Chapter I. Smooth Manifolds

Abstract

This chapter introduces just enough differential topology to serve as a suitable framework for Stokes's Theorem. The subject matter is the elementary structure of smooth manifolds, which is a topic in real analysis that sits at the intersection of algebraic topology and differential geometry.

Section 1 presents the beginning definitions and results about smooth manifolds, tangent vectors and vector fields, cotangent vectors and differential 1 forms, derivatives of smooth mappings, and differentials.

Section 2 defines the exterior algebra of a finite dimensional real vector space. Tensor algebras, which are discussed in Chapter VI of the author's Basic Algebra, are taken as known.

Section 3 introduces differential forms and their pullbacks under smooth maps. It shows how to compute pullbacks, and it establishes some properties of them.

Section 4 introduces the exterior derivative, which is the differentiation operator to be used with differential forms, and shows that it satisfies a number of properties.

Section 5 contains the construction of a smooth partition of unity, which is a device making it unnecessary in many cases to cut manifolds into pieces when treating integration problems.

Section 6 introduces the notion of an oriented smooth manifold and integration of top-degree differential forms on it. The section shows also the relationship between integration and pullback.