How does one characterize $n$-dimensional semi-Riemannian spaces of constant curvature? By "characterize," I mean giving both a definition and some insight into how the possibilities work out in low-dimensional spaces with signature $(1,n-1)$, which are the ones of interest in relativity. Googling is giving me lots of information on the Riemannian case, but not the semi-Riemannian one. I'm also having some trouble interpreting the info I find online for the Riemannian case because a lot of it is written in index-free notation, but I'm only really familiar with index-gymnastics notation. The sources that I'm finding give some criteria, but don't explain why they're valid or whether they're both necessary and sufficient.
Is the correct criterion the vanishing of the covariant derivative of the Riemann tensor, $\nabla_a R_{bcde}=0$? Is it sufficient for the covariant derivative of the Ricci tensor to vanish, $\nabla_a R_{bc}=0$? Why? Are these conditions equivalent to simply counting Killing vectors and getting $n(n+1)/2$ of them? (The WP article on the Riemannian case http://en.wikipedia.org/wiki/Constant_curvature seems to be saying this, but doesn't say why it's valid, or why $n(n+1)/2$ is the magic number.) What is the lowest $n$ for which there are constant-curvature spaces with signature $(1,n-1)$ that are not flat, and what do the possibilities look like for this $n$?