Hmmm... let me think about this one for a while...

Okay. Let's try this:

A manifold is basically a fancy formal mathematical treatment of what we would call "space" (in the sense of the volume we occupy and move around in, not outer space)*. A Riemannian manifold is just a manifold with a bunch of properties that make it nice in a formal mathematical sense, specifically which allow you to measure things like length and volume in a sensible way. (A Riemannian manifold is actually a simpler object to understand than a general manifold, because it behaves the way we intuitively think space should behave.)

Now, a manifold can either be Euclidean or Non-Euclidean. These fancy words refer to the ancient Greek mathematician and geometer Euclid, who formulated pretty much all those rules of geometry you learnt and promptly forgot at school. The most important one for this discussion is the idea that parallel lines never meet or intersect. Pretty basic and obvious stuff, since parallel lines are always the same distance apart. If you're using Euclidean geometry, that is...

Non-Euclidean geometry involves straight lines that stay the same distance apart if you look at them here, but if you look at them a few miles, or light years, down the road, they actually get closer together (or further apart). Not because they're drawn inaccurately or curved, but because the space (or the manifold) in which they exist is itself curved.

Think of it this way: If you have a pair of parallel lines that never intersect, and you want to make them intersect a few miles away, you can either curve the lines, or leave the lines straight and curve the space they pass through. Yes, it's basically as simple as that.

Now, if you're going to all this bother of curving space, there are two different directions you can curve it. You can curve it so the parallel lines intersect; this is called positive (or elliptical) curvature. You can also curve it the other way, so the parallel lines get further apart; this is called negative (or hyperbolic) curvature.

Still with me? We're nearly there. The interesting thing about space is that it goes forever. Ah... Well, it does if it's Euclidean (i.e. those parallel lines stay the same distance apart, remember). It also does if it's hyperbolic. But if it's elliptical, something odd happens. If you follow a straight line forever, eventually you "wrap around" and come back to where you started from, from the other direction. Seriously. Just like a video game.

What this means is, if you live in a manifold which has a positive non-Euclidean curvature, space is not infinite. Because no matter how far you travel, you never get more than a certain distance away from your starting position before you start getting closer again and eventually return.

The mind-boggling thing is that all this isn't just mathematical gibberish. The space we actually live in is a Riemannian manifold. And it may well be curved. We're not sure if it is curved yet (and if so, positively or negatively), because if it is, then the curvature is so tiny that it's very difficult to measure. We can only measure it by looking at objects a really long way away. This is what some astronomers do... They make observations to try to measure the curvature of our universe.

If you ever wondered just what the heck astronomers are trying to achieve with their research funding - now you know.

* Manifolds don't have to be three-dimensional like the space we're familiar with, but that's not important here.

Evidently Jamie forgot that you need delta-pinched positive curvature to conclude that a surface is diffeomorphic to S². That is there are constants c and δ so that at every point the curvature k satisfies

0 < δc < k < c

Consider the paraboloid z = a(x²+y²), along with the obvious map from the x-y plane up to it. Define a metric on the x-y plane by pulling back the metric induced on the paraboloid by the standard Euclidean metric in R³. This has everywhere-positive curvature bounded above by 4a², but it gets arbitrarily small as you move out.

Actually, that might make the origin a good place to put the desk...

It is possible to have a manifold with positive curvature everywhere, complete, and not compact, thus infinite. The only detail is that the curvature cannot have a positive lower bound - as you go out far enough, it approaches 0; if the curvature is bounded below by some positive constant, the manifold (and its universal cover) must be compact.

Of course, the "Infinite Featureless Plane of Death" looks fairly homogeneous, implying the same curvature everywhere, but it could simply be a very slight nonzero curvature, and it could vary, imperceptibly by human standards. And why should Deaths know differential geometry?