Submersions, Immersions, and Embeddings
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 injecti
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Submersions, Immersions, and Embeddings
Because the differential of a smooth map is supposed to represent the “best linear approximation” to the map near a given point, we can learn a great deal about a map by studying linear-algebraic properties of its differential. The most essential property of the differential—in fact, just about the only property that can be defined independently of choices of bases—is its rank (the dimension of its image). In this chapter we undertake a detailed study of the ways in which geometric properties of smooth maps can be detected from their differentials. The maps for which differentials give good local models turn out to be the ones whose differentials have constant rank. Three categories of such maps play special roles: 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. In the first section of the chapter, we prove the rank theorem and some of its important consequences. 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.
Maps of Constant Rank The key linear-algebraic property of a linear map is its rank. In fact, as Theorem B.20 shows, the rank is the only property that distinguishes different linear maps if we are free to choose bases independently for the domain and codomain. Suppose M and N are smooth manifolds with or without boundary. Given a smooth map F W M ! N and a point p 2 M; we define the rank of F at p to be the rank of the linear map dFp W Tp M ! TF .p/ N ; it is the rank of the Jacobian matrix J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, DOI 10.1007/978-1-4419-9982-5_4, © Springer Science+Business Media New York 2013
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Submersions, Immersions, and Embeddings
of F in any smooth chart, or the dimension of Im dFp TF .p/ N . If F has the same rank r at every point, we say that it has constant rank, and write rank F D r. Because the rank of a linear map is never higher than the dimension of either its domain or its codomain (Exercise B.22), the rank of F at each point is bounded above by the minimum of fdim M; dim N g. If the rank of dFp is equal to this upper bound, we say that F has full rank at p, and if F has full rank everywhere, we say F has full rank. The most important constant-rank maps are those of full rank. A smooth map F W M ! N is called a smooth submersion if its differential is surjective at each point (or equivalently, if rank F D dim N ). It is call
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