Vector Bundles
In this chapter, we introduce an important generalization of tangent bundles: if M is a smooth manifold, a vector bundle over M is a collection of vector spaces, one for each point in M, glued together to form a manifold that looks locally like the Cartes
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Vector Bundles
In Chapter 3, we saw that the tangent bundle of a smooth manifold has a natural structure as a smooth manifold in its own right. The natural coordinates we constructed on TM make it look, locally, like the Cartesian product of an open subset of M with Rn . This kind of structure arises quite frequently—a collection of vector spaces, one for each point in M; glued together in a way that looks locally like the Cartesian product of M with Rn , but globally may be “twisted.” Such structures are called vector bundles, and are the main subject of this chapter. The chapter begins with the definition of vector bundles and descriptions of a few examples. The most notable example, of course, is the tangent bundle of a smooth manifold. We then go on to discuss local and global sections of vector bundles (which correspond to vector fields in the case of the tangent bundle). The chapter continues with a discussion of the natural notions of maps between bundles, called bundle homomorphisms, and subsets of vector bundles that are themselves vector bundles, called subbundles. At the end of the chapter, we briefly introduce an important generalization of vector bundles, called fiber bundles. There is a deep and extensive body of theory about vector bundles and fiber bundles on manifolds, which we cannot even touch. We introduce them primarily in order to have a convenient language for talking about the tangent bundle and structures like it; as you will see in the next few chapters, such structures exist in profusion on smooth manifolds.
Vector Bundles Let M be a topological space. A (real) vector bundle of rank k over M is a topological space E together with a surjective continuous map W E ! M satisfying the following conditions: (i) For each p 2 M; the fiber Ep D 1 .p/ over p is endowed with the structure of a k-dimensional real vector space. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, 249 DOI 10.1007/978-1-4419-9982-5_10, © Springer Science+Business Media New York 2013
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Vector Bundles
Fig. 10.1 A local trivialization of a vector bundle
(ii) For each p 2 M; there exist a neighborhood U of p in M and a homeomorphism ˚ W 1 .U / ! U Rk (called a local trivialization of E over U ), satisfying the following conditions (Fig. 10.1): U ı ˚ D (where U W U Rk ! U is the projection); for each q 2 U , the restriction of ˚ to Eq is a vector space isomorphism from Eq to fqg Rk Š Rk . If M and E are smooth manifolds with or without boundary, is a smooth map, and the local trivializations can be chosen to be diffeomorphisms, then E is called a smooth vector bundle. In this case, we call any local trivialization that is a diffeomorphism onto its image a smooth local trivialization. A rank-1 vector bundle is often called a (real) line bundle. Complex vector bundles are defined similarly, with “real vector space” replaced by “complex vector space” and Rk replaced by C k in the definition. We have no need to treat complex vector bundles in this book, so all of our
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