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\begin{document}

\begin{center} {\Large
\underline{\bf Exam}}
\\[1.5cm]
{\large\bf M.Sc.\ in Quantum Fields and Fundamental Forces}
\\[2cm]
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 \put(1,1){\framebox(148,8){\bf Differential Geometry}}
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\\[3cm]
 2:00 -- 5:00, Tuesday May 3, 2011
\\[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]
 **********************************
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% 1 *******************************************************************
\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 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 Use your answer to compute the Lie bracket of two left invariant vector fields, $V$ and $W$ in $GL(n,\mathbb{R})$, giving the components of $[ V, W ]$ in the above basis.  


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




\item Show that $f_\star (X_V) = X_{f_\star V}$.


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



\item  Use your answer to part iv) and the fact that $f_\star [ X_V, X_W ] = [ X_{f_\star V}, X_{f_\star W} ]$ to show how one may use your explicit calculation of the components of the Lie bracket for left invariant vector fields on $GL(n,\mathbb{R})$ to compute the structure constants of other Lie groups that may be embedded in  $GL(n,\mathbb{R})$.

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




\vspace{1cm}

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

\end{enumerate}




% 2 *******************************************************************
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\nextq


\begin{enumerate}

\item Consider a chain - when is it a \emph{cycle} and when is it a \emph{boundary}? What is \emph{homology}?


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


\item Give a simplicial complex that triangulates the 2-sphere topological space - \emph{Hint:} you may use a complex containing only \emph{four} 2-simplices.

{\hspace{14cm} [2 marks]}

\item Construct the boundary operators $\partial_2$ and $\partial_1$ as a matrices for your triangulation of $S^2$. 


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


\item Using your triangulation confirm that the boundary operator is nilpotent operator. 


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


\item Explicitly compute the Betti numbers $b_0, b_1$ and $b_2$ of $S^2$. Give a basis for the homology vector spaces in terms of your triangulation.

{\hspace{14cm} [1 marks]}


\vspace{1cm}


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



\end{enumerate}










% 3 *******************************************************************
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\nextq


\begin{enumerate}

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


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



\item Use Stereographic projection to give an Atlas on the $n$-sphere, $S^n$.


{\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 Show the matrix group $SO(n+1)$ has a \emph{transitive} action on $S^{n}$. 
[ You may assume the fact that for vectors $\mathbf{u}, \mathbf{v}$ in $\mathbb{R}^{n+1}$ such that $\mathbf{v} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{u}$ one may always find a matrix $\mathbf{M} \in SO(n+1)$ so that $\mathbf{v} = \mathbf{M}\cdot \mathbf{u}$. ]

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


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

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


\vspace{1cm}


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


\end{enumerate}






% 4 *******************************************************************
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\nextq



\begin{enumerate}


\item Using a coordinate basis define the exterior derivative $d$ of an $r$-form $\omega$. Show explicitly that $d$ is nilpotent.


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


\item Take $\mathcal{M}$ to be an $m$-dimensional Riemannian manifold. Using a coordinate basis define the Hodge star $\star$ of an $r$-form $\omega$. Show that $\star \star \omega = \pm \omega$ where you should determine the sign $\pm$ in terms of $m$, $r$.


Hint: $(\det{g_{\mu\nu}}) \epsilon^{\alpha_1 \ldots \alpha_r \alpha_{r+1} \ldots \alpha_m} \epsilon_{\alpha_1 \ldots \alpha_r \beta_{r+1} \ldots \beta_m} = r! (m-r)! \delta^{[ \alpha_{r+1}}_{~\beta_{1+r}}\delta^{ \alpha_{r+2}}_{~\beta_{r+2}} \ldots \delta^{\alpha_m]}_{~\beta_m} $



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


\item Use differential forms to write the equations of electromagnetism. Show these equations imply the current is co-closed - what does this imply physically?

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



\item Given an $(m-2)$-boundary chain $s$ we may define a quantity $Q \equiv \int_s \star F$. What condition is placed on $j$ if $Q$ vanishes?


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


\item Given a $2$-cycle $c$ we may define a quantity $Q' \equiv  \int_c  F$. Suppose $Q'$ vanishes for any choice of $c$ ...

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



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


\end{enumerate}







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