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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 6}

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 \noindent 1. Let $(M,g)$ be an oriented Riemannian manifold and $X$ a vector field. Use a local orthonormal frame which is parallel in one point to show that \begin{equation*}
L_X dvol_g=div(X)dvol_g.
\end{equation*}     
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 \noindent 2.  Let $(M,g)$ be a closed oriented Riemannian manifold and $X$ a Killing field. Show that:   

a) The flow of $X$ operates through isometries. 

b) The adjoint $L^*_X$ of the map $L_X:\Omega^k(M) \rightarrow \Omega^k(M)$ with respect to the $L^2$-norm is $-L_X$.

c) If $\omega \in \Omega^k(M)$ is harmonic then $L_X \omega=0$. 
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 \noindent 3. Let $(M,g)$ be a closed connected oriented Riemannian manifold. 

a) Show that for smooth functions $f$ and $h$
\begin{equation*}
\Delta(fh)=(\Delta f)h+f(\Delta h)+2g(grad\,f,grad\,h).
\end{equation*}
Suppose that $f$ is a smooth function with $\Delta f \geq 0$. Use Stokes' theorem to prove (without using Hodge theory) that:

b) $\Delta f=0$.

c) $f$ is constant. 
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 \noindent 4. Let $(M,g)$ be a closed oriented Riemannian manifold with $Ric \geq 0$. Suppose that $b_1(M)=k$. Show that the Riemannian universal covering $(\tilde{M},\tilde{g})$ is isometric to $(N,h) \times (\mathbb{R}^k,g_{stand})$ for some Riemannian manifold $(N,h)$.
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\noindent Please hand in your solutions on Monday, the 14th of June, 
 in the lecture. 




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