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 \begin{Large}

 \noindent Mathematisches Institut  

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 \noindent Universit"at M"unchen
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 \end{Large}

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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 4}

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 \noindent 1. Consider $\mathbb{R}^n$ with the standard Euclidean metric $g_0$.  The group of affine isometries is defined by 
\begin{equation*}
Iso^{aff}(\mathbb{R}^n,g_0) = \{\phi:\mathbb{R}^n\rightarrow \mathbb{R}^n \mid \phi(v)=A\cdot v +t,\,\mbox{for some}\,A \in O(n), t\in \mathbb{R}^n\}.
\end{equation*}

a) Show that the group of translations is a normal subgroup in $Iso^{aff}(\mathbb{R}^n,g_0)$, with quotient isomorphic to $O(n)$. 

b) Prove that every isometry of $(\mathbb{R}^n,g_0)$ is affine, hence  $Iso^{aff}(\mathbb{R}^n,g_0) = Iso(\mathbb{R}^n,g_0)$.  
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 \medskip

 \noindent 2. Consider $\mathbb{R}^{n+1}$ with the quadratic form $Q(v,w)=-v_0w_0+\sum_{i=1}^nv_iw_i$. 

a) Show that $H^n=\{v\in \mathbb{R}^{n+1}\mid Q(v,v)=-1,v_0>0\}$ is an $n$-dimensional submanifold of $\mathbb{R}^{n+1}$ and the restriction of $Q$ to $TH^n$ induces a Riemannian metric $g_Q$ on $H^n$. 

b) Prove that $(H^n,g_Q)$ is isometric to {\it hyperbolic n-space} $(\mathbb{H}^n,g)$, where $\mathbb{H}^n=\{x \in \mathbb{R}^n\mid x_n >0\}$ and $g$ is given by 
\begin{equation*}
g_{ij}=\frac{1}{x_n^2}\delta_{ij}. 
\end{equation*}
({\it Hint:} You can do this in two steps. First project $H^n$ to the open ball  $\mathring{D^n}=\{x=(0,x_1,...,x_n) \mid |x|<1\}$ along the lines through $(-1,0,...,0)$ and then map $\mathring{D^n}$ onto the upper half-plane $\mathbb{H}^n$ by inversion at $t=(0,....,0,-1)$, i.e. by the map $x \mapsto t+2\frac{(x-t)}{|x-t|^2}$.)
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 \medskip

 \noindent 3. Consider hyperbolic $n$-space $(\mathbb{H}^n,g)$ as in Ex. 2.

 a) Show that $(\mathbb{H}^n,g)$ has constant sectional curvature $-1$. 

b) Prove that the isometry group of $(\mathbb{H}^n,g)$ is isomorphic to $O^+(1,n) = \{\phi \in End(\mathbb{R}^{n+1})\mid Q(\phi v,\phi w)=Q(v,w),\, \phi(H^n)\subset H^n\}$, with $Q$ as above.  
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\medskip


 \noindent 4. a) Let $M,N$ be differentiable manifolds and $\phi:N \rightarrow M$ a surjective differentiable map of constant rank. Show that $\phi$ is a submersion. 

b) Let $G \times M \rightarrow M$ be a smooth Lie group action. Suppose that $M$ is connected and the action is transitive. Show that the induced action of the identity component $G_0$ is still transitive.
\bigskip

 \noindent Please hand in your solutions on {\bf Monday}, the 24th of May, 
 in the lecture. 




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