This chapter should be viewed as a point of transition between the considerations of affine spaces of the first half of the book and those of the more general spaces—differentiable manifolds—of the second. The surfaces under consideration are those smooth 2-dimensional surfaces, sensible to sight and touch, of 3-dimensional Euclidean space with which everyone is familiar: sphere, cylinder, ellipsoid … In the first instance the metrical properties of such surfaces are deduced from those of the surrounding space. One of the main geometrical tasks is to formulate a definition and measure of the curvature of a surface. One such measure is the Gaussian curvature; Gauss, for whom it is named, discovered that it is in fact an intrinsic property of the surface, which is to say that it can be calculated in terms of measurements carried out entirely within the surface and without reference to the surrounding space. This is a most important result, because it renders possible the definition and study of surfaces in the abstract and, by a rather obvious process of generalisation to higher dimensions, of so-called Riemannian and pseudo-Riemannian manifolds, of which the space-times of general relativity are examples.

We shall show in this chapter how the machinery of earlier chapters is used to study the differential geometry of 2-surfaces in Euclidean 3-space, and so pave the way to the study of manifolds in later chapters.