Frame Bundles and Spacetimes

Abstract

An n-dimensional smooth manifold X is locally diffeomorphic to ℝⁿ and the study of such manifolds generally necessitates piecing together well-understood local information about ℝⁿ into global information about X. The basic tool for this piecing together operation is a partition of unity. In this section we will prove that these exist in abundance on any manifold and use them to establish the global existence of two useful objects that clearly always exist locally (Riemannian metrics and connection forms). The same technique will be used in Section 4.3 to obtain a convenient reformulation of the notion of orientability while, in Section 4.6, a theory of integration is constructed on any orientable manifold by piecing together Lebesgue integrals on coordinate neighborhoods.