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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 17.06.2013\\
Dr. Mihaela Pilca\\

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 10} $ $\\
\end{center}



{\bf Exercise 1}\\
Determine $\cC_p^{\rm tan}M$,  and $\cC_pM$ for\\[-6mm]
\ben
\item $M=\mR^2/\Gamma$, where $\Gamma$ is the subgroup of $\mR^2$ 
generated  by $\begin{pmatrix}1\\0\end{pmatrix}$ and 
$\begin{pmatrix}0\\2\end{pmatrix}$, and $p:=[0]$.
\item $M=\mR P^m=S^m/\{\pm 1\}$  with the quotient metric, and $p:=[e_1]$.
\een
\ \\[-4mm]

{\bf Exercise 2}\\
Let $M=\{(x,y,z)\in\mR^3\,|\,x^2+y^2=e^{-z^2}\}$. Show that $M$ is a smooth surface, and that $M$ is complete, ${\rm vol}(M)<\infty$, $\injrad(M)=0$, $\diam(M)=\infty$.
\ \\

{\bf Exercise 3}\\
Let $M$ be a complete connected Riemannian manifold, $p\in M$ fixed. We  
define $\diam M:=\sup\{d(x,y)\,|\, x,y\in M\}$.
Show 
\ben
\item $\diam M= \sup_{X\in SM} s(X)$
\item $\injrad(p) = \min_{X\in S_pM}s(X)$
\item $\injrad(M) = \inf_{X\in SM}s(X)$
\item $\sup_{X\in SM} s(X)=\infty$ if and only if there is for all $p\in M$ an $X\in S_pM$ with $s(X)=\infty$.\\
{\it Hint: Use  Exercise no. 3 on Sheet no. 9 of Differential Geometry I}
\item Give an example of a complete Riemannian manifold such that 
$\sup_{X\in S_pM} s(X)$ depends on $p$.
\een


{\bf Exercise 4}\\
We consider $S^3\subset \mC^2$ endowed with the standard metric, and $\Gamma:=\{1,i,-1,-i\}$ which acts
freely und isometrically on $S^3$. Let $M:=S^3/\Gamma$, $\pi:S^3\to M$ the corresponding projection 
and $p:=\pi(e_1)= e_1 \mod \Gamma\in M$.
Show that for the cut locus $\cC_p$ the following holds:
  $$\cC_p=\{\pi(x)\,|\, x\in S^3 \mbox{ with $d(x,e_1)=d(x,ie_1)$}\}$$
    $$=\left\{\pi\left.\left(\frac{(1+i)r}{\sqrt{2}}e_1+v e_2\right)\,\right|\, r\in [0,1],\quad v\in\mC\mbox{ with } r^2+|v|^2=1\right\}.
   $$
Answer without justification: Where are the minima and maxima of the function $s:S_pM\to (0,\infty)$?\\ 
Bonus question: Where is $\cC_p$ a smooth hypersurface and where not?











\vskip1cm
{\sl Hand in the solutions on
\textbf{Monday, June 24, 2013} before the lecture}.\\
\end{document}



