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\begin{document}
\pagestyle{empty}
Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 3.12.2012\\
Dr. Nicolas Ginoux\\

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

{\bf Exercise 1}\\
Let $(M^n,g)$ be a smooth $n$-dimensional Riemannian manifold and $f:M\to\mR$ be a smooth function on $M$.
\ben
 \item Show that there exists a unique smooth tangent vector field -- which we denote by ${\rm grad}f$ and call the \emph{gradient vector field} of $f$ -- on $M$ such that
\[\partial_Xf=g({\rm grad}f,X)\]
holds for every $X\in\mathfrak{X}(M)$.
\item Given any $p\in M$ and any orthonormal basis $\{e_1,\ldots,e_n\}$ of $T_pM$ (i.e., $g_p(e_i,e_j)=\delta_i^j$), prove that ${\rm grad}f{|_p}=\sum_{j=1}^n\partial_{e_j}f\cdot e_j$.
\item Given any chart $\varphi:U_\varphi\to V_\varphi$ of $M$, show that
\[{\rm grad}f_{|_{U_\varphi}}=\sum_{i,j=1}^ng^{ij}\frac{\partial f}{\partial\varphi^i}\cdot\frac{\partial}{\partial\varphi^j},\]
where $\left(g^{ij}\right)_{1\leq i,j\leq n}$ is the inverse matrix of the matrix $\left(g_{kl}\right)_{1\leq k,l\leq n}$ of $g$ in the chart $\varphi$.
\end{enumerate}


{\bf Exercise 2} {\it (Poincar\'e half-space)}\\
Let $H:=\left\{(x,y)\in\mR^2\,,\,y>0\right\}$ denote the upper half-space and define the Riemannian metric $g$ on $H$ by 
\[g:=\frac{dx^2+dy^2}{y^2}.\]
\ben
 \item Compute the Christoffel symbols of the Levi-Civita connection in the canonical coordinates $x,y$ of $H$.
\item Let $c:\mR\to H$, $t\mapsto(t,1)$ and $v_0:=(0,1)\in\mR^2\cong T_{(0,1)}H$.
Show that, for the parallel vector field $v$ along $c$ with $v(0)=v_0$, the vector $v(t)$ makes an angle equal to $t$ with the $y$-axis, for all $t$.
\end{enumerate}

{\bf Exercise 3} {\it (Surfaces of Revolution and Clairaut's theorem)}\\
For an interval $I$ and a positive smooth function $f:I\to \mR^+$ we define
  $$F(x,\phi):=\begin{pmatrix}x\\f(x) \cos(\phi)\\f(x)\sin(\phi)\end{pmatrix}.$$
The surface of revolution generated by $f$ is 
  $$M_f:=\left\{ F(x,\phi)\,\Big|\, x\in I,\phi\in\mR\right\}.$$ 
Then $\partial F/\partial x$ and  $\partial F/\partial \phi$ define vector 
fields along $F:I\times \mR\to M_f$.
\ben
\item Construct vector fields $X_x$, $X_\phi\in \mathcal{X}(M_f)$ such that 
$X_x\circ F= \partial F/\partial x$ and 
$X_\phi\circ F= \partial F/\partial \phi$. Show that $X_x$ and  $X_\phi$ are 
everywhere orthogonal.
 \item Show that $c_{x_0}:\mR\to M_f$, 
$\phi\mapsto  F(x_0,\phi)$ is a geodesic if and only if $f'(x_0)=0$. 
{\it Hint: Exercise 2 from sheet no. 7 is helpful.}
\item Let $\gamma(t)=F(x(t),\phi(t))$ be a curve in $M_f$. Verify
\begin{equation}\label{eq}
g\Bigl(\dot\gamma(t),\frac{\pa F}{\pa \phi}(x(t),\phi(t))\Bigr)= f(x(t))^2\dot\phi(t)
\end{equation} 
\item Verify  the formula
  $$\frac{d}{dt}\left(g\Bigl(\dot\gamma(t),\frac{\pa F}{\pa \phi}\Bigr)\right)
    = g\Bigl(\frac{\nabla}{dt}\dot\gamma(t),\frac{\pa F}{\pa \phi}\Bigr)
     + \dot\phi(t) g\Bigl(\dot\gamma(t),\frac{\pa^2 F}{\pa^2 \phi}\Bigr)
     + \dot x(t) g\Bigl(\dot\gamma(t),\frac{\pa^2 F}{\pa \phi\,\pa x}\Bigr)$$
where we suppressed $(x(t),\phi(t))$ in the notation.
\item Show that both $\|\dot\gamma(t)\|$ and 
$f(x(t))^2\dot\phi(t)$ are constant if $\gamma$ is a geodesic. 
{\it Hint: Derive both sides of \eqref{eq} 
with respect to $t$ and use (d).}
\item (Extra question, 2 bonus points.)
Does ``only if'' hold in (e)?
\een

{\bf Exercise 4}\\
Let $\nabla$ be any connection on a smooth manifold $M$.
Let $c:I\to M$ be any smooth curve and denote by $P_{s,t}:T_{c(s)}M\to T_{c(t)}M$ the parallel transport along $c$.
Show that, for any smooth vector field $X$ along $c$ and any $t_0\in I$, we have
\[\frac{\nabla X}{dt}(t_0)=\lim_{t\to t_0}\left(\frac{P_{t,t_0}(X(t))-X(t_0)}{t-t_0}\right).\]


\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 10.12.2012} vor der Vorlesung}.\\
\end{document}



