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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 2}

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 \noindent 1. a) Let $M, N$ be differentiable manifolds and $f:M\rightarrow N$ a smooth map of constant rank $k$. Show that the connected components of the preimages $f^{-1}(y)$, where $y$ runs over $f(M)$, form the leaves of a codimension $k$ foliation on $M$.

b) Let $f:\mathbb{R}^3\rightarrow \mathbb{R}$ be the map
\begin{equation*}
f(x,y,z)=(1-x^2-y^2)e^z.
\end{equation*}
Check that this is a submersion. By a), the preimages form the leaves of a codimension 1 foliation $\mathcal{F}$ on $\mathbb{R}^3$. This foliation is called the {\it Reeb foliation}. Sketch a picture of $\mathcal{F}$. Denote the preimage of $s \in \mathbb{R}$ by $L_s$. Prove that $\mathcal{F}$ is invariant under translations in the $z$-direction and that $L_0 \cong S^1 \times \mathbb{R}$ and $L_s \cong \mathbb{R}^2$ for $s >0$.  
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 \noindent 2.  Consider the 1-form $\alpha=dx+zdy$ on $\mathbb{R}^3$ with coordinates $(x,y,z)$. 

a) Prove that $\mbox{ker}\alpha=\{X \in T\mathbb{R}^3\mid i_X\alpha=0\}$ is a smooth rank 2 distribution $\mathcal{D}$ on $\mathbb{R}^3$. 

b) Show that $\mathcal{D}$ is nowhere integrable.
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 \noindent 3. Consider the Lie group $SL(2,\mathbb{R})$ with Lie algebra $sl(2,\mathbb{R})$. 

a) Show that $sl(2,\mathbb{R})=\{A \in \mbox{Mat}(2\times 2, \mathbb{R}) \mid \mbox{Tr}A=0\}$. 

b) Prove that the exponential map of $SL(2,\mathbb{R})$ is not surjective. ({\it Hint}: You can consider $\mbox{Tr}\,\mbox{exp}A$ for $A \in sl(2,\mathbb{R})$.)
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 \noindent 4. Let $\phi:H \rightarrow G$ be a smooth homomorphism between Lie groups. Prove that $\phi(H)$ is a Lie subgroup of $G$.
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 \noindent Please hand in your solutions on Thursday, the 6th of May, 
 in the lecture. 




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