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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 17.12.2012\\
Dr. Nicolas Ginoux\\

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

{\bf Exercise 1}\\Show that the map $p:S^n\to \mRP^n$, $x\mapsto \mR x$ is a 
local diffeomorphism, i.e.\ every $x\in S^n$ is in an open set $U$ such that 
$p(U)$ is open in $\mRP^n$ and such that 
$p|_U$ is a diffeomorphism from $U$ to $p(U)$. Show that  $\mRP^n$ carries
a metric $g_0$ such that $p^*g_0$ is the standard metric on $S^n$. 
This metric $g_0$ is called the \emph{standard metric of\/}  $\mRP^n$.
Determine the injectivity radius of $(\mRP^n,g_0)$.\\

{\bf Exercise 2}\\
The \emph{tautological bundle} on the $n$-dimensional real projective space $\mRP^n$ is given by $L:=\left\{(\ell,y)\in\mRP^n\times\mR^{n+1}\,,y\in \ell\right\}$ together with the projection map $\pi:L\to\mRP^n$, $(\ell,y)\mapsto \ell$.
%\ben\item Show that $(L,p,\mRP^n)$ is a smooth real vector bundle of rank $1$.
%\item 
Prove that there does not exist any continuous and nowhere vanishing section $s$ of $\pi:L\to \mRP^n$.
\emph{(Hint: Interprete such a section as a map $\mRP^n\to\mR^{n+1}\setminus\{0\}$; considering the composition with the map $S^n\to\mRP^n$, get a map $S^n\to S^n$ which has to be $\pm\mathrm{Id}$; conclude.)}\\

%\een

{\bf Exercise 3}\emph{(Geodesics and distance function on products)}
\ben\item Let $\gamma:[a,b]\to M$ be a piecewise 
$C^1$ curve on a smooth Riemannian manifold $(M,g)$.
Prove that $\gamma$ minimizes the energy functional $E:c\mapsto\frac12\int_a^bg(\dot{c},\dot{c})\,dt$ among all piecewise $C^1$ curves $c:[a,b]\to M$ with $c(a)=p$ and $c(b)=q$ iff $\gamma$ is a minimal geodesic.
\item From now on let $(M,g):=(M_1\times M_2,g_1\oplus g_2)$, where $(M_i,g_i)$ is a smooth Riemannian manifold and the product manifold $M_1\times M_2$ (see Exercise no. 1 in Sheet 2) is equipped with the \emph{product metric} $g_1\oplus g_2$, which is defined at $p=(p_1,p_2)\in M_1\times M_2$ by: 
\[(g_1\oplus g_2)|_{(p_1,p_2)}\left((X_1,X_2),(Y_1,Y_2)\right):=g_1{}|_{p_1}(X_1,Y_1)+g_2{}|_{p_2}(X_2,Y_2)\]
for all $X_i,Y_i\in T_{p_i}M$, $i=1,2$.
Show that, if $\gamma_i:[a,b]\to M_i$ is a piecewise $C^1$ curve, $i=1,2$, then $\gamma:=(\gamma_1,\gamma_2):[a,b]\to M_1\times M_2$ is a piecewise $C^1$ 
curve with $E(\gamma)=E(\gamma_1)+E(\gamma_2)$.
\item Show that $\gamma$ is a minimal geodesic iff $\gamma_1$ and $\gamma_2$ are minimal geodesics.
\item Deduce that the distance function $d$ associated to $g=g_1\oplus g_2$ is given by
\[d\left((x_1,x_2),(y_1,y_2)\right)=\sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2},\]
for all $(x_1,x_2),(y_1,y_2)\in M_1\times M_2$, where $d_i$ is the distance function associated to the metric $g_i$ on $M_i$. 
\een

\newpage
{\bf Exercise 4}\emph{(Sufficient criterion for the existence of a line)}\\
Let $(M,g)$ be a complete smooth Riemannian manifold.
\ben
\item Let $(X_k)_{k\in\mathbb{N}}$ be a sequence in $TM$ converging to some $X$ and $a,b\in\mR$ with $a<b$.
Show that, if $\gamma_{X_k}{}_{|_{[a,b]}}:[a,b]\to M$ is a shortest curve, then so is $\gamma_X{}_{|_{[a,b]}}:[a,b]\to M$.
Here and as usual, for any $Y\in TM$, we denote by $\gamma_Y:\mR\to M$ the unique geodesic with $\gamma_Y(0)=\pi(Y)\in M$ and $\dot{\gamma}_Y(0)=Y$.
\item Assume the existence of two sequences $(x_k)_{k\in\mathbb{N}}$, 
$(y_k)_{k\in\mathbb{N}}$ in $M$, of a point $p\in M$ and of an 
$R\in\mo]0,\infty\mc[$ 
with $d(x_k,p)\buil{\lra}{k\to\infty}\infty$, 
$d(y_k,p)\buil{\lra}{k\to\infty}\infty$ and such that every shortest curve 
from $x_k$ to $y_k$ meets the ball $B_R(p)$.
Show that there exists a line in $(M,g)$.\\
\emph{(Hint: construct a limit of a sequence of shortest curves. Exercise no. 3 of Sheet 9 may be helpful.)}
\een
\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 7.1.2013} vor der Vorlesung}.\\

{\sl Wir w\"unschen allen Teilnehmerinnen und Teilnehmern frohe Weihnachten und einen guten Rutsch!}
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