Smooth surfaces

We will study − contemplate − the next simplest objects after curves, i.e. surfaces. We studied curves essentially in the plane, whereas surfaces appear in the Euclidean three-dimensional space $\mathbb{E}^3$ . However, we will see soon enough the necessity of considering abstract surfaces see Sect. V.XYZ. We didn’t encounter this problem for curves, for the only abstract curves are the line and the circle, and we can always visualize them, with their internal geometry, as situated in the plane. This impossibility of visualizing certain surfaces has already been encountered in Sect. I.7 with regard to the projective plane and in Sect. V.14 with regard to elliptic curves. We will encounter it once more in Sect. VI.4 below with regard to hyperbolic geometry. The word smooth is usual for saying differentiable, having a differential, requiring the existence of a tangent plane at the very least. In another direction there are the polyhedra, that will be treated amply in Chap. VIII.