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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 7}

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 \noindent 1. a) Let $N$ be a closed manifold. Show that $N \times S^2$ admits a metric of positive scalar curvature. 

b) Find a closed 4-manifold $X$ which admits a metric with $scal >0$ but which does not admit a metric with $Ric \geq 0$.
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 \medskip

 \noindent 2. Let $(M,g)$ be a Riemannian manifold and $\omega \in \Omega^*(M)$ a differential form. Fix a point $p \in M$, an orthonormal basis $\{X_i\}$ of $T_pM$ with dual basis $\{\theta_i\}$ and let $\lambda_\alpha$ be the eigenvalues of the curvature operator $\mathfrak{R}_p:\Lambda^2T_pM\rightarrow \Lambda^2T_pM$ with $\Theta_\alpha \in \Lambda^2T^*_pM$ the duals of the corresponding eigenvectors. Show that 
\begin{equation*}
\sum_{i,j=1}^ng([\theta_i\cdot \theta_j,R(X_i,X_j)\omega],\omega)=-\sum_\alpha \lambda_\alpha|[\Theta_\alpha,\omega]|^2.
\end{equation*}
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Hint: Prove first that the left hand side equals 
\begin{equation*}
-\sum_{i<j,k<l}^n g(\mathfrak{R}(X_i\ \wedge X_j),X_k\wedge X_l)g([\theta_k\cdot \theta_l,\omega],[\theta_i\cdot \theta_j,\omega])
\end{equation*}
and then show that this expression is independent of the orthonormal basis of $\Lambda^2T_pM$.  
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 \medskip

 \noindent 3. Let $(M^n,g)$ be a Riemannian manifold and $\omega$ a differential form of degree $k$ where $1\leq k\leq n-1$1. With the notation as in Ex. 2 prove that each of the following two conditions implies that $\omega$ vanishes identically:

a) $[\theta_i\cdot\theta_j,\omega]=0$ for all $i,j$. 
 
b) $[\Theta_\alpha,\omega]=0$ for all $\alpha$. 
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\medskip


 \noindent 4. Let $(X,g)$ be a closed oriented Riemannian 4-manifold. Prove the following Weitzenb\"ock formula for $2$-forms $\omega \in \Omega^2(X)$:
\begin{equation*}
\Delta_H\omega = \nabla^*\nabla\omega -2\,W(\omega) + \frac{1}{3}scal\, \omega.
\end{equation*} 
Here $W$ denotes the {\it Weyl tensor} which acts on 2-forms via 
\begin{equation*}
W(\omega) = W(\sum_{a,b=1}^4\omega_{ab}\theta_a \wedge \theta_b)=\frac{1}{2}\sum_{a,b,i,k=1}^4W_{abik}\omega_{ab}\theta_i\wedge \theta_k
\end{equation*}
and has components
\begin{equation*}
W_{ijkl} = R_{ijkl} + \frac{1}{2}(\delta_{li}R_{kj}-\delta_{lj}R_{ki}-\delta_{ki}R_{lj}+\delta_{kj}R_{li}) - \frac{1}{6}(\delta_{li}\delta_{kj}-\delta_{lj}\delta_{ki})scal,
\end{equation*}
where 
\begin{eqnarray*}
R_{ijkl} & = & g(R(E_i,E_j)E_k,E_l) \\
R_{ij} & = & \sum_{k} R_{ikjk} \\
scal & = & \sum_{k} R_{kk},
\end{eqnarray*} 
in a local orthonormal frame $\{E_i\}$ with dual frame $\{\theta_i\}$. 
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\noindent Hint: Prove first the identity
\begin{eqnarray*}
\frac{1}{2}\sum_{i,j=1}^4\theta_i\cdot\theta_j\cdot R(E_i,E_j)(\theta_a\cdot\theta_b)  & = &  \sum_{i=1}^4(-R_{ia}\theta_i\cdot\theta_b+R_{ib}\theta_i\cdot\theta_a) \\
& & +\sum_{i,k=1}^4(R_{ibak}-R_{iabk})\theta_i\cdot\theta_k.
\end{eqnarray*}
\bigskip
\bigskip

\noindent Please hand in your solutions on Monday, the 21st of June, in the lecture. 




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