Compact hyperbolic three-manifold - a question In a recent paper the authors considered a spacetime described by
$$AdS_4 \times \Sigma_3 \times \mathcal{I}_r \times S^2$$
where $\mathcal{I}_r$ is an interval of the $r$-coordinate and the two-sphere $S^2$ is fibered around the manifold $\Sigma_3$. For consistency reasons of the construction, the manifold $\Sigma_3$ has to be a compact hyperbolic three-manifold. However they do not give the metric of that subspace. I was wondering how I could write the invariant line element of a compact hyperbolic three-manifold. That is, how to write
$\begin{equation}
ds^2_{\Sigma_3} = g_{ab} dx^a dx^b = \cdots 
\end{equation}$
explicitly.
In a later section of the paper they also say that they fix the Ricci scalar of $\Sigma_3$ to be equal to minus six; $R=-6$. This is another question that I have; does this uniquely determine which compact hyperbolic thee manifold $\Sigma_3$ has to be?
Any explicit answers or suggestions to literature would be greatly appreciated. Please keep in mind that while I do have some maths background, my main discipline is theoretical physics.
 A: (This is packaged together from a series of comments.)
Hyperbolic manifolds of a fixed dimension are locally identical (not just topologically, which is true for all manifolds, but metrically). If you want a good local picture of 3-manifolds, look at the Poincaré half-space, which has a very simple model. The space is $\{ (x,y,z) \mid z > 0\}$, and the metric is:
$$ (ds)^2 = \frac{(dx)^2 + (dy)^2 + (dz)^2}{z^2}.$$
Intuitively, as one gets closer to the equator $\{z = 0\}$ distances grow. A nice way of thinking about hyperbolic geometry is Google Maps. Supposing the earth is flat (as some believe), let the $z$ coordinate be distance from the ground. The metric will then be the amount of time it takes to scroll from one point to another. As anyone who has used Google Maps can confirm, the fastest way from Marrakesh to Maine is to zoom out, pan across, and zoom back in. This is what hyperbolic geodesics look like too: distances are huge close up, so the fastest path is to back up and then re-approach.
Since the geometry is the same everywhere, what differentiates different hyperbolic manifolds is how they are glued together. For $n=2$, there is a rich gluing theory, called Teichmüller theory. For $n\geq 3$, Mostow's rigidity theorem tells us that topology determines geometry (for complete, finite-volume manifolds, and hence, in particular, compact ones).
A nice large-scale geometric property of hyperbolic manifolds is that their geodesic flow is ergodic, so if you start at a random point and start walking in a random direction, you'll very likely visit almost all of the space in a statistically nice way. The "universal" half-space model of Hyperbolic space has the feature that if you and a friend start walking from the same point in different directions, you end up far apart much more quickly than in flat, Euclidean space (this is the "exponential divergence" of geodesics in hyperbolic space). Said another way, the volume of balls in hyperbolic space grow exponentially, although this is only true locally in any given compact space, where the volume is ultimately bounded.
In general, visualizing a particular hyperbolic 3-manifold is tough. There are, however, a variety of concrete constructions that give more intuition. You can build hyperbolic 3-manifolds out of (a pair of) hyperbolic surfaces, you can build a hyperbolic 3-manifold by crossing a surface with an interval and gluing the ends with a very fancy twist (actually, a random twist works most of the time!), you can build a hyperbolic 3-manifold by cutting a link out of the $3$-dimensional sphere and gluing it back with a flourish (Dehn surgery, Lickorish-Wallace theorem), you can build a hyperbolic $3$-manifold by taking limits of the objects described above, and there are other combinatorial and algebraic constructions that I find a little harder to picture.
