

%
%
%  Final version, 2nd May 2008.
%
%


% Shell for LaTeX2e MSc exam scripts
% TSE 25/3/02
\documentclass[12pt]{article}


% ----------------------------------------------------------------------------
% Below are needed for special fonts and symbols.  amsfonts
% package is not sufficient for the bold symbols but is for the
% general symbols including the blackboard bold for the set of reals etc.

%\usepackage{amsmath}
\usepackage{amssymb}
%\usepackage{amsfonts}

% ----------------------------------------------------------------------------






% ----------------------------------------------------------------------------
% --- Package to display keys.  Use this to show citation and equation labels

%\usepackage{showkeys}

% ----------------------------------------------------------------------------




% ----------------------------------------------------------------------------
% --- page style, headers etc.
 \pagestyle{myheadings}
 \markboth{Differential Geometry Exam 2008,cont.}{Differential Geometry Exam 2008, cont.}
 \thispagestyle{empty}
% ----------------------------------------------------------------------------



% ---------------------------------------------------------------
% --- Increasing width and height of text
% A4 paper is 29.5cm high and 21cm wide.
% Letter paper is 28cm by 21.5cm
%\setlength{\textwidth}{15.8cm} % OK for both Letter and A4
 \setlength{\textwidth}{18cm}
 \setlength{\oddsidemargin}{-1.5cm}  %margins = 1inch +
                                 %top/odd/even-sidemargin
 \setlength{\evensidemargin}{-1.5cm} %  ditto
 \setlength{\topmargin}{-2.0cm}
 \setlength{\headheight}{0.6cm}
 \setlength{\headsep}{0.4cm}
 \setlength{\topskip}{0pt}  %see pp155 also about baselineskip
 \setlength{\textheight}{23.0cm}
 \setlength{\footskip}{0.7cm}
% ---------------------------------------------------------------



% ----------------------------------------------------------------------------
% --- Question Numbering System                                            ---
%
% see pp.175
\newcounter{qnumber}
\renewcommand{\theqnumber}{\arabic{qnumber}}

% Use this command to start a new question, then use begin{enumerate} etc within question
% see pp173 and 57
\newcommand{\nextq}{ \stepcounter{qnumber} \setcounter{equation}{0}
{\strut{} \\ \large\bf Question (\theqnumber)} \vspace{0.5cm}}

% The following changes the question part labeling
% See pp. 175 and 165-166.
\renewcommand{\theenumi}{(\theqnumber.\roman{enumi})}
\renewcommand{\theequation}{\theqnumber.\arabic{equation}}

% Use this to break question over two pages
\newcommand{\tseexamnp}{\vspace{0.5cm}{\large\bf Question
\theqnumber \mbox{ } continues on next page $\Rightarrow$}
\newpage {\bf (Question \theqnumber \mbox{ } continued)}
\vspace{0.25cm}}

% ----------------------------------------------------------------------------






\begin{document}

\begin{center} {\Large
\underline{\bf Exam}}
\\[1.5cm]
{\large\bf M.Sc.\ in Quantum Fields and Fundamental Forces}
\\[2cm]
\setlength{\unitlength}{1mm}
\begin{picture}(150,10)(0,0)
 \put(0,0){\framebox(150,10){}}
 \put(1,1){\framebox(148,8){\bf TP.2
--- Differential Geometry}}
\end{picture}
\\[3cm]
 2:00 -- 5:00, Friday May 2, 2008
\\[2.6cm]
 **********************************
\\[0.5cm]
 Answer {\bf THREE} out of the four questions.
%\\[1cm]
Use a separate booklet for each question. Make sure that each
booklet carries your name, the course title, and the number of the
question attempted.
\\[0.5cm]
 **********************************
\end{center}

%\centerline{We use the signature $(-+++)$ and the conventions
%$\hbar = 1$, $c=1$.}





% 1 *******************************************************************
\newpage

\nextq

This question concerns induced maps and Lie derivatives.

\begin{enumerate}

\item Give the definition of a \emph{differentiable map} between 2 real manifolds. Give the definition of a \emph{diffeomorphism}. Define an \emph{embedding}.

{\hspace{15cm} [5 marks]}


\item Given a manifold $\mathcal{M}$, a vector field $X  \in \mathcal{J}^1_0(\mathcal{M})$ and  a $(0,2)$ tensor field $T \in \mathcal{J}^0_2(\mathcal{M})$, give the definition of the Lie derivative $\mathcal{L}_X T$ without using coordinates. Now using coordinates explicitly compute the components of $\mathcal{L}_X T$ in the coordinate basis.

{\hspace{15cm} [7 marks]}


\item The 4 dimensional de Sitter manifold, ${dS}$, can be embedded into $\mathbb{R}^5$ with coordinates $T, X^i,W$, with $i=1,2,3$, as the hyperboloid $-T^2 + X^i X^i + W^2 = 1$. A chart $(U,\psi)$ on ${dS}$ uses the coordinates $t, x^i$ where $- \infty < x^i < \infty$ and $0 < t < \infty$ and the embedding into $\mathbb{R}^5$ is defined by
\begin{eqnarray}
T & = & \frac{1}{2 \, t} \left( 1 - t^2 - t^2 x^2 \right) \nonumber \\
W & = &  \frac{1}{2 \, t} \left( 1 + t^2 - t^2 x^2 \right) \nonumber \\
X^i & = & t \, x^i \; . \nonumber 
\end{eqnarray}
Show that this chart covers half of the hyperboloid.

{\hspace{15cm} [2 marks]}


\item
The de Sitter metric $g_{dS}$ on ${dS}$ is given by the pull-back of the Minkowski metric on $\mathbb{R}^5$, $g_{Mink} = - dT \otimes dT + dX^i \otimes dX^i + dW \otimes dW$. Explicitly give the components of $g_{dS}$ in the chart $(U,\psi)$ by computing this pull-back.

{\hspace{15cm} [6 marks]}


\item
The Lorentz group acts on $\mathbb{R}^5$ leaving the metric $g_{Mink}$ invariant. It also leaves the hyperboloid invariant and hence induces a diffeomorphism on the de Sitter manifold $dS$. Consider the Lorentz transformation,
\begin{eqnarray}
T' & = & T \cosh{\beta} + W \sinh{\beta} \nonumber \\
W' & = & W \cosh{\beta} + T \sinh{\beta}  \nonumber \\
X'^i & = & X^i \nonumber 
\end{eqnarray}
with `boost' parameter $\beta$. Give the induced diffeomorphism on $dS$ in our chart $(U,\psi)$. Give the vector field, $V$, on $dS$ that generates the infinitesimal version of this diffeomorphism. Show that $\mathcal{L}_V g_{dS} = 0$.

{\hspace{15cm} [5 marks]}

\vspace{1cm}

{\hspace{14cm}[Total 25 marks]}

\end{enumerate}


% 2 *******************************************************************
\newpage

\nextq

This question discusses manifolds and group actions.

\begin{enumerate}

\item Give the definition of a \emph{complex manifold}.

{\hspace{15cm} [5 marks]}

\item $\mathbb{CP}^n$ is the space of complex undirected lines in $\mathbb{C}^{n+1}$. Construct $\mathbb{CP}^n$ as a complex manifold by giving an Atlas and showing that it satifies the definition given in part i).

{\hspace{15cm} [5 marks]}

\item By considering the matrix group $U(n+1)$ and its action on $\mathbb{C}^{n+1}$ show that $U(n+1)$ has a transitive group action on $\mathbb{CP}^n$. 
[ You may assume the fact that for complex $k$-vectors $\mathbf{u}, \mathbf{v}$ such that $\mathbf{v}^\dagger \cdot \mathbf{v} = \mathbf{u}^\dagger \cdot \mathbf{u}$ one may always find a matrix $\mathbf{M} \in U(k)$ so that $\mathbf{v} = \mathbf{M}\cdot \mathbf{u}$. ]

{\hspace{15cm} [5 marks]}

\item Let a Lie group $G$ have a transitive action on a compact manifold $\mathcal{M}$. Take a point $p_0 \in \mathcal{M}$, and its stabilizer subgroup $H_{p_0}$. By constructing a diffeomorphism between $\mathcal{M}$ and the coset manifold $G/H_{p_0}$ show that these manifolds are in fact the same. The manifold $G/H_{p_0}$ is called a \emph{homogeneous space}.

{\hspace{15cm} [5 marks]}

\item Construct $\mathbb{CP}^n$ as a \emph{homogeneous space}. 

{\hspace{15cm} [5 marks]}


\vspace{1cm}


{\hspace{14cm} [Total 25 marks]}



\end{enumerate}



% 3 *******************************************************************
\newpage

\nextq

This question involves cohomology, electromagnetism and Hodge theory.

\begin{enumerate}

\item Define \emph{closed} and \emph{exact} forms. What is \emph{cohomology}?

{\hspace{15cm} [5 marks]}

\item For a compact $m$-dimensional Riemannian manifold, $\mathcal{M}$, give the definition of the Hodge star $\star$.
Using this we may define the inner product between 2 $r$-forms $\alpha, \beta$ as $(\alpha, \beta) \equiv \int \alpha \wedge \star \beta$. Show that this inner product is \emph{symmetric} and \emph{positive definite}. 

{\hspace{15cm} [4 marks]}

\item Give the equations of electromagnetism in the language of differential forms, in terms of the forms $F$ and $j$, the field strength and current respectively. Show $d^\dagger j = 0$.

{\hspace{15cm} [3 marks]}

\item Define \emph{co-closed}, \emph{co-exact} and \emph{harmonic} forms. Show that on a compact Riemannian manifold a harmonic form is closed and co-closed.

{\hspace{15cm} [4 marks]}

\item State the \emph{Hodge decomposition theorem} for a compact Riemannian manifold. Using this, show that there is a unique harmonic representative for cohomology.

{\hspace{15cm} [4 marks]}

\item Suppose we are given a closed 2-form $\omega$ and a current $j$ on  $\mathcal{M}$ and are instructed to find a solution of electromagnetism such that $[ F ] = [ \omega ]$. Using the Lorentz gauge show that we may reduce the problem to finding the solution of the Poisson equation $\triangle A = j$ for a 1-form $A$ where $\triangle$ is the Laplacian.

{\hspace{15cm} [5 marks]}


\vspace{1cm}


{\hspace{14cm} [Total 25 marks]}


\end{enumerate}




% 4 *******************************************************************
\newpage

\nextq

This question concerns Lie groups and Riemannian geometry.

\begin{enumerate}


\item Give the definition of a \emph{Lie group}. Define a \emph{left invariant vector field}. Define the \emph{structure constants} of a Lie group.

{\hspace{15cm} [5 marks]}


\item For a general Lie group $G$, we denote the basis of left invariant vector fields as $\{ \xi_a \}$. 
A set of one-forms, $\{ \theta^a \}$, can be constructed such that $< \theta^a , \xi_b > |_g = \delta^a_{~b}$ for all $g \in G$. Show that these one-forms satisfy,
\begin{eqnarray*}
L_g^* ( \theta^a |_g ) = \theta^a |_e ,
\end{eqnarray*}
where $L_g$ is the left action of $G$.

{\hspace{15cm} [4 marks]}

\item 
Using the coordinate free expression for the exterior derivative of a one form field $\alpha$, 
\begin{eqnarray}
d\alpha(X,Y) \equiv X[\alpha(Y)] - Y[\alpha(X)] - \alpha([X,Y]) \nonumber
\end{eqnarray}
for any vector fields $X,Y$, derive the Cartan-Maurer equation,
\begin{eqnarray*}
d \theta^c + K_{ab}^{~~c} \theta^a \wedge \theta^b = 0
\end{eqnarray*}
where you must specify what the constants $K_{ab}^{~~c}$ are. 

{\hspace{15cm} [4 marks]}


\item 
Consider a manifold $\mathcal{M}$ with metric $g$ and metric connection $\nabla$. Give a basis independent definition of the \emph{torsion tensor} and the \emph{Riemann curvature tensor}. 

{\hspace{15cm} [4 marks]}

\item
Define the connection 1-form, $\omega_a^{~b}$,  the torsion 2-form, $T^a$, and the curvature 2-form $R_a^{~b}$ on $\mathcal{M}$.
State the two \emph{Cartan structure equations} relating these quantities. 

{\hspace{15cm} [3 marks]}

\item 
Consider the metric $g = \delta_{ab} \, \theta^a \otimes \theta^b$ on the group manifold $SU(2)$ where $\{ \theta^a \}$ give a basis of left invariant one forms.  For $SU(2)$ we may choose a basis $\{ \theta^a \}$ so that $K_{ab}^{~~c} = \epsilon_{abd} \delta^{cd}$. 
Using a suitable non-coordinate basis compute the curvature 2-form in the {absence of torsion}. Show that the \emph{Ricci tensor}, $\emph{Ric}$, the $(0,2)$ tensor given by the (non-trivial) contraction of the Riemann tensor, is proportional to the metric so that $Ric = k \, g$ for some constant $k$.

{\hspace{15cm} [5 marks]}


\vspace{1cm}


{\hspace{14cm} [Total 25 marks]}



\end{enumerate}





\end{document}

