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\begin{document}
\pagestyle{empty}
Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 12.11.2012\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 5} $ $\\
\end{center}

{\bf Exercise 1}\\
Let $M$ be a smooth $n$-dimensional manifold and, for each point $p\in M$, $g_{|_p}$ be a Euclidean inner product on $T_pM$.
Show that the following statements are equivalent:
\begin{enumerate}
\item For any smooth tangent vector fields $X,Y$ on $M$, the map $M\to\mR$, $p\mapsto g_{|_p}(X(p),Y(p))$, is smooth.
\item For any chart $\varphi:U_\varphi\to V_\varphi$ of $M$ and all $1\leq i,j\leq n$, the function $g_{ij}^\varphi:V_\varphi\to\mR$ defined in the lecture is smooth.\\
\end{enumerate}

{\bf Exercise 2}\\
Let $M^m$, $N^n$ be smooth manifolds and $\phi:M\to N$ be an immersion, that is, $\phi$ is a smooth map and $d\phi_{|_p}:T_pM\to T_{\phi(p)}N$ is injective for any $p\in M$.
Show that, for any Riemannian metric $h$ on $N$, the map $p\mapsto (d\phi_{|_p})^*h_{|_p}$ introduced in the lecture defines a Riemannian metric on $M$.\\

{\bf Exercise 3}\\
Let $M$ be a smooth $n$-dimensional manifold.
Recall that a \emph{derivation} on $M$ is a linear map $\delta:C^\infty(M)\to C^\infty(M)$ which satisfies the product rule: for all $f_1,f_2\in C^\infty(M)$,
\[\delta(f_1f_2)=(\delta f_1)f_2+f_1(\delta f_2).\]
Let $X,Y$ are two smooth tangent vector fields on $M$.
\begin{enumerate}
 \item Show that $[\partial_X,\partial_Y]:=\partial_X\circ\partial_Y-\partial_Y\circ\partial_X$ defines a derivation on $M$.
Here, $\partial_X$ is the derivation associated to $X$ as in the lecture.
\item Deduce that there exists a unique smooth tangent vector field on $M$, which we denote by $[X,Y]$, such that $\partial_{[X,Y]}=[\partial_X,\partial_Y]$.
\item Show that, for any $f\in C^\infty(M)$, one has $[X,fY]=\partial_Xf\cdot Y+f[X,Y]$.
\item Show that, if $\varphi:U_\varphi\to V_\varphi$ is a chart of $M$, then $[\frac{\partial}{\partial\varphi^i},\frac{\partial}{\partial\varphi^j}]=0$ for all $1\leq i,j\leq n$.
Deduce that, if $X_{|_{U_\varphi}}=X^i\frac{\partial}{\partial\varphi^i}$ and $Y_{|_{U_\varphi}}=Y^i\frac{\partial}{\partial\varphi^i}$, then 
\[[X,Y]_{|_{U_\varphi}}=\bigl(\partial_X(Y^i)-\partial_Y(X^i)\bigr)\frac{\partial}{\partial\varphi^i}=
\Bigl(X^j\frac{\partial Y^i}{\partial\varphi^j}-Y^j\frac{\partial X^i}{\partial\varphi^j}\Bigr)\frac{\partial}{\partial\varphi^i}.\]
\end{enumerate}


\newpage
{\bf Exercise 4}\\
Let $\lala\cdot\,,\cdot\rara$ denote the following bilinear form on $\mR^{n+1}$:
\[\lala x,y\rara:=-x_0y_0+\sum_{j=1}^nx_jy_j\]
for all $x=(x_0,x_1,\ldots,x_n)$ and $y=(y_0,y_1,\ldots,y_n)$ in $\mR^{n+1}$.
\begin{enumerate}
\item Show that $\lala\cdot\,,\cdot\rara$ defines a non-degenerate symmetric bilinear form of index $1$ on $\mR^{n+1}$.
\item Let $\mathbb{H}^n:=\left\{x\in\mR^{n+1}\,,\,\lala x,x\rara=-1\textrm{ and }x_0>0\right\}\subset\mR^{n+1}$.
Show that $\mathbb{H}^n$ is a smooth $n$-dimensional submanifold of $\mR^{n+1}$.
\item Prove that, for any $p\in\mathbb{H}^n$, the tangent space of $\mathbb{H}^n$ at $p$ can be canonically identified with $E_p:=\left\{X\in\mR^{n+1}\,,\,\lala X,p\rara=0\right\}$.
\item Show that $\lala\cdot\,,\cdot\rara_{|_{E_p\times E_p}}$ is positive-definite and deduce that $\lala\cdot\,,\cdot\rara$ induces a Riemannian metric on $\mathbb{H}^n$.
\end{enumerate}


\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 19.11.2012} vor der Vorlesung}.\\
\end{document}



