Submanifolds
Many familiar manifolds appear naturally as smooth submanifolds, which are smooth manifolds that are subsets of other smooth manifolds. As you will soon discover, the situation is quite a bit more subtle than the analogous theory of topological subspaces.
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Submanifolds
Many familiar manifolds appear naturally as subsets of other manifolds. We have already seen that open subsets of smooth manifolds can be viewed as smooth manifolds in their own right; but there are many interesting examples beyond the open ones. In this chapter we explore smooth submanifolds, which are smooth manifolds that are subsets of other smooth manifolds. As you will soon discover, the situation is quite a bit more subtle than the analogous theory of topological subspaces. We begin by defining the most important type of smooth submanifolds, called embedded submanifolds. These have the subspace topology inherited from their containing manifold, and turn out to be exactly the images of smooth embeddings. As we will see in this chapter, they are modeled locally on linear subspaces of Euclidean spaces. Because embedded submanifolds are most often presented as level sets of smooth maps, we devote some time to analyzing the conditions under which level sets are smooth submanifolds. We will see, for example, that level sets of constantrank maps (in particular, smooth submersions) are always embedded submanifolds. Next, we introduce a more general kind of submanifolds, called immersed submanifolds, which turn out to be the images of injective immersions. An immersed submanifold looks locally like an embedded one, but globally it may have a topology that is different from the subspace topology. After introducing these basic concepts, we address two crucial technical questions about submanifolds: When is it possible to restrict the domain or codomain of a smooth map to a smooth submanifold and still retain smoothness? How can we identify the tangent space to a smooth submanifold as a subspace of the tangent space of its ambient manifold? Then we show how the theory of submanifolds can be generalized to the case of submanifolds with boundary.
Embedded Submanifolds Suppose M is a smooth manifold with or without boundary. An embedded submanifold of M is a subset S M that is a manifold (without boundary) in the subspace topology, endowed with a smooth structure with respect to which the inclusion map J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, DOI 10.1007/978-1-4419-9982-5_5, © Springer Science+Business Media New York 2013
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Embedded Submanifolds
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S ,! M is a smooth embedding. Embedded submanifolds are also called regular submanifolds by some authors. If S is an embedded submanifold of M; the difference dim M dim S is called the codimension of S in M , and the containing manifold M is called the ambient manifold for S . An embedded hypersurface is an embedded submanifold of codimension 1. The empty set is an embedded submanifold of any dimension. The easiest embedded submanifolds to understand are those of codimension 0. Recall that in Example 1.26, for any smooth manifold M we defined an open submanifold of M to be any open subset with the subspace topology and with the smooth charts obtained by restricting those of M . Proposition 5.1 (Open Submanifolds). Su
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