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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 10.12.2012\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 9} $ $\\
\end{center}

{\bf Exercise 1}\\
Let $\gamma:[a,b]\lra M$ be a piecewise $C^1$ curve on a smooth 
Riemannian manifold $(M,g)$.
\ben\item Prove that $L[\gamma]^2\leq 2(b-a)\cdot E[\gamma]$, where $E[\gamma]:=\frac12 \int_a^bg(\dot{\gamma},\dot{\gamma})dt$ is the \emph{energy} of the curve $\gamma$.
\item Show that $L[\gamma]^2=2(b-a)\cdot E[\gamma]$ holds iff $\gamma$ is parametrized proportionally to arc-length.
\een

{\bf Exercise 2}\\
Let $M$ be a smooth submanifold of $\mR^k$.
\ben\item Show that, if $M$ is closed, then $M$ is complete.
\item Show that the converse statement is wrong.
\een

{\bf Exercise 3}\\
Let $(M,g)$ be a connected complete non-compact Riemannian manifold and $p\in M$ be a point.
\ben
\item Show that there exists a sequence $(p_i)_{i\in\mathbb{N}}$ in $M$ such that 
$d(p,p_i)\buil{\lra}{i\to\infty} \infty$. 
\item Show that, for each $i\in\mathbb{N}$, there exist $X_i\in T_{p}M$ and $r_i\in[0,\infty[$ with $g_p(X_i,X_i)=1$ and $p_i=\exp_p(r_iX_i)$.
\item Show that the sequence $\{X_i\}_{i\in\mathbb{N}}$ admits a converging subsequence and deduce that there exists a ray $\gamma:[0,\infty)\rightarrow M$ 
in $(M,g)$ with $\gamma(0)=p$.
\een

{\bf Exercise 4}\\Let $M$ be a connected $m$-dimensional manifold, and assume 
that $N\subset M$ is an $n$-dimensional submanifold, 
i.e.\ for every $p\in N$ there is a chart 
$\phi:U\to V\subset \mathbb{R}^m$, $p\in U$ such that 
$\phi(U\cap N)=V\cap(\mathbb{R}^n\times\{0\})$. 
Let $g$ be a Riemannian metric on $M$, such that $(M,g)$ is complete, and assume
that $N$ is a closed (as a subset of $M$).
Fix a point $q\in M$.
\ben\item Show the existence of a point $p\in N$ with $d(q,p)=d(q,N)$, 
where $d(q,N):=\buil{\inf}{x\in N}\{d(q,x)\}$.
Is $p$ unique?
Justify your answer.
\item Prove that there is a geodesic $\gamma$ from $q$ to $p$ with length $L[\gamma]=d(q,p)$.
\item Show that $\gamma$ meets $N$ orthogonally.
\een

\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 17.12.2012} vor der Vorlesung}.\\
\end{document}



