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 \begin{Large}

 \noindent Mathematisches Institut  

 \smallskip

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

 \vspace*{0.2cm}

 \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 1}

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 \noindent 1. Let $G$ be a Lie group. Show that the identity component, i.e. the connected component containing the identity element, is a normal subgroup of $G$. 
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 \medskip

 \noindent 2. The {\it special unitary group} is defined as 
\begin{equation*}
SU(n)=\{A \in \mbox{Mat}(n\times n,\mathbb{C}) \mid A\cdot \bar{A}^T = I, \quad  \mbox{det}A=1\}.
\end{equation*}
Show that $SU(n)$ is a compact Lie group of dimension $n^2-1$ and determine its Lie algebra $su(n)$. Prove that, as a manifold, $SU(2)$ is diffeomorphic to $S^3$. 
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 \medskip

 \noindent 3. a) Let $G$ be a Lie group and $X,Y \in \mathfrak{g}$. Show that 
the curve 
\begin{equation*}
exp(tX)exp(tY), \quad t \in \mathbb{R}
\end{equation*}
 has initial velocity vector $X+Y$. 

b) Find an explicit example where $exp(X)exp(Y) \neq exp(X+Y)$. 
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\medskip


 \noindent 4. Consider the general linear group $GL(n,\mathbb{R})$ with Lie algebra $gl(n,\mathbb{R})$. Show that as a vector space, $gl(n,\mathbb{R})$ can be canonically identified with $\mbox{Mat}(n\times n,\mathbb{R})$. Prove that the (abstract) exponential mapping is given by 
\begin{equation*}
exp(X) = \sum_{k=0}^\infty \frac{X^k}{k!} \quad \forall X \in gl(n,\mathbb{R}),
\end{equation*}
and that 
\begin{equation*}
[X,Y]=XY-YX \quad \forall X,Y \in gl(n,\mathbb{R}).
\end{equation*}
\bigskip

 \noindent Please hand in your solutions on Thursday, the 29th of April, 
 in the lecture. 




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