Vector Bundles

Abstract

In the preceding chapter we saw that the tangent bundle of a smooth manifold has a natural structure as a smooth manifold in its own right. The standard coordinates we constructed on TM make it look, locally, like the Cartesian product of an open subset of M with ℝⁿ. As we will see later in the book, 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 a structure is called a vector bundle.