Now in this chapter we have discussed only non-Euclidean geometries of two dimensions. Riemann’s geometry and Lobatchewski’s geometry are more particularly these same geometries, extended to the case of three dimensions. We can easily see how Riemann’s geometry of three dimensions would arise.
We have only to suppose that, as contrasted with Euclidean solids, all material bodies expand as did the shadows when displaced from a fixed centre. But the Riemannian observer of course would have no realisation of this expansion, owing to the modified laws of light propagation and owing to his own body’s expanding in company with all other bodies. So he would be of the opinion that all these bodies maintained their same shape and size, that is, remained congruent when displaced. In fact, if he applied the test of practical congruence he would find that these bodies which the Euclidean observer would claim were expanding were, so far as he himself was concerned, always the same both in visual appearance and as regards exploration by the sense of touch. If our Riemannian being were to view bodies which the Euclidean man considered rigid and undeformable, he would assert without hesitation that those bodies were decreasing in size as they approached a certain point. Once again maintenance of shape and size is essentially relative.
All the conclusions which we reached when discussing a Riemannian world of two dimensions can be extended to the case of three dimensions. Thus, a Riemannian universe or spherical space of three dimensions is finite but unbounded. Travelling for a sufficient length of time along a geodesic or straight line, we should return finally to our starting point, and rays of light which follow geodesics would circle round our space.
Just as in the case of two-dimensional geometry, we may also conceive of a three-dimensional non-Euclidean space as of constant or variable curvature. Prior to Einstein’s discoveries mathematicians concerned themselves more especially with the homogeneous types of geometry, because it was assumed (notwithstanding Riemann’s premonitions) that whatever the practical geometry of real space might turn out to be, it would always remain the same throughout, there being no palpable reason for it to vary here from there. But with the advent of the material or metrical field, conditioned by a more or less capricious distribution of matter, the possibility of a variation in the geometry of space from place to place had to be taken into consideration, and it then became impossible to limit our analysis to spaces of constant curvature. It has been proved that the variable non-Euclideanism of the space surrounding matter in Einstein’s theory—say, over the equatorial plane—would be represented by the geometry of a surface of variable curvature, which turns out to be that of a paraboloid of revolution.
There is one additional aspect of Riemann’s geometry which it may be of interest to mention on account of its possible bearing on the shape of the universe in Einstein’s theory. We refer to elliptical space. The type of Riemann’s geometry which we have discussed so far is known as Riemann’s spherical geometry, because in the case of two dimensions it turns out to be the Euclidean geometry of a spherical surface. But there exists another type of Riemannian geometry discovered by Klein. It is called elliptical space, though it has nothing to do with the surface of an ellipsoid. It corresponds to exactly the same geometry as that of Riemann proper, the only difference consisting in the connectivity of the space. That is to say, the paths of continuous passage from a point
to a point
are different in the two spaces.
In a similar way the geometry of the cylinder is Euclidean, as is that of the plane (since a plane sheet of paper can be rolled round a cylinder); but its connectivity is different since we can go round the cylinder by following a geodesic and yet return to our starting point.