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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 *******************************************************************
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\nextq



\begin{enumerate}

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

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


\item Consider the Lie group $GL(n,\mathbb{R})$ so that $g \in GL(n,\mathbb{R})$ is an $n \times n$ matrix (with non-zero determinant). We may take the matrix components $\{ x^{ij}(g) \}$ as coordinates. Using such 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 components of an $n \times n$ matrix. Use this to 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 $f_\star (X_V) = X_{f_\star V}$. Also show that $f_\star [ X_V, X_W ] = [ X_{f_\star V}, X_{f_\star W} ]$.


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



\item  Use your answer to part iv) to briefly explain how one may use your explicit calculation above 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 Define \emph{cycle} and \emph{boundary} chains? 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 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 *******************************************************************
\newpage

\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 $G/H_{p_0}$ argue that these manifolds are in fact the same. 



{\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 *******************************************************************
\newpage

\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 $(m-2)$-chain $b$ on the manifold $\mathcal{M}$ we may define a quantity $Q \equiv \int_b \star F$. Suppose $b$ is a boundary chain and that $Q$ vanishes. Then what condition is placed on $j$?


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


\item Given a $2$-chain $c$ we may define a quantity $Q' \equiv  \int_c  F$. Suppose we take $\mathcal{M} = \mathbb{R}^m$ and $c$ to be a cycle chain, then show that $Q'$ always vanishes. Suppose we removed the origin point so that $\mathcal{M} = \mathbb{R}^m - \{ 0 \}$ then would $Q'$ still vanish for all choices of $c$?

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



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


\end{enumerate}







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