Submersions, Immersions, and Embeddings

Abstract

In this chapter we study three classes of smooth maps whose local behavior is accurately modeled by the behavior of their differentials: smooth submersions (whose differentials are surjective everywhere), smooth immersions (whose differentials are injective everywhere), and smooth embeddings (injective smooth immersions that are also homeomorphisms onto their images). Smooth immersions and embeddings, as we will see in the next chapter, are essential ingredients in the theory of submanifolds, while smooth submersions play a role in smooth manifold theory closely analogous to the role played by quotient maps in topology. The engine that powers this discussion is the rank theorem, a corollary of the inverse function theorem, which we prove in the first section of the chapter. Then we delve more deeply into smooth embeddings and smooth submersions, and apply the theory to a particularly useful class of smooth submersions, the smooth covering maps.