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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 15.04.2013\\
Dr. Mihaela Pilca\\

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 1} $ $\\
\end{center}
{\bf Exercise 1}\\
Assume $(M,g)$ and $(M',g')$ are surfaces with Riemannian metrics 
with negative Gau{\ss} curvature. Does the product metric on 
$M\times M'$ has everywhere negative sectional curvature?\\

{\bf Exercise 2}\\
Let $(M,g)$ be a Riemannian manifold, $p\in M$. For $r<\mathrm{injrad}(p)$, we define the chart $\varphi:=(\mathrm{exp}_p|_{B_r(p)})^{-1}$, 
which defines the normal coordinates centered in $p$. As usual, we set
\[g_{ij}(x):=g_x(\frac{\partial}{\partial \varphi^i}|_x, \frac{\partial}{\partial \varphi^j}|_x), \quad \text{ for } x\in B_r(p).\]
\begin{itemize}
 \item[i)] Show that if $X=\sum_i X^i\frac{\partial}{\partial\varphi^i}$, then $\dot{\gamma}_X(t)=\sum_i X^i \frac{\partial}{\partial\varphi^i}|_{\gamma_{X}(t)}$.
 \item[ii)] Show that the associated Christoffel symbols satisfy $\Gamma_{ij}^{k}(p)=0$. (Hint: use the geodesic equation $\nabla_{\dot{\gamma}_X} \dot{\gamma}_X=0$
 to show that $\sum_{i,j}X^i X^j\Gamma_{ij}^{k}(p)=0$, for any $k$ and any $(X^1, \dots, X^n)\in\mathbb{R}^n$). 
 \item[iii)] Deduce that there exists $c\in\mathbb{R}$ such that $|g_{ij}(x)-\delta_{ij}|\leq c\cdot(d(x,p))^2$, for all $x\in B_{\frac{r}{2}}(p)$. (Hint: use the Koszul formula for $\Gamma_{ij}^k$). 
\end{itemize}




{\bf Exercise 3}\\
Let $(M,g)$ be a Riemannian manifold, $p,q\in M$. Assume that \mbox{$\gamma_i:[0,L]\to M$}, $i=1,2$, are two different shortest curves from $p$ to $q$, parametrized by arc\-length. 
Extend each geodesic $\gamma_i$ to its maximal domain.
\begin{itemize}
 \item[i)] Show that $\dot{\gamma_1}(L)\neq \dot{\gamma_2}(L)$.
 \item[ii)] Show that $\gamma_1|_{[0,L+\varepsilon]}$ is not a shortest curve for any $\varepsilon>0$. (Hint: construct a shorter path from $p$ to $\gamma_1(L+\varepsilon)$ by using a chart around $q$ and the geodesic $\gamma_2$). 
\end{itemize}



{\bf Exercise 4}\\
Show that the following groups with the manifold structure induced from $\mathbb{R}^{n\times n}\cong \mathbb{R}^{n^2}$ are Lie groups and determine their Lie algebras: 
\[\mathrm{SO}(n), \mathrm{GL}(m,\mathbb{C}), \mathrm{U}(m), \mathrm{SU}(m), \text{ where } n=2m.\]
Also determine the adjoint representations.
Which of these Lie groups have a bi-invariant Riemannian metric?

\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 22.04.2012} vor der Vorlesung}.\\
\end{document}



