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\begin{document}
\pagestyle{empty}
Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 01.07.2013\\
Dr. Mihaela Pilca

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 12} $ $\\
\end{center}

{\bf Exercise 1}\\
Let $(M,g)$ be a Riemannian manifold, whose sectional curvature $K$ 
satisfies the inequalities:
\[0< L\leq K\leq H,\]
for some positive constants $L$ and $H$. For a geodesic 
$\gamma:[0,\ell]\to M$, parametrized by arclength,
we define 
  $$d:=\min \{t>0\,|\, \mbox{$\gamma(t)$ is conjugated to 
$\gamma(0)$ along $\gamma|_{[0,t]}$}\}.$$
Show 
%that the distance $d$ along $\gamma$ between
%two consecutive conjugate points of $\gamma$ satisfies the inequalities:
\[\frac{\pi}{\sqrt{H}}\leq d\leq \frac{\pi}{\sqrt{L}}.\]

\emph{Hint: Use the First Rauch Comparison Theorem.}\\

{\bf Exercise 2}\\
Let $(M,g)$ be a complete Riemannian manifold with sectional curvature \mbox{$K\geq 0$}. Let $\Gamma$ be a discrete group without $2$-torsion (\emph{i.e.} $\gamma^2\neq e$, for any $\gamma\in\Gamma\setminus\{e\}$, where $e$ is the identity element of $\Gamma$), acting isometrically, \mbox{freely} and properly on $M$. For a point $p\in M$, let $\gamma_0\in \Gamma$ be an element with $d(p,\gamma_0 p)=\underset{\gamma\in\Gamma\setminus\{e\}}{\min} d(p,\gamma p)$. 

We choose a minimal geodesics $c_1$ which connects $p$ 
to $\gamma_0 p$, and a geodesic $c_2$ which connects $p$ to
$\gamma_0^{-1} p$. Show that $c_1$ and $c_2$ form at $p$ an angle 
$\alpha\geq\frac{\pi}{3}$.\\


{\bf Exercise 3}\\
Let $(M,g)$ be a complete Riemannian manifold  with sectional curvature \mbox{$K\geq 0$} and let $\gamma,\sigma:[0,\infty)\to M$ be two geodesics, parametrized by arclength, with $\gamma(0)=\sigma(0)$. We assume that $\gamma$ is a ray and that $\alpha:=\sphericalangle(\dot{\gamma}(0), \dot{\sigma}(0))<\frac{\pi}{2}$. 

Show that $\underset{t\to \infty}{\lim} d(\sigma(0), \sigma(t))=\infty$.

\emph{Hint:  Using the triangle inequality, show first that it is enough to prove: $\underset{s\to\infty}{\lim}(d(\gamma(s),\sigma(t))-d(\gamma(s),\gamma(0)))\geq t\cos\alpha$, for any fixed $t\geq 0$. Then apply Toponogov's Theorem (A).}\\








\vskip1cm
{\sl Hand in the solutions on
\textbf{Monday, July 15, 2013} before the lecture}.\\
\end{document}



