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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 24.06.2013\\
Dr. Mihaela Pilca

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 11} $ $\\
\end{center}



{\bf Exercise 1}\\
Let $M$ be a complete Riemannian manifold; let $N$ be a submanifold and 
a closed subset of $M$. For any $p_0\in M$ we define its distance to $N$ as 
$d(p_0,N):=\underset{q\in N}{\inf} d(p_0,q)$. Show the following:
\begin{itemize}
 \item[i)] There exists a point $q_0\in N$, such that $d(p_0,N)=d(p_0,q_0)$.
 \item[ii)] If $p_0\in M\setminus N$, then a minimizing geodesic joining 
$p_0$ and $q_0$ is orthogonal to $N$ at $q_0$.\\
 \emph{Hint: Use a variation of the geodesic with curves starting at $p_0$ and ending at points in $N$.} \\
\end{itemize}


{\bf Exercise 2}\\
Let $N$ be a submanifold of a Riemannian manifold $(M,g)$.
The normal exponential map of $N$, $\exp^{\perp}: TN^\perp \to M$ is defined as the restriction
of the exponential map $\exp: TM \to M$, $(p,v)\mapsto \exp_p v$ to points $q\in N$ and vectors $w\in (T_q N)^\perp$.
Show that $p\in M$ is a focal point of $N\subset M$ if and only if $p$ is a critical value of $\exp^\perp$.

\emph{Hint: For  ``$\Rightarrow$'' consider for a suitable variation $\gamma:(-\varepsilon,\varepsilon)\times [0,\ell] \to M$
with $\alpha(s):=\gamma(s,0)\subset N$ and $V(s):=\frac{\nabla}{dt}\gamma |_{(s,0)}$ the curve $c(s):=(\alpha(s), \ell V(s))$.}

\emph{For ``$\Leftarrow$'' consider for a suitable curve $c(s)=(\alpha(s), \ell V(s))$ in $TN^{\perp}$ the variation $\gamma(s,t)=\exp_{\alpha(s)}(tV(s))$.}\\



{\bf Exercise 3}\\
Let $N$ be a submanifold of a flat manifold $(M,g)$ and $\gamma$ be a geodesic in $M$ with $\gamma(0)\in N$ and $\dot{\gamma}(0)\perp T_{\gamma(0)}N$.
Show that $\gamma(\frac{1}{\lambda})$ is a focal point of $N$ if and only if $\lambda$ is a non-zero eigenvalue of $S_{\dot{\gamma}(0)}$. 

%Here we use the following definition: Let $p\in N$ and let $\eta$ 
%be a unit normal vector $\eta\in T_pN^\perp$. 
%Then $S_\eta\in\End(T_pN)$ is defined
%as follows: $g_p(S_{\eta}(v), w)=g_p(\mathrm{I}\!\mathrm{I}(v,w), \eta)$, 
%for all $p\in N$, $v,w\in T_p N$. 

\emph{Hint: For  ``$\Rightarrow$'' consider $X(t):=(1-\lambda t)E(t)$, where $E$ is a parallel vector field along $\gamma$ and $S_{\dot{\gamma}(0)}(E(0))=\lambda E(0)$.}




\vskip1cm
{\sl Hand in the solutions on
\textbf{Monday, July 1, 2013} before the lecture}.\\
\end{document}



