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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 27.05.2013\\
Dr. Mihaela Pilca\\

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 7} $ $\\
\end{center}



{\bf Exercise 1}\\
The \emph{Killing form} of a Lie algebra $\mathfrak{g}$ is the function defined by:
\[B:\mathfrak{g}\times\mathfrak{g}\to \mathbb{R}, \quad B(X,Y):=\mathrm{\tr}(\mathrm{ad}(X) \circ \mathrm{ad} (Y)).\]
Show the following properties of the Killing form:
\begin{itemize}
 \item[i)] $B$ is a symmetric bilinear form on $\mathfrak{g}$.
 \item[ii)] If $\mathfrak{g}$ is the Lie algebra of the Lie group $G$, then $B$ is $\mathrm{Ad}$-invariant:
 \[B(\mathrm{Ad}(\sigma)X, \mathrm{Ad}(\sigma)Y)=B(X,Y), \quad \forall \sigma\in G, \,\forall X,Y\in\mathfrak{g}.\]
 Hint: Show first that if $\alpha$ is an automorphism of $\mathfrak{g}$, i.e. a linear isomorphism $\alpha$ satisfying $\alpha([X,Y])=[\alpha(X),\alpha(Y)]$ for all $X,Y\in\mathfrak{g}$, then $\mathrm{ad}(\alpha(X))=\alpha\circ\mathrm{ad}(X)\circ\alpha^{-1}$, for any $X\in\mathfrak{g}$.  
 \item[iii)] For each $Z\in\mathfrak{g}$, $\mathrm{ad}(Z)$ is skew-symmetric with respect to $B$:
 \[B(\mathrm{ad}(Z)X, Y)=-B(X,\mathrm{ad}(Z)X), \forall X,Y\in\mathfrak{g}.\]
\end{itemize}


{\bf Exercise 2}\\
Let $(M,g)$ be a Riemannian manifold of constant sectional curvature $\kappa$ and let $\gamma: [0,\ell]\to M$ be a geodesic parametrized by arc-length. Let $J$ be a vector field along $\gamma$, normal to $\gamma'$. 
\begin{itemize}
\item[i)] Show that the Jacobi equation can be written as $J''+\kappa J=0$.
\item[ii)] Let $V$ be a parallel unit vector field along $\gamma$ normal to $\gamma'$. Determine the Jacobi vector field $J$ satisfying the initial conditions $J(0)=0$ and $J'(0)=V(0)$.\\
\end{itemize}



%{\bf Exercise 3}
%\begin{itemize}
%\item[i)] Let $V$ and $W$ be two $m$-dimensional real vector spaces and $A_t$ a% smooth family of
%homomorphisms, where $t$ is a real parameter. Let $A'_t=\frac{d}{dt}A_t$. Assum%e that
%$$\mathrm{Im}(A_0)\oplus A'_0(\mathrm{Ker}(A_0))=W.$$
%Show that there exists an $\varepsilon>0$, such that $A_t$ has rank $m$ for all% $t\in(-\varepsilon, 0)\cap(0,\varepsilon)$.
%\item[ii)] Let $J_1$ and $J_2$ be two Jacobi vector fields along a geodesic on %a Riemannian manifold. Show that the function 
%$$ t\mapsto \langle J_1(t), J'_2(t)\rangle - \langle J'_1(t), J_2(t)\rangle$$
%is constant.
%\item[iii)] Let $\gamma:[0,b)\to M$ be a geodesic on a Riemannian manifold. Sho%w that the set
%\[\{t\in [0,b) \, \vert \, t \text{ is conjugated to } 0\}\]
%is closed and discrete in $[0,b)$. Hint: Use i) and ii).\\
%\end{itemize}


{\bf Exercise 3}
\begin{itemize}
\item[i)] Let $(M,g)$ be a Riemannian manifold and $\gamma:I\to M$ a geodesic. Show that if $M$ is $2$-dimensional, then the relation for points of $\gamma$ to be conjugated to each other along $\gamma$ is transitive. More precisely, for any $t_i\in I$, $i=1,2,3$, such that $\gamma(t_1)$ is conjugated to $\gamma(t_2)$ and $\gamma(t_2)$ is conjugated to $\gamma(t_3)$, it follows that $\gamma(t_1)$ is conjugated to $\gamma(t_3)$.
\item[ii)] Show that the statement in i) is not true for higher dimensions, by considering for instance the Riemannian manifold $(S^2\times S^2, g_{std}\oplus g_{std})$, that is the Riemannian product of two spheres with the standard metric
and the following geodesic $\gamma(t)=(\cos(t), 0, \sin(t), \cos(\pi t), 0, \sin(\pi t))\in S^2\times S^2\subset \mathbb{R}^3\times \mathbb{R}^3= \mathbb{R}^6$.
\end{itemize}





\vskip1cm
{\sl Hand in the solutions on
\textbf{Monday, June 3, 2013} before the lecture}.\\
\end{document}



