

%
%
%  Final version, 30th April 2009.
%
%


% 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 2009,cont.}{Differential Geometry Exam 2009, 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}{16cm}
 \setlength{\oddsidemargin}{-0.5cm}  %margins = 1inch +
                                 %top/odd/even-sidemargin
 \setlength{\evensidemargin}{-0.5cm} %  ditto
 \setlength{\topmargin}{-1.0cm}
 \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 Differential Geometry}}
\end{picture}
\\[3cm]
 2:00 -- 5:00, Friday May 1, 2009
\\[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

\begin{enumerate}

\item Give the definition of an \emph{integral curve} of a vector field $V$. What is a \emph{flow}? Explain how a diffeomorphism $\sigma_V(\lambda)$ is generated by $V$.

{\hspace{14cm} [5 marks]}

\item  Consider a chart where the coordinates of a point $p$ are $x^\mu(p)$ and may write a vector field, $V$, in the coordinate basis as $V = V^\mu(x) \partial / \partial x^\mu$.
Show that for small  $\lambda$, we may write the coordinates of a point $\sigma_V(\lambda) \cdot p$  as,
\begin{eqnarray*}
x^\mu\left( \sigma_V(\lambda) \cdot p \right) = x^\mu\left( p \right) + \lambda V^\mu(p) + O( \lambda^2 ) \, .
\end{eqnarray*}


{\hspace{14cm} [5 marks]}

\item Given a tensor field $g$, we may define a new tensor field $g'$ as,
\begin{eqnarray*}
g'(p) \equiv (\sigma_V(\lambda))^* \cdot g(\sigma_V(\lambda) \cdot p) \, .
\end{eqnarray*}
Show that for small $\lambda$,  $g'$ can be written in terms of $g$ using the Lie derivative, $\mathcal{L}$. Explain why we might say the tensor $g'$ is the \emph{active coordinate transformation} of $g$. 

{\hspace{14cm} [5 marks]}

\item Let $g$ be a $(0,2)$ tensor field which is given explicitly as $g = g_{\mu\nu}(x) dx^\mu \otimes dx^\nu$. Use the Lie derivative Leibniz rule and the fact that,
\begin{eqnarray*}
\mathcal{L}_V \left( dx^\mu \right) = \frac{\partial V^\mu(x)}{\partial x^\nu} dx^\nu ,
\end{eqnarray*}
to determine the components of $\mathcal{L}_V g$ in the coordinate basis.

{\hspace{14cm} [5 marks]}


\vspace{1cm}


{\hspace{13cm} [Total 20 marks]}



\end{enumerate}





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

\nextq

\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{14cm} [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-form fields, $\{ \theta^a \}$, can be constructed such that $< \theta^a , \xi_b > |_g = \delta^a_{~b}$ for all $g \in G$. Show that these one-form fields satisfy,
\begin{eqnarray*}
L_g^* ( \theta^a |_g ) = \theta^a |_e ,
\end{eqnarray*}
where $L_g$ is the left action of $G$. The $\{ \theta^a \}$ are a basis of \emph{left invariant one-forms}.

{\hspace{14cm} [4 marks]}


\item Consider the General Linear matrix group $GL(n,\mathbb{R})$. An element $g \in GL(n,\mathbb{R})$ is an $n \times n$ matrix with non-zero determinant and we may take the components of this matrix $\{ x^{ij}(g) \}$ to give coordinates on the group manifold. Using these coordinates, show that the vector field
\begin{eqnarray*}
V =  v^{ij}  x^{ki}(g) \frac{\partial}{\partial x^{kj}(g) } 
\end{eqnarray*} 
is left invariant, where $v^{ij}$ are the components of a matrix. Hence write a basis for the left invariant vector fields on $GL(n,\mathbb{R})$.


{\hspace{14cm} [6 marks]}



\item Show that a basis for the left invariant one-forms on $GL(n, \mathbb{R})$ is given by,
\begin{eqnarray*}
\theta^{ij} = x^{ik}(g^{-1}) \; d x^{kj}(g) \, .
\end{eqnarray*} 

{\hspace{14cm} [5 marks]}



\vspace{1cm}

{\hspace{13cm}[Total 20 marks]}

\end{enumerate}


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

\nextq

\begin{enumerate}

\item Define \emph{closed} and \emph{exact} differential forms and explain how these give rise to \emph{cohomology}.

{\hspace{14cm} [6 marks]}


\item Write down the equations of electromagnetism in terms the field strength 2-form $F$ and the current 1-form $j$ in the language of differential forms. What condition must $j$ satisfy?

{\hspace{14cm} [3 marks]}

\item State Poincare's Lemma. Using this show that $F$ may be written in terms of a 1-form $A$ as $F = d A$ for electromagnetism on the manifold $\mathbb{R}^n$. 

{\hspace{14cm} [5 marks]}


\item For electromagnetism on $\mathbb{R}^n$ a \emph{gauge transformation} is defined by $A \rightarrow A' = A + d \lambda$ where $\lambda$ is a function. Check that $F$ is invariant under this transformation. Show that we may always find a gauge so that $d^\dagger A = 0$ -- \emph{you may assume that we can always solve $\triangle f = \rho$ with $\triangle$ being the Laplacian and $f, \rho$ being functions}. Hence show that electromagnetism on $\mathbb{R}^n$ can be written as $\triangle A = j$.


{\hspace{14cm} [6 marks]}



\vspace{1cm}

{\hspace{13cm}[Total 20 marks]}

\end{enumerate}



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

\nextq


\begin{enumerate}


\item 
Consider a Riemannian manifold $\mathcal{M}$ with metric $g$ and connection $\nabla$. Give a definition of the \emph{torsion map} and explain how it is related to the \emph{torsion tensor}. What is meant by a \emph{metric connection}. What is the \emph{Levi-Civita connection}?

{\hspace{14cm} [7 marks]}

\item
Given a \emph{non-coordinate} basis $\{ {e}_a \}$ and its dual $\{ \theta^a \}$, define the \emph{connection 1-form}, $\omega_a^{~b}$, and  the \emph{torsion 2-form}, $T^a$, which obey the  \emph{Cartan structure equation},
\begin{eqnarray*}
T^b & = & d \theta^b + \omega_a^{~b} \wedge \theta^a \, .
\end{eqnarray*}


{\hspace{14cm} [3 marks]}


\item
Show that for a metric connection this connection 1-form is antisymmetric, so that $\omega_{ab} = - \omega_{ba}$.

{\hspace{14cm} [5 marks]}



\item
Consider a 2-dimensional Riemannian manifold with metric,
\begin{eqnarray*}
g = \frac{1}{(1+ X^2 + Y^2)^2} \left( dX\otimes dX + dY\otimes dY \right) \, ,
\end{eqnarray*}
written in a coordinate basis for coordinates $\{ X, Y \}$.  Write this metric in a non-coordinate dual basis $\{ \theta^a \}$. Consider the Levi-Civita connection of this metric and explicitly deduce its connection 1-form in the $\{X,Y\}$ coordinate basis using the antisymmetry property and Cartan structure equation discussed above. 


{\hspace{14cm} [5 marks]}

\vspace{1cm}


{\hspace{13cm} [Total 20 marks]}



\end{enumerate}



% End page *******************************************************************

 
\newpage

\markboth{}{}


.
\vspace{10cm}

\centerline{Blank page}




\end{document}



