To build a foundation for geometry we first define the structures where geometry can take place. Euclid tells us in Definition 5 of the Elements that “a surface is that which has length and breadth only.” A plane is a special kind of surface and taking this point of view a little further, we can expect that the non-Euclidean “plane” will not be like a Euclidean plane and non-Euclidean “lines” will not be the familiar lines y = mx + b of the Cartesian plane. The insight of Euclid's definition is echoed in the preceding quote of Euler – the essence of a surface is its two-dimensionality, that it is describable by two variables, length and breadth.

Our goal in developing the classical topics of differential geometry is to discover the surfaces where non-Euclidean geometry holds. On the way to this goal we examine the major themes of differential geometry which include the notions of intrinsic properties, curvature, geodesies, and abstract surfaces.

We begin with the theory of surfaces in ℝ3. This theory may be said to have been founded by Euler in his 1760 paper Recherches sur la courbure des surfaces (see Struik (1933)). It was developed extensively by the French school of geometry led by Gaspard Monge (1746–1818). We will consider the early contributions to the subject in Chapter 9. We first introduce the relevant objects, structures, and examples.