Smooth Manifolds

Abstract

In this chapter, we begin by introducing the simplest type of manifolds, the topological manifolds, which are topological spaces with three special properties that encode what we mean when we say that they “locally look like ℝⁿ.” We then prove some important topological properties of manifolds that we use throughout the book. In the second section we introduce an additional structure, called a smooth structure, that can be added to a topological manifold to enable us to do calculus. Following the basic definitions, we introduce a number of examples of manifolds, so you can have something concrete in mind as you read the general theory. At the end of the chapter we introduce the concept of a smooth manifold with boundary, an important generalization of smooth manifolds that will have numerous applications throughout the book.