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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 10}

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\noindent 1. Let $\pi:(\bar{M},\bar{g})\rightarrow (M,g)$ be a submersion between Riemannian manifolds. The tangent bundle $T\bar{M}$ splits into the vertical and horizontal bundle, $T\bar{M}=V\oplus H$, where $V=T\pi$ is the tangent bundle along the fibres and $H=V^\perp$. For a vector field $X$ on $M$ denote by $\bar{X}\in \mathfrak{X}(\bar{M})$ the {\it horizontal lift}, i.e. the unique horizontal vector field on $\bar{M}$ such that $D\pi(\bar{X})=X$. The submersion $\pi$ is called {\it Riemannian} if $D\pi:H \rightarrow TM$ is an isometry on each fibre. Let $\pi$ be a Riemannian submersion. Show that the Levi-Civita connections $\bar{\nabla},\nabla$ of $\bar{M}$ and $M$ are related by
\begin{equation*}
\bar{\nabla}_{\bar{X}}\bar{Y} = \overline{\nabla_X Y}+\frac{1}{2}[\bar{X},\bar{Y}]^v,
\end{equation*}
for all vector fields $X,Y$ on $M$, where $Z^v$ denotes the vertical component of $Z \in \mathfrak{X}(\bar{M})$. 
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\noindent 2. Let $\pi:(\bar{M},\bar{g})\rightarrow (M,g)$ be a Riemannian submersion. Denote the sectional curvatures of $\bar{M},M$ by $\bar{K}$ and $K$. Prove O'Neill's formula: 
\begin{equation*}
K(X,Y) = \bar{K}(\bar{X},\bar{Y}) + \frac{3}{4}|[\bar{X},\bar{Y}]^v|^2,
\end{equation*}
for all orthonormal vector fields $X$ and $Y$ on $M$.
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\noindent 3. Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. A {\it bi-invariant metric} is a Riemannian metric on $G$ which is invariant under all left- and right-translations. (One can show that every compact Lie group admits a bi-invariant metric). Suppose $G$ admits a bi-invariant metric $g$. Denote the Levi-Civita connection of $g$ by $\nabla$. Prove the following formulas for $X,Y,Z,W \in \mathfrak{g}$:
\begin{itemize}
\item $g([Y,X],Z)+g(Y,[Z,X])  = 0$. 
\item $\nabla_X Y = \frac{1}{2}[X,Y]$. 
\item $g(R(X,Y)Z,W) = \frac{1}{4}g([X,Y],[Z,W])$.
\end{itemize}
\noindent Hint: Recall that $[X,Y]=\frac{d}{dt}|_{t=0}(Ad(exp(tX))Y)$, where $Ad(h)=D_e(L_hR_{h^{-1}})$.
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\noindent 4. Let $G$ be a Lie group.

a) Show that if $G$ admits a bi-invariant metric and $\mathfrak{g}$ has trivial center, then $G$ is compact and $\pi_1(G)$ is finite. Prove that $SL(2,\mathbb{R})$ does not admit a bi-invariant metric. 

b) Show that if $G$ is compact and non-abelian and $M$ any compact manifold, then $M\times G$ admits a metric of positive scalar curvature.
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\noindent Please hand in your solutions on Monday, the 12th of July, 
 in the lecture. 




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