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\begin{document}
\pagestyle{empty}
Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 28.01.2013\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 14} $ $\\
\end{center}


{\bf Exercise 1}\\
Let $(M^n,g)$ be a connected Riemannian manifold and $f:M\to M$ be an isometry.
Show that, if at a point $p\in M$, one has $f(p)=p$ and $d_pf={\rm Id}_{T_pM}$, then $f={\rm Id}_M$.\\

{\bf Exercise 2}\\
We denote as usual by $\mRP^n$ the $n$-dimensional real projective space.
\ben\item Given any $p\in S^n$, we define $\omega\in \Gamma(\Lambda^nT_p^*S^n)$
as $\om_p(V_1,V_2,\ldots,V_n)=\det(p,V_1,\ldots,V_n)$. Show that 
$\omega$ defines a trivialization of $\Lambda^nT_p^*S^n$ and an orientation of
$S^n$.
%declare a basis $\{V_1,\ldots,V_n\}$ of $T_pS^n\cong p^\perp$ to be positively %oriented iff $\det(V_1,\ldots,V_n,p)>0$.
%Show that this defines an orientation on $S^n$.
\item Show $(-{\rm Id}_{S^n})^*\omega = (-1)^{n+1}\omega$. In other words, 
show that 
  $$\omega_{-p}(d(-{\rm Id}_{S^n})(V_1),d(-{\rm Id}_{S^n})(V_2),\ldots,d(-{\rm Id}_{S^n})(V_n))=  (-1)^{n+1}\omega_p(V_1,V_2,\ldots,V_n)$$
for all $p\in S^n$ and all $V_1,\ldots,V_n\in T_pS^n$.   
\item Deduce that $\mRP^n$ is orientable iff $n$ is odd.\\
\emph{(Hint: The proof that it is non-orientable for $n$ even 
can be carried out similarly to Exercise 2 of Sheet no. 10.)}
\een

{\bf Exercise 3}\\
Let $M$ be a compact (orientable) surface in  $\mR^3$. Let 
$\overline{B}_r(0)$ be the closed ball of radius $r$ around $0$ in $\mR^3$, and
let $S_r(0)= \partial \overline{B}_r(0)$ be its boundary.
\ben\item Show that the infimum 
$R:=\inf \{r>0\,|\, M\subset \overline{B}_r(0)\}>0$ is attained, 
and conclude that $M\cap S_R(0)$ is not empty.
\item Show that $T_xM$ is the orthogonal complement of $x$
for any $x\in M\cap S_R(0)$. Show for any such $x$, that the two principal 
curvatures in $x$ have the same sign (for a given choice of a unit normal field). 
\item Deduce that the Gauss curvature of $M$ is positive in any  
$x\in M\cap S_R(0)$.
\item Are there compact minimal surfaces $M$ in $\mR^3$? Justify your answer.
\emph{(We want to recall the fact that a surface
is called minimal if the mean curvature $H$ vanishes on all of $M$.)}
\een

\newpage
{\bf Exercise 4}\\
Let $M\subset\mR^3$ be a compact and connected surface. We accept without proof that $\mR^3\setminus M$ has two connected components, one bounded and one unbounded.
\ben
\item Show the existence of a smooth map $N:M\to S^2$ with $N(p)\perp T_pM$ for all $p\in M$.
\item Deduce that $M$ is orientable.
\item
Show that $N$ is surjective.\\
\emph{(Hint: show that any $2$-dimensional vector subspace of $\mR^3$ is the tangent space of $M$ at two or more points of $M$.)}\\
\emph{Comment: Such a map $N$ is called ``the'' Gau\ss{} map of the surface. It is unique up to a sign.}\\
\een

\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 4.2.2013} vor der Vorlesung}.
\end{document}

{\bf Exercise 3}\\
Let $M$ be a surface in $\mR^3$ (here and in the following 
in the sense of a 2-dimensional submanifold), 
and assume that $M$ is a diffeomorphic to $S^2$.
Using the Gau\ss{}-Bonnet theorem, show that the surface $M$ has an umbilical point.\\
\emph{(Hint: Assuming this were not the case, prove the existence of at least one continuous tangent vector field on $M\cong S^2$ without zeros.)}\\


We add another exercise to give you the opportunity to obtain bonus points:
{\bf Exercise 5}\\
