Questions about how the curvature of universe is described with a single number I often read of physics talking about "the" curvature of the universe, giving it some letter $K$, and claiming that we have seen that $K = 0$ within such and such certainty. I have three questions: what do we mean by describing curvature with one number? I'm really only familiar with the differential geometry of surfaces, and there if we were only using one number to describe curvature it'd probably be Gaussian curvature. Is it some kind of generalization of this to a three- manifold?
My second question is as follows: we are talking about the curvature of space, right? $K$ is the curvature of some slice $t= t_0$ of spacetime, not the curvature of the whole 4-manifold, right?
My final question is: why can we talk about $K$ as some constant, or at least of constant sign? I can sort of get the assumption that $K$ not vary with time, although I'd like that explained, but the assumption that $K$ not vary sign with space (as on a torus) baffles me.
If I'm just exposing my ignorance about the definition of what $K$ actually is, that'd be good to know as well.
 A: As CuriousOne comments, the reason we can describe the curvature of the Universe with one single value of $k$ is that we assume that the cosmological principle holds — i.e. that space is homogenous and isotropic. If this is true, then the curvature must be the same everywhere, and thus there only exists three possible geometries, namely negative, positive, or zero curvature. The magnitude of the curvature parameter can be folded into the scale factor (the "size") of the Universe, so rather than any value of $k$, we can restrict ourselves to the three cases $k = -1, 0,$ or $+1$.
A torus has negative curvature on the inside and positive curvature on the outside. If, as observations suggest, the cosmological principle holds true, we don't live in a 3-torus.
As you suspect, $k$ measures the curvature of space, and does not change with time. What does change with time, is its contribution to the geometry of the Universe, since not only curvature but also energy density affects the geometry. This is described by the Friedmann equation, which gives the relation between the expansion rate $H$ of the Universe (the Hubble parameter), its total energy density $\rho_\mathrm{tot}$, and the curvature, as a function of time or, equivalently, the scale parameter:
$$
H^2 = \frac{8\pi G}{3}\rho_\mathrm{tot} - \frac{k c^2}{a^2}.
$$
Here you see that expansion is halted by high densities and positive curvature.
The energy density is the sum of several components (radiation, matter, and dark energy). For instance, for matter, the energy density in some volume is given by the total energy of matter inside that volume. But the measured volume of, say, a sphere depends on the curvature of space, just like the area inside a circle depends on whether you draw that circle on a table top, a ball, ot a saddle. This is why the "intrinsic" curvature of space affects the evolution of the Universe; volumes evolve differently for different curvatures. And since the curvature of a sphere is proportional to its radius squared, the contribution of the Universe's curvature to the geometry depends on the scale factor squared, as seen from the Friedmann equation.
