Integration on Manifolds

Abstract

The purpose of this chapter is to generalize the theory of integration known for functions defined on open subsets of $\mathbb {R}^n$ to manifolds. As a first step, we explain how differential forms defined on an open subset of $\mathbb {R}^n$ are integrated. Then, if M is a smooth manifold of dimension n, and if ω is an n-form on M (with compact support), the integral ∫_M ω is defined by patching together the integrals defined on small-enough open subsets covering M using a partition of unity. If (U, φ) is a chart such that the support of ω is contained in U, then the pullback (φ ⁻¹)^∗ ω of ω is an n-form on $\mathbb {R}^n$, so we know how to compute its integral ∫_φ(U)(φ ⁻¹)^∗ ω. To ensure that these integrals have a consistent value on overlapping charts, we need for M to be orientable. Actually, there is a more general notion of integration on a manifold that uses densities instead differential forms, but we do not need such generality.