\documentclass[a4paper,12pt]{article}
%%%%%%%%%%
% Deutsch
%%%%%%%%%%
\usepackage{ngerman}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
%\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}

%%%
\usepackage{latexsym,amssymb,amsmath}
%\usepackage{mltex}
\usepackage{graphicx}
\usepackage{hyperref}
%\usepackage{amstext}
%\usepackage[english]{babel}
\usepackage[matrix,arrow]{xy}
\usepackage{enumerate}

\usepackage{bbold}


\parindent0mm

\frenchspacing
%\vfuzz=2pt \hfuzz=2pt
% \def \be{\begin{eqnarray*}}
% \def \ee{\end{eqnarray*}}
\def \ben{\begin{enumerate}[{\rm (a)}]}
\def \een{\end{enumerate}}
\def \beit{\begin{itemize}}
\def \eeit{\end{itemize}}
\def \bui#1#2{\mathrel{\mathop{\kern 0pt#1}\limits^{#2}}}
%\[\bui{\rightarrow}{b}\]
\def \buil#1#2{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}}}
\def \bfll{\begin{flushleft}}
\def \efll{\end{flushleft}}
\def \bflr{\begin{flushright}}
\def \eflr{\end{flushright}}
\def \ligne{\noindent \rule{\textwidth}{0.5mm} \\}
\def \findemo{\hfill$\square$\\}
\def \fintdemo{\hfill$\surd$\\}
\def \grad{\mathop{\mathrm{grad}}}
\def \diam{\mathop{\mathrm{diam}}}
\def \ker{\mathop{\mathrm{ker}}}
\def \im{\mathop{\mathrm{im}}}
\def \injrad{\mathop{\mathrm{injrad}}}
\def \lra{\longrightarrow}
\def \lmt{\longmapsto}
\def \ovl{\overline}
\def \udl{\underline}
\def \wih{\widehat}
\def \wit{\widetilde}
\def \wnabla{\wit{\nabla}}
\def \cdotM{\buil{\cdot}{M}}
\def \diag{\mathrm{diag}}
\def \Id{\mathrm{Id}}
% \def \Q{\mathrm{Q}}
\def \ric{\mathrm{ric}}
\def \SO{\mathrm{SO}}
\def \so{\mathfrak{so}}
\def \Spin{\mathrm{Spin}}
\def \SU{\mathrm{SU}}
\def \su{\mathfrak{su}}
\def \Sym{\mathrm{Sym}}
\def \tr{\mathrm{tr}}
\def \T{\mathrm{T}}
\def \lala{\langle\!\langle}
\def \rara{\rangle\!\rangle}
\def\II{{\rm I\!I}}

\def\End{\mathop\mathrm{End}}

\def\cC{\mathcal{C}}
\def\cF{\mathcal{F}}
\def\cK{\mathcal{K}}
\def\cS{\mathcal{S}}

\def\mR{\mathbb{R}}
\def\mC{\mathbb{C}}
\def\mZ{\mathbb{Z}}
\def\mN{\mathbb{N}}

\let\mR\mR
\let\C\mC

\newcommand{\lquot}[2]{\raisebox{-0.5ex}{$#2$}\backslash\!\raisebox{0.5ex}{$#1$}}
\newcommand{\rquot}[2]{\raisebox{0.5ex}{$#1$}\!/\!\raisebox{-0.5ex}{$#2$}}
\newcommand{\pa}[1]{\left(#1\right)}
%\pa{argument}
\newcommand{\page}[1]{\ind \begin{minipage}[t]{15cm} #1 \end{minipage}}
% \newcommand{\la}{\langle}
% \newcommand{\ra}{\rangle}
\newcommand{\mnote}[1]{\marginpar{\tiny\em #1}}

\newtheorem{ethm}{Theorem}[section]
\newtheorem{edefi}[ethm]{Def\mbox{}inition}
\newtheorem{elemme}[ethm]{Lemma}
\newtheorem{erem}[ethm]{Note}
\newtheorem{erems}[ethm]{Notes}
\newtheorem{ecor}[ethm]{Corollary}
\newtheorem{prop}[ethm]{Proposition}
\newtheorem{eexemple}[ethm]{Example}
\newtheorem{eexemples}[ethm]{Examples}
\setlength{\textheight}{247mm}
%\setlength{\oddsidemargin}{-20mm}
%\setlength{\evensidemargin}{-20mm} \setlength{\textwidth}{200mm}
\setlength{\topmargin}{-1.5cm}

\let\al\alpha
\let\de\delta
\let\ep\epsilon
\let\ga\gamma
\let\la\lambda
\let\si\sigma

\def \Id{\mathrm{Id}}
\def \Q{\mathrm{Q}}
\def \SO{\mathrm{SO}}
\def \so{\mathfrak{so}}
\def \Spin{\mathrm{Spin}}
\def \SU{\mathrm{SU}}
\def \su{\mathfrak{su}}
\def \Sym{\mathrm{Sym}}
\def \tr{\mathrm{tr}}
\def \T{\mathrm{T}}

\def\mRP{\mR\mathrm P}
\def\mCP{\mC\mathrm P}

\let\<\langle
\let\>\rangle
\def\lala{\langle\!\langle}
\def\rara{\rangle\!\rangle}

\let\witi\widetilde
\let\ti\tilde
\let\na\nabla
\let\pa\partial
\let\setminus\smallsetminus

\def \bui#1#2{\mathrel{\mathop{\kern 0pt#1}\limits^{#2}}}
%\[\bui{\rightarrow}{b}\]
\def \buil#1#2{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}}}

\let\mo\mathopen
\let\mc\mathclose

\let\wihat\widehat

\begin{document}
\pagestyle{empty}
Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 21.01.2013\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 13} $ $\\
\end{center}


{\bf Exercise 1}\\
Let $(M^n,g)$ be a connected $n$-dimensional Riemannian manifold.
Assume $n\geq3$ and that, for each $p\in M$ and any two planes $E,E'\subset T_pM$, we have $K(E)=K(E')$, where $K(E)$ denotes the sectional curvature of the plane $E$.
\ben\item Show the existence of a smooth function $\kappa:M\to\mR$ such that, for all $X,Y,Z,T\in\mathfrak{X}(M)$,
\[\<R(X,Y)Z,T\>=\kappa\cdot\Bigl(g(Y,Z)(gX,T)-g(X,Z)g(Y,T)\Bigr)\]
holds on $M$.
\item Deduce that ${\rm ric}=(n-1)\kappa g$ and that $(M^n,g)$ has constant sectional curvature, i.e.~that $\kappa$ is constant.
\een

{\bf Exercise 2} \emph{(M\"obius strip)}\\
Let $F:\mR\times \mo]-1,1\mc[\to\mR^3$ be the map defined 
by 
  $$F(x,y):=\left(\begin{array}{c}(1+\frac{y}{2}\cos(\frac{x}{2}))\cos(x)\\(1+\frac{y}{2}\cos(\frac{x}{2}))\sin(x)\\\frac{y}{2}\sin(\frac{x}{2})\end{array}\right)$$ and let $M:=F(\mR\times \mo]-1,1\mc[)\subset\mR^3$.
\ben\item Show that $M$ is a smooth $2$-dimensional submanifold of $\mR^3$.
\item Show that, for every $(x,y)\in\mR\times \mo]-1,1\mc[$, the vector $\frac{\frac{\partial F}{\partial x}(x,y)\times\frac{\partial F}{\partial y}(x,y)}{\|\frac{\partial F}{\partial x}(x,y)\times\frac{\partial F}{\partial y}(x,y)\|}\in\mR^3$ has unit norm and is orthogonal to $T_{F(x,y)}M$. 
Here ``$\times$'' denotes the cross product for vectors in $\mR^3$.
\item Show that no continuous unit normal field exists on $M$ and deduce that $M$ is not orientable.
\een

{\bf Exercise 3}\\
Let $C:=\left\{x=(x_0,x_1,x_2,x_3,x_4)\in\mR^5\,|\,x_1^2+x_2^2+x_3^2+x_4^2=\cosh(x_0)^2\right\}$. 
Compute the second fundamental form (in $\mR^5$), the Ricci-tensor and the scalar curvature of $(M,g)$, where $g$ is the Riemannian metric induced by the standard Euclidean inner product.\\
\emph{Hint: Show that the second fundamental form has pointwise two eigenvalues, $\kappa$ with multiplicity $1$ and $-\kappa$ with multiplicity $3$.}


\newpage
{\bf Exercise 4}\\
Let $M\subset\wihat{M}$ be a submanifold of the Riemannian manifold $\wihat{M}$.
It is called totally geodesic iff $\II\equiv 0$.
\ben
\item Show that $M$ is a totally geodesic iff 
every geodesic of $M$ is also a geodesic of $\hat{M}$.
\item Assume additionally that $M$ is complete. Show that $M$ is totally geodesic iff every geodesic $\gamma:I\to\hat{M}$, $0\in I$, of $\hat{M}$ with 
$\dot{\gamma}(0)\in TM$ is contained in $M$.
\een

\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 28.1.2013} vor der Vorlesung}.
\end{document}



