Integration on Manifolds

Abstract

We shall consider integration of p-forms over differentiable singular p-chains in n-dimensional manifolds, and integration of n-forms over regular domains in oriented n-dimensional manifolds. For both of these situations we shall prove a version of Stokes’ theorem. This is a generalization of the Fundamental Theorem of Calculus and is undoubtedly the single most important theorem in the subject. We shall also consider integration on Riemannian manifolds and on Lie groups. Finally, we shall introduce the de Rham cohomology groups and shall prove the Poincaré lemma, from which we will conclude that the de Rham cohomology groups of Euclidean space are trivial. This lemma will be of central importance for the de Rham theorem, which is stated at the end of this chapter and proved in Chapter 5.