Vector Bundles

Abstract

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 Cartesian product of M with ℝⁿ, but globally may be “twisted.” 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). At the end of the chapter, we discuss the natural maps between bundles, called bundle homomorphisms, and subsets of vector bundles that are themselves vector bundles, called subbundles.