2 - Manifolds

Summary

This chapter is intended as a gentle introduction to Chapter 3. In Chapter 3 we will define schemes; by way of preparation, before we begin the technicalities, it might be helpful to take a close look at a related concept that could already be somewhat familiar, that of a manifold. The reader might be outraged, and complain that differential geometry was not listed among the prerequisites for this book. How dare I presume that the reader will find manifolds familiar?

My answer is twofold: first, this chapter was written in such a way that it should be readable, even by the reader who has never before met a manifold. And second, we do only a very minimal amount of differential geometry. In this chapter we will not go beyond the definition of a manifold, and the definition of C^k–functions on C^k–manifolds. Even a reader without much background in differential geometry might have seen this much.

The idea of a scheme, which will occupy us from Chapter 3 on, mimics that of a manifold; but to make the parallel transparent it helps to start with the right definition of a manifold. With the right definitions the formalisms really are precisely the same, not just similar. In this chapter we treat manifolds. We will start with the traditional definition of a manifold, then modify it slightly.