ such that 
its fixed point set is the plane spanned by $x\in\mathbb{H}^n$ and 
$V\in x^\perp\setminus\{0\}$. Show that its restriction to $\mathbb{H}^n$ defines an isometry 
$\phi:\mathbb{H}^n\to \mathbb{H}^n$. What is the fixed point set?
\item Conclude that $\frac{\nabla}{dt}\dot\gamma_{x,V}(t)=f(t)\dot\gamma_{x,V}(t)$ for all $t\in \mathbb{R}$ and for a suitable function $f$.
\item Show that $\gamma_{x,V}$ is a geodesic. 
{\it Hint: Calculate $\|\dot\gamma_{x,V}(t)\|$}. Are all non-constant geodesics in  
$\mathbb{H}^n$ of this form?
\end{enumerate}



{\bf Exercise 4}\\
Does there exist a Riemannian metric 
\begin{enumerate}\item on $\mR^2$ such that all circles can be parametrized as geodesics?
\item on $\mR^2\setminus\{0\}$ such that all circles centered at $0$ can be parametrized as geodesics?
\item on $\mR^2\setminus\{0\}$ such that all circles centered at $0$ can be parametrized as geodesics but \emph{no} ray through $0$ is a geodesic?
\end{enumerate}
Justify each of your answers.

\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 3.12.2012} vor der Vorlesung}.\\
\end{document}



