Computing Christoffel Symbols HELP!! (Relativity / Maths)

Could someone also explain how the got the metrics. Many thankss

Well, from the very little that I've just learnt through Google - one of the ways to get the 2-sphere is to imaging the 3-sphere then restrict it to constant radius.

Since, the 3-sphere has $ds^2 = dr^2 +r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2$ then restricting $r=R$ (the unit 2-sphere):

$ds^2 = R^2 d \theta^2 + R^2 \sin^2 \theta d\phi^2$, as a matrix, you can represent this as:

$\displaystyle g_{ij} = \begin{pmatrix} R^2 & 0 \\ 0 & R^2 \sin^2 \theta\end{pmatrix}$ which reduces to your matrix if we take $R=1$ (i.e: the unit 2-sphere)

The inverse is readily obtainable as $g^{ij} = \begin{pmatrix} \frac{1}{R^2} & 0 \\ 0 & \frac{1}{R^2 \sin^2 \theta} \end{pmatrix}$.

Although there are more intuitive ways to get this metic, such as specifying the symemtries you require, three independent rotations and then finding the 2-sphere via that. Or you could look for 2-dim spaces with constant curvature or compute the metric for a general 2-dim geometry and then impose constant curvature that gives a set of DE's which you can then use to get this metric.

That's fantastic! Could you possibly explain how you represented it as the matrix? Thanks a million for taking the time to help!!