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 \begin{Large}

 \noindent Mathematisches Institut  

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 \noindent Universit"at M"unchen
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 \noindent Mark Hamilton  
 
 \noindent Office hour: Mon 1-2 pm, Room 308 \hspace*{5cm} SS 04
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 {\bf \large Geometry of Manifolds II (Prof. Kotschick) \\ \medskip Problem sheet 11}

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\noindent 1. a) Let $(M,g)$ be a Riemannian manifold, $\gamma$ a geodesic and $J(t)$ a Jacobi field along $\gamma$ with $J(0)=0$. Prove that $J(t)$ is orthogonal to $\dot{\gamma}(t)$ for all $t$ if and only if $\dot{J}(0)\perp \dot{\gamma}(0)$. 

b) Suppose $(M,g)$ has constant curvature $K$. Let $w(t)$ be a parallel vector field along the geodesic $\gamma$ which is orthogonal to $\dot{\gamma}$ and has length $1$. Solve in each case $K>0,K=0,K<0$ the Jacobi equation for $J(t)$ with initial conditions $J(0)=0,\dot{J}(0)=w(0)$. 
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\noindent 2. Let $(M,g)$ be a Riemannian manifold, $p\in M$ be a point and $v,w \in T_pM$. Consider a geodesic $\gamma(t)$ with $\gamma(0)=p$, $\dot{\gamma}(0)=v$. Show that the vector field $J$ along $\gamma$ given by $J(t)=(D_{tv}exp_p)(tw)$ solves the Jacobi equation 
\begin{equation*}
\ddot{J}+R(\dot{\gamma}, J)\dot{\gamma}=0,
\end{equation*}
with initial values $J(0)=0, \dot{J}(0)=w$.
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\noindent 3.  a) Let $M$ be a manifold which admits a complete Riemannian metric with $Ric\geq k >0$. Show that $M$ does not admit a metric of non-positive  sectional curvature.  

b)  Let $M,N$ be closed manifolds of positive dimension. Show that $M\times N$ does not admit a metric of negative sectional curvature.
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\noindent 4. Let $(M,g)$ be a closed Riemannian manifold of negative sectional curvature. Show that any two commuting elements of the fundamental group $\pi_1(M)$ belong to a common cyclic subgroup and that $\pi_1(M)$ is a uniquely determined union of infinite cyclic subgroups which intersect pairwise only in the identity element.
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\noindent Please hand in your solutions on Monday, the 19th of July, 
 in the lecture. 




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