.
In this chapter we have mentioned only the Gaussian curvature, for, as Gauss discovered, it is this type of curvature alone which characterises the geometry of the surface when explored with Euclidean measuring rods. This discovery is a direct consequence of the following considerations:
Two surfaces which have the same Gaussian curvature can be superposed on each other without being torn or stretched, whereas two surfaces which have different Gaussian curvatures can never be applied on each other unless we tear them or stretch them. Obviously the metric relations of a surface are not disturbed so long as we do not stretch the surface; and this is why surfaces which have the same Gaussian curvature and which can therefore be superposed without being stretched, must necessarily possess the same geometry.
A few illustrations may be helpful: A plane sheet of paper has a zero Gaussian curvature and a zero mean curvature at every point. A cylinder, a cone or a roll has a zero Gaussian curvature but a positive mean curvature. It is the differences in the mean curvatures that cause these surfaces to appear to us visually as differing from the plane. But owing to the fact that the plane, the cylinder, the cone and the roll have the same zero Gaussian curvature, we can wrap the plane on to the cylinder or cone without stretching it.
For this reason all these surfaces have the same Euclidean geometry (Euclidean measuring rods being used). On the other hand, a minimal surface (such as the surface defined by a soap film, stretched from wires situated in different planes) has a zero curvature just like the plane, but in contradistinction to the plane it has a negative Gaussian curvature; hence, cannot be applied to a plane, and its geometry is therefore non-Euclidean.
For the same reason a sphere, whose Gaussian curvature is a constant positive number, or that saddle-shaped surface called the pseudosphere whose Gaussian curvature is a constant negative number, can neither of them be flattened out on to a plane without being stretched. As a result their geometries differ from the Euclidean geometry of the plane. We have seen that these geometries are Riemannian and Lobatchewskian, respectively.
[30] If we were to represent non-Euclidean geometry as arising from the behaviour of our measuring rods, squirming when displaced as compared with Euclidean rods, we should see that Euclideanism in the infinitesimal implies that when our rods are of infinitesimal length and are displaced over infinitesimal distances, they behave in the same way as rigid Euclidean rods.
[31] These invariant types of curvature are given by
and