We place the sphere onto the plane, so that its South Pole is touching the surface, and then we draw lines from the North Pole to the plane. After leaving N, each line intersects one (and only one) point on the spherical surface, and one (and only one) point on the spherical plane. Every point on one on one of the two surfaces has its corresponding point on the other. As long as we don't mind making a vanishingly-small pinprick in the spherical surface at its North Pole, the two surfaces are topologically identical … we can take our pin-mark, stretch it to a finite-sized hole, and then stretch the resulting bowl-shaped surface to cover the full infinite plane.
We can also imagine this as a simple optical projection – if the sphere is a hollow transparent surface and we place a lightsource at N, then anything drawn on the sphere will project shadows onto the plane.
Einstein used the projection in his "Geometry and Experience" lecture, as an aid to visualising the idea of a closed finite universe:There's also a nice Riemann Sphere animation on YouTube, courtesy of the American Mathematical Society, and a nice image at Encyclopaedia Britannica.
Now although we don't usually want to make this sort of projection (unless we're working on something a bit abstract, like Moebius transformations), the "Riemann Sphere" projection was psychologically important for physics, because the thing was fairly easy to visualise, and because it had such far-reaching implications for geometrical physics.
Thanks to the projection, we know that any physics described in sphereland has to have an exact counterpart description in flatland, as long as we scale all our definitions to match. When we lay rulers over the surface of the sphere, rulers near the North Pole have projections onto the plane that tend towards becoming infinitely large, so the plane's surface appears (to its occupants) to be finite, just like the sphere. Similarly, a constant-speed light-pulse travelling around the sphere has a "shadow" on the plane whose speed tends to infinity as the corresponding position on the sphere approaches N . If we take objects and structures whose internal equilibrium is maintained by signals travelling at the speed of light, then as we move these objects away from S, they enlarge. So it takes us the same number of tiles to pave the infinite plane as the sphere. And to the plane's inhabitants, there's no obvious way of telling which tile is the central tile – the internal physics of the plane and sphere are precisely the same.
But the intrinsic geometry of a blank plane, on its own, is not the same as that of a sphere. We need to add something – a density-map. In order to recreate the sphere's properties , we need to either project a helpful scaling grid from the sphere onto the plane to describe how scalings need to vary across the plane's surface, or attach a value to each point on on the plane to describe the local scaling. This "density" parameter varies smoothly over the surface, so we're entitled to describe it as a field. We can then say that it's this density-field that deflects light and matter in the plane towards the region of highest density (S), by Huygens' principle. But as Newton and Einstein both pointed out, a variation in the density of an underlying medium, and the associated variation in the speed of light, can both be considered as expressions of the action of a gravitational field.
As a crude first approximation, we can say that the unscaled plane description includes a gravitational field that doesn't exist in the sphere description – and yet both descriptions are equivalent.
So ... the implication of the Riemann projection is that gravitational fields aren't absolute. We can take a physical description that works, and stretch and squash our reference-grid in weird and silly ways, and as long as we invent compensating gravitational fields that vary in sympathy with our fictitious distortions (causing space's contents to nominally stretch and squash and distort to fill exactly the same region as before), the final predictions should be identical, regardless of which grid we use.
Within a space defined by that grid, these fields are physically real. And, said Einstein, we could also run the process backwards. We can place an observer in a genuine gravitational field, and allow them freefall acceleration, and for them, that field will no longer exist in their local physics ("a freefalling observer feels no gravity"). If Eötvös' Principle (that everything falls at the same rate in a gravitational field) was right, and gravity affected everything equally, then we had to be able to produce a geometrical description of gravitational effects ... and by allowing space to be warped, we could then eliminate gravitational fields from our description as a separate effect. The background gravitational field was simply space(-time), and what we normally thought of as conventional gravity was simply the result of curvature, and of curvature-related variations in projected density.
In practice, things were a little more difficult than this: Riemann and co couldn't get their curved-space models to work using curvature in just three dimensions, so a geometrical theory of gravity had to wait until Einstein had noticed the argument for gravitational time dilation, and that it led to curvature in four dimensions.
Einstein also decided to use a "frame-based" approach, which led to some simplified geometries being cross-mapped and projected that sometimes didn't correspond to actual physics, or to the shapes that more general principles said ought to be there.
I'll deal with the topological failure of the current default version of the general theory of relativity in a future post (or two). If anyone can't wait, it's in the book.