Chapter 4 - Smooth Manifolds

Abstract

In this chapter we first introduce the notion of a manifold, and then of a smooth structure on a manifold, a manifold together with a smooth structure on it being a smooth manifold. In fact we introduce the more general notion of a (smooth) manifold with boundary. We see how to generalize the basic notions we have introduced for regions in Rn to general smooth manifolds with boundary: the notions of tangent vectors and differential forms. We see how to do calculations with differential forms on smooth manifolds with boundary: using the notion of a pull-back, we reduce these to computations in Rn. We introduce the notion of an orientation for a smooth manifold with boundary, and, in the case of an oriented smooth manifold with boundary, the notion of the induced orientation on the boundary. We develop the technical tool of a partition of unity, and conclude by proving the converse of Poincaré’s Lemma in the case of forms defined on a smoothly contractible smooth manifold with boundary.