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.

• Any introduction to the FLRW metric should cover these issues. Have you studied/read about this online/in some book? – AccidentalFourierTransform Mar 12 '16 at 19:24
• The most physical question here is why we are talking about it as one number. That's because of the model assumption that the universe is homogeneous and isotropic. The local curvature does, of course, depend on the local matter distribution. For the cosmological solution, however, we pretend that the average curvature over sufficiently large areas of the universe is identical. This may or may not be so. Within current observational limits it's a valid assumption. Could K vary with time? Absolutely, but, again, we don't seem to have any observational evidence for that. – CuriousOne Mar 12 '16 at 20:05

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 $+1$.
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.