ABSTRACT
An important discovery of the nineteenth century was that nonorientable surfaces exist. The goal of the present chapter is to gain an understanding of such surfaces. We begin in Section 11.1 by saying precisely what it means for a regular surface to be orientable, and defining the associated Gauss map. Examples of nonorientable surfaces are first described in Section 11.2 by means of various identifications of the edges of a square. We obtain topological descriptions of some of the simpler nonorientable surfaces, without reference to an underlying space like ℝ n with n ≥ 3. The analytical theory of surfaces described in such an abstract way is developed in Chapter 26.
