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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 10.06.2013\\
Dr. Mihaela Pilca\\

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 9} $ $\\
\end{center}



{\bf Exercise 1}\\
Let $(M,g)$ be a connected, complete and simply-connected Riemannian manifold with sectional curvature $K\leq 0$.
Show that there is a unique geodesic between any two points on $M$. Hint: use Cartan-Hadamard Theorem.\\



{\bf Exercise 2}\\
Let $M$ be a connected manifold and $p\in M$. We consider the map defined in the lecture between the fundamental group of $M$ and the set of free homotopy classes of loops:
\[F: \pi_1(M,p) \to \pi_o \mathcal{L}(M),\] 
\[ [\gamma]\mapsto [\gamma]_{\text{free}}.\]
Show the following:
\begin{itemize}
 \item[i)] $F$ is surjective.
 \item[ii)] $F$ induces a well-defined map on the set of conjugacy classes in $\pi_1(M,p)$, that is $[\gamma\tau\gamma^{-1}]_{\text{free}}=[\tau]_{\text{free}}$, for any $\gamma,\tau\in \pi_1(M,p)$.
 \item[iii)] The map induced by $F$ on the set of conjugacy classes in $\pi_1(M,p)$ is injective.\\
\end{itemize}


{\bf Exercise 3}\\
We consider the Hopf fibration and the Fubini-Study metric on $\mathbb{C}P^n$ introduced in Exercise 2, (iii) on Sheet no. 8. We use the same notation as in this exercise, and again $X^v$ is the vertical part of $X$. 
The vertical vectors of the Hopf fibration in the point $z\in S^{2n+1}$ 
are of the form $\lambda i z$, $\lambda\in\mathbb{R}$. 

For $X,Y\in\mathbb{C}^{n+1}$, we define $\langle X,Y\rangle_{\mathbb{C}}:=\overset{n+1}{\underset{j=1}{\sum}} X_j \overline Y_j$ and $\langle X,Y\rangle_{\mathbb{R}}:=\mathrm{Re}(\overset{n+1}{\underset{j=1}{\sum}} X_j \overline Y_j)$. Then it holds $\langle X,Y\rangle_{\mathbb{C}}=\langle X,Y\rangle_{\mathbb{R}}+i\langle X,iY\rangle_{\mathbb{R}}$. Show the following:

\begin{itemize}
\item[i)] For any $\widetilde X_0\in \mathbb{C}^{n+1}$, the map $w\mapsto \widetilde X_w:=\widetilde X_0 - \langle \widetilde X_0, w\rangle_{\mathbb{C}}w$ is a well-defined vector field on $S^{2n+1}$.
\item[ii)] $\widetilde X$ is horizontal everywhere.
\item[iii)] Each point $p\in \mathbb{C}P^n$ admits an open neighborhood $U$ and a smooth map $f: \pi^{-1}(U)\to S^1$, such that $f(\lambda z)=\lambda f(z)$, for all $z\in \pi^{-1}(U)$ and $\lambda\in S^1$.
\item[iv)] $f\widetilde X$ is a horizontal lift of a vector field $X\in\Gamma(TU)$.
\item[v)] For a fixed $z\in S^{2n+1}$ assume that $\langle \widetilde X_0, z\rangle_{\mathbb{C}}=\langle \widetilde Y_0, z\rangle_{\mathbb{C}}=0$. For the Levi-Civita connection $\nabla$ of $S^{2n+1}$ it holds:
\[\nabla_{\widetilde Y_w} \widetilde X_w |_{w=z}=-(\mathrm{Im}(\langle \widetilde X_0,  \widetilde Y_0\rangle_{\mathbb{C}}))iz\]
\item[vi)] Choose $f$ such that $f(z_0)=1$ for a $z_0\in\pi^{-1}(p)$.
Conclude that 
$[f\widetilde Y, f\widetilde X]^v |_{z_0}=-2(\mathrm{Im}\langle \widetilde X_0,  \widetilde Y_0\rangle_{\mathbb{C}})iz_0$. 
\item[vii)] The sectional curvature $K$ of $\mathbb{C}P^n$ satisfies: $1\leq K\leq 4$. For which planes is $K=4$ and for which planes is $K=1$?
\end{itemize}










\vskip1cm
{\sl Hand in the solutions on
\textbf{Monday, June 17, 2013} before the lecture}.\\
\end{document}



