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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 9}

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 \noindent 1. Let $M,N$ be differentiable manifolds. Recall that a smooth map $\phi:M\rightarrow N$ is a {\it covering} if $\phi$ is surjective and for every $p\in N$ there exists a neighbourhood $V$ such that $\phi^{-1}(V)=\dot{\bigcup}_{i\in I} U_i$ and $\phi:U_i\rightarrow V$ is a diffeomorphism. Prove that if $M$ is compact, $N$ connected and $\phi$ a local diffeomorphism, then $\phi$ is a covering.  
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\noindent 2. Let $(M,g)$ be an $n$-dimensional Riemannian manifold, $f$ a smooth function and $p\in M$. Consider an orthonormal basis $\{e_i\}$ in $T_pM$ and geodesics $\gamma_i$ with $\gamma_i(0)=p$ and $\dot{\gamma}_i(0)=e_i$. Prove the following formula:
\begin{equation*}
(\Delta f)(p)=\sum_{i=1}^n\frac{d^2}{dt^2}|_{t=0}f(\gamma_i(t)).
\end{equation*}
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 \noindent 3. Let $(M,g)$ be a closed oriented Riemannian manifold and $\Delta_H = d^*d$ the Hodge-Laplace operator on functions (note that $\Delta_H f = -\Delta f =-div\,\nabla f$). A smooth function $f$ is called {\it eigenfunction} of $\Delta_H$ with eigenvalue $\lambda \in \mathbb{R}$ if 
\begin{equation*}
\Delta_H f =\lambda f, \quad f \not\equiv 0.
\end{equation*}

a) Show that the eigenvalues of $\Delta_H$ are non-negative.

b) Let $M=S^n$ with the standard metric. Fix a vector $v \in \mathbb{R}^{n+1}\setminus 0$ and define the smooth function $f_v:S^n\rightarrow \mathbb{R}, p\mapsto \langle p,v\rangle$, where $\langle,\rangle$ denotes the Euclidean scalar product on $\mathbb{R}^{n+1}$. Prove that $f_v$ is an eigenfunction of $\Delta_H$ on $S^n$ with eigenvalue $n$. 
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\noindent 4. Let $(M,g)$ be a closed oriented $n$-dimensional Riemannian manifold. Prove the following theorem of Lichnerowicz: if the Ricci curvature of $M$ is greater or equal to $k>0$ (i.e. $Ric(X)\geq k|X|^2$ for all tangent vectors $X$) then the non-zero eigenvalues of $\Delta_H$ are bounded from below by $nk$. Show that this bound is optimal for $S^n$ with the standard metric.    
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\noindent Hint: Consider the function $h=\frac{1}{2}|\nabla f|^2$ and use a Bochner formula for $\Delta h$ from the lecture. Show that $|\nabla(\nabla f)|^2\geq (\Delta f)^2/n$.
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\noindent Please hand in your solutions on Monday, the 5th of July, 
 in the lecture. 




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