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 \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 5}

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 \noindent 1. Let $\tilde{M}$ be a complete simply-connected $n$-dimensional Riemannian manifold of constant sectional curvature $K=1$. Let $p,p'$ be points on $S^n$ with antipodal points $q,q'$ such that all four points are different. Use suitably defined local isometries $f:S^n\setminus \{q\}\rightarrow \tilde{M}$, $f':S^n\setminus \{q'\} \rightarrow \tilde{M}$ to show that $\tilde{M}$ is isometric to $S^n$. 
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 \noindent 2. Let $M$ be a differentiable manifold and $G$ a group which acts {\it properly discontinously} on $M$, i.e. for each $p\in M$ there exists a neighbourhood $U \subset M$ such that $U \cap g\cdot U = \emptyset$ for all $g \ne e$. Show that the quotient $M/G$ is a manifold, the projection $M \rightarrow M/G$ is a covering map and the number of sheets equals $|G|$. 
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 \noindent 3. a) Let $\Gamma \subset O(2m+1)$ be a subgroup which acts properly discontinously on $S^{2m}$ and let $\gamma \in \Gamma$. Consider the eigenvalues of $\gamma$ to show that if $det \,\gamma =1$ then $\gamma=Id$ and if $det \,\gamma =-1$ then $\gamma=-Id$. 

b) Let $M^{2m}$ be an even dimensional complete Riemannian manifold with constant sectional curvature $K=1$. Prove that $M$ is isometric to $S^{2m}$ or $\mathbb{R}P^{2m}$. 
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 \noindent 4. Let $S^3=\{(z,w)\in \mathbb{C}^2 \mid |z|^2+|w|^2=1\}$, $p$ a prime number and $q$ an integer coprime to $p$. Consider the action 
\begin{equation*}
\mathbb{Z}_p \times S^3 \rightarrow S^3, \,[k]\cdot(z,w) = (e^{2\pi ik/p}z, e^{2\pi i kq/p}w).
\end{equation*}

a) Show that the action is properly discontinous and isometric. 

\noindent The quotient space $L(p,q)=S^3/\mathbb{Z}_p$ is a manifold and is called a {\it $(p,q)$-lens space}. The standard Riemannian metric on $S^3$ induces on $L(p,q)$ a metric of constant sectional curvature $K=1$. 

b) Prove that two lens spaces $L(p,q)$ and $L(p,q')$ are isometric if and only if $qq'\equiv \pm 1 \,\mbox{mod}\, p$ or $q \equiv \pm q'\,\mbox{mod}\,p$. (Use without proof that two manifolds $S^3/\Gamma_1$ and $S^3/\Gamma_2$ with $\Gamma_1,\Gamma_2 \subset O(4)$ are isometric if and only if $\Gamma_1,\Gamma_2$ are conjugate subgroups.)  
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 \noindent Please hand in your solutions on Thursday, the 3rd of June, 
 in the lecture. 




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