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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 5.11.2012\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 4} $ $\\
\end{center}

{\bf Exercise 1}\\
Let $X$ be a tangent vector field on a smooth manifold $M$, that is, $X$ is a map $M\to TM$ with $\pi\circ X=\mathrm{Id}_M$, where $\pi:TM\to M$ is the projection map.
Recall that, given a chart $\varphi:U\to V$ of $M$, the associated coordinate vector fields $\{\frac{\partial}{\partial\varphi^1},\ldots,\frac{\partial}{\partial\varphi^n}\}$ form a basis of $TM$ in each point of $U$, in particular any tangent vector field $X$ on $M$ can be written in the form $X=X^i\frac{\partial}{\partial\varphi^i}$ on $U$, where $X^1,\ldots,X^n:U\to\mR$ are functions.\\
Show that $X$ is smooth as a map between manifolds if and only if the functions $X^1,\ldots,X^n:U\to\mR$ are smooth for any chart.\\

{\bf Exercise 2}\\
Let $f:M\to\mR$ be a $C^1$ map on a compact smooth $n$-dimensional manifold, where $n\geq1$.
\begin{enumerate}
 \item Let $p\in M$ be a point.
Show that, if $f$ reaches a maximum or a minimum at $p$, then $d_pf=0$.
\item Show that the differential map of $f$ vanishes in at least two points in $M$.
\item Show that $f$ has exactly one critical value if and only if $f$ is constant.
\end{enumerate}
% Let $M$ and $N$ be open subsets of $\mR^m$ and $\mR^n$ respectively.
% Let $F:M\to N$ be a $C^1$ map and $p\in M$ be a point.
% Show that, via the canonical isomorphisms $T_p M\cong\mR^m$ and 
% $T_{F(p)}N\cong\mR^n$ the differential map $d_pF$ from the lecture coincides with the one you already know from previous lectures.\\

{\bf Exercise 3}\\
Let $M$ be a compact smooth $n$-dimensional manifold.
By definition, a \emph{one-parameter group of diffeomorphisms} on $M$ is a smooth map $\varphi:M\times\mR\to M$, $(x,t)\mapsto\varphi_t(x)$, with $\varphi_0=\mathrm{Id}_M$ and  $\varphi_t\circ\varphi_s=\varphi_{t+s}$ for all $s,t\in\mR$.
\begin{enumerate}
 \item Show that, given any one-parameter group of diffeomorphisms $(\varphi_t)_t$ on $M$, the map $X(x):=\frac{d}{dt}|_{t=0}(\varphi_t(x))$ defines a smooth tangent vector field on $M$.
\item Conversely, show that, given any smooth vector field $X$ on $M$, there exists a unique one-parameter group of diffeomorphisms $(\varphi_t)_t$ on $M$ such that $\frac{d}{dt}|_{t=0}(\varphi_t(x))=X(x)$ for all $x\in M$.\\
{\it Hint: First construct $\varphi_t(x)$ for fixed $x$ and $t$ close to $0$ using the theorem of Picard-Lindel\"off; then show that $t\mapsto\varphi_t(x)$ can be extended on $\mR$.}
\end{enumerate}





\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 12.11.2012} vor der Vorlesung}.\\
\end{document}



