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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill SoSe 2013\\
Prof. Dr. Bernd Ammann\hfill 03.06.2013\\
Dr. Mihaela Pilca\\

\begin{center}
{\bf Differential Geometry II \\
  Exercise Sheet no. 8} $ $\\
\end{center}



{\bf Exercise 1}\\
Let $G$ be a Lie group which acts isometrically, freely and properly on a 
Riemannian manifold $(M,g)$. (An action is \emph{isometric}  
if $l_\sigma$ is an isometry for any $\sigma\in G$.)
Show that there exists a metric on the 
quotient manifold $G\backslash M$ such that the projection 
$\pi: M\to G\backslash M$ is a Riemannian submersion.
(A submersion $\pi:M\to N$ between Riemannian manifolds 
is called a \emph{Riemannian submersion}
if $d_x\pi$ is an isometry from the orthogonal complement 
of $\ker d_x\pi$ in $T_xM$ to $T_{\pi(x)}N$ for any $x\in M$.)\\


{\bf Exercise 2}\\
Let $\pi: (M,g)\to (N,h)$ be a Riemannian submersion. 
\begin{itemize}
\item[i)] Let $\gamma$ be a geodesic in $(N,h)$. Show that any horizontal lift of $\gamma$ is a geodesic in $(M,g)$.
\item[ii)] Let $\tau:[a,b]\to M$ be a geodesic in $(M,g)$ such that $\dot{\tau}(a)$ is horizontal. Show that $\dot{\tau}(t)$ is horizontal for all $t\in[a,b]$. Conclude that if a horizontal lift $\widetilde{\gamma}$ of a curve $\gamma$ is a geodesic in $(M,g)$, then $\gamma$ is a geodesic in $(N,h)$.
\item[iii)] Let $\pi: S^{2n+1}\to \mathbb{C}P^n$ be the projection $z\mapsto [z]$, which defines the so-called \emph{Hopf fibration}. 
Consider on $\mathbb{C}P^n$ the Riemannian metric that makes $\pi$ a Riemannian submersion, where $S^{2n+1}$ carries the standard metric. This means $\mathbb{C}P^n$ carries the metric defined vai Exercise 1. This metric on $\mathbb{C}P^n$ 
is called the \emph{Fubini-Study} 
metric of $\mathbb{C}P^n$.\\
Show that the geodesics parametrized by arclength 
in $\mathbb{C}P^n$ are of the form $\gamma(t)=[\cos t\, v+ \sin t\, w]$, where $v,w\in S^{2n+1}\subset \mathbb{C}^{n+1}$ with $\overset{n+1}{\underset{j=1}{\sum}} v_j \overline w_j=0$. Show furthermore that in $\mC P^1$ the points $[(1,0)]$ and $[(0,1)]$ are conjugated along a geodesic.
\end{itemize}


{\bf Exercise 3}
\begin{itemize}
\item[i)] Let $V$ and $W$ be two $m$-dimensional real vector spaces and $A_t$ a smooth family of
homomorphisms, where $t$ is a real parameter. Let $A'_t=\frac{d}{dt}A_t$. Assume that
$$\mathrm{Im}(A_0)\oplus A'_0(\mathrm{Ker}(A_0))=W.$$
Show that there exists an $\varepsilon>0$, such that $A_t$ has rank $m$ for all $t\in(-\varepsilon, 0)\cup(0,\varepsilon)$.
\item[ii)] Let $J_1$ and $J_2$ be two Jacobi vector fields along a geodesic on a Riemannian manifold. Show that the function 
$$ t\mapsto \langle J_1(t), J'_2(t)\rangle - \langle J'_1(t), J_2(t)\rangle$$
is constant.
\item[iii)] Let $\gamma:[0,b)\to M$ be a geodesic on a Riemannian manifold. Show that the set
\[\{t\in [0,b) \, \vert \, t \text{ is conjugated to } 0\}\]
is closed and discrete in $[0,b)$. Hint: Use i) and ii).\\
\end{itemize}

{\bf Exercise 4}\\
Let $\pi:\,  (M,g) \to (N,h)$ be a Riemannian submersion. The vectors in the kernel of $d\pi$ are called vertical. For each $X\in\Gamma(TN)$, let $\overline X$ denote the horizontal lift of $X$, \emph{i.e.} $\overline X\in\Gamma(TM)$ such that $d\pi \circ \overline X=X\circ \pi$ and $\overline X$ is orthogonal in each point to the kernel of $d\pi$.

\begin{itemize}
\item[i)] Show that the vertical part of $[\overline X,\overline Y]$ in $p\in M$, denoted by  $[\overline X,\overline Y]^v_p$, depends only on $\overline X(p)$ and $\overline Y(p)$.
\item[ii)] Let $X\in\Gamma(TN)$, $\eta\in\Gamma(TM)$ and $\eta$ is vertical. Show that $[\eta, \overline X]$ is vertical.
\item[iii)]  Compute $\overline{[X,Y]}-[\overline X, \overline Y]$ and $\nabla^M_{\bar X} \overline Y-\overline{\nabla^N_{X}Y}$, for $X,Y\in\Gamma(TN)$.
\item[iv)] Assume that $\overline{X}(p)$ and $\overline{Y}(p)$ are orthonormal. Let $E$ be the plane spanned by $X(\pi(p))$ and $Y(\pi(p))$ and $\overline{E}$ be the plane spanned by $\overline X(p)$ and $\overline Y(p)$. Show the following formula for the sectional curvatures of $(M,g)$ and $(N,h)$:
\[K^{N,h}(E)=K^{M,g}(\overline E)+\frac{3}{4}\| [\overline X, \overline Y]^v_p\| ^2.\]
\end{itemize}










\vskip1cm
{\sl Hand in the solutions on
\textbf{Monday, June 10, 2013} before the lecture}.\\
\end{document}



