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\begin{document}
\pagestyle{empty}
Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 4.02.2013\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 15} $ $\\
\end{center}


{\bf Exercise 1}\\
Right or wrong?
Justify shortly each of your answers.
In the whole exercise, $V$ and $W$ denote smooth $\mathbb{K}$-vector bundles over a smooth manifold $M$, where $\mathbb{K}=\mR$ or $\mC$.
\ben 
\item If $V$ and $W$ are trivial, then so are $V\oplus W$, $V\otimes W$, ${\rm Hom}(V,W)$ and $\Lambda^2V$.
\item For any $n\geq3$, there exists an $n$-dimensional compact smooth manifold whose tangent bundle is trivial.
\item Each complex vector bundle with connection over a $1$-dimensional manifold is flat.
\item Each real vector bundle over $S^1$ is trivial.
\item If $V$ and $V\oplus W$ are trivial, then so is $W$.
\item Every complex line bundle with connection which admits a nowhere-vanishing parallel section is flat.
\item Every real line bundle with connection is flat.
\item There exists on every manifold a vector bundle with flat connection.
\item The tangent bundle of every Riemannian manifold admits a unique metric connection.
\item If a surface $M\subset\mR^3$ contains a line (i.e., if there exists $p,v\in\mR^3$, $v\neq0$, with $p+\mR\cdot v\subset M$), then $K\leq0$ along that line.\\
\een

{\bf Exercise 2}\\
Let $M:=S^2\times\mR\subset\mR^3\times\mR=\mR^4$.
Compute (after choosing a smooth unit normal field) the Weingarten map, the mean curvature, the Ricci-tensor and the scalar curvature of $M$.\\

{\bf Exercise 3}\\
Let $M$ and $M'$ be compact surfaces.
\ben\item Assume $M$ and $M'$ to carry metrics with negative sectional (or Gau\ss{}) curvature.
Does the connected sum $M\sharp M'$ carry such a metric?
Justify your answer.
\item Same question if $M$ and $M'$ carry a metric with positive sectional curvature.
\een
%\newpage
{\bf Exercise 4}\\
Let $V\to M$ be a real or complex vector bundle over an arbitrary smooth
manifold. Show that there is a scalar product on $V$. \\
\emph{Hint: Construct at first a scalar product on trivial vector bundles. 
Then use local trivializations and a partition of unity (Lemma IV.6.2)
to prove the general case.}\\

{\bf Exercise 5}\\
Given $n\in\mathbb{N}$, $n\geq2$, let $\pi_N:S^n\setminus\{e_{n+1}\}\to\mR^n$, $x\mapsto\frac{1}{1-x_{n+1}}(x_1,\ldots,x_n)$ and  $\pi_S:S^n\setminus\{-e_{n+1}\}\to\mR^n$, $x\mapsto\frac{1}{1+x_{n+1}}(x_1,\ldots,x_n)$, denote the stereographic projections from the North and South pole respectively. Recall the definition
$f_*X= df\circ X\circ f^{-1}$ for a diffeomorphism $f:M\to N$ and a vector field
$X$ on $M$.
\ben
% \item Given $w\in S^{n-1}$, show that the degree of the map $S^{n-1}\bui{\lra}{\ovl{X}}S^{n-1}$, $x\mapsto w-2\langle w,x\rangle x$ is equal to $1+(-1)^n$.\\
% \emph{(Hint: Show that, for any $y\in S^{n-1}$, the preimages of $y$ under $\ovl{X}$ are of the form $\{-x,x\}$ for some $x\in S^{n-1}$; then show that $\ovl{X}$ preserves the standard orientation of $S^{n-1}$ iff $n$ is even.)}
\item Given $v\in\mR^n\setminus\{0\}$, define the vector field $X$ on $S^n\setminus\{e_{n+1}\}$ by $X:=\pi_N^*v$, that is, $X_{|_p}:=(d_p\pi)^{-1}(v)$ for all $p\in S^n\setminus\{e_{n+1}\}$.
Show that $\left((\pi_S)_*X\right)_{|_x}=|x|^2\cdot\left(v-2\langle v,\frac{x}{|x|}\rangle\frac{x}{|x|}\right)$ for all $x\in\mR^n\setminus\{0\}$.
\item Deduce that $X$ can be extended uniquely as a smooth vector field $\wit{X}$ on $S^n$, and determine its zero points.
\item From now on let $n=2$.
Apply the Poincar\'e-Hopf theorem to compute the index of $\wit{X}$ in $e_{n+1}$.
\item In the case $n=2$ check the result of (c) by a direct computation using part (a).
\een


\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 11.2.2013} bei Nicolas Ginoux (M122)}.
\end{document}



