Smooth Functions and Smooth Maps

The main reason for introducing smooth structures was to enable us to define smooth functions on manifolds and smooth maps between manifolds. In this chapter we carry out that project. We begin by defining smooth real-valued and vector-valued functions, and then generalize this to smooth maps between manifolds. We then focus our attention for a while on the special case of diffeomorphisms, which are bijective smooth maps with smooth inverses. If there is a diffeomorphism between two smooth manifolds, we say that they are diffeomorphic. The main objects of study in smooth manifold theory are properties that are invariant under diffeomorphisms. At the end of the chapter, we introduce a powerful tool for blending together locally defined smooth objects, called partitions of unity. They are used throughout smooth manifold theory for building global smooth objects out of local ones.