vector bundle

Vector bundles provide a rigorous geometric formulation of the concept of a vector quantity which varies smoothly from point to point on a manifold. I describe the most familiar case, namely finite-dimensional vector bundles over finite-dimensional smooth (C^∞) manifolds; this is the case you are likely to encounter first. Of course one can get considerably more exotic.

This is a hard technical definition; if you want a more intuitive description try reading about tangent bundles and differential forms first.

Let X be a smooth manifold of dimension n. A vector bundle over X is a surjective smooth mapping p, from another smooth manifold E to X, together with a collection Φ of mappings, so that

• for a fixed integer m, and every point x of X, there is an open neighborhood U of x so that Φ contains a diffeomorphism φ_U: U x R^m → p^-1(U). This means that locally (in sufficiently small neighborhoods of any point) the bundle charts Φ represent E as the Cartesian product of X with R^m. The integer m is naturally called the dimension of the vector bundle p;
• the charts φ_U are coherent in the following sense: Suppose you have φ_U and φ_V, and the intersection W = U ∩ V is nonempty. Define the bundle transition map ψ_UV: W x R^m → W x R^m to be the restriction to W x R^m of φ_V^-1 o φ_U. Then ψ_UV must be given by a smoothly varying family of linear transformations, i.e. there is a smooth mapping g_UV: W → GL(R^m) so that ψ_UV(x, v) = (x, g_UV(x) v).

Here GL(R^m) is the general linear group of the vector space R^m, that is, the group of invertible m x m matrices. It is actually enough to require that the values g_UV(x) of the transition map be linear, because requiring the bundle charts to be diffeomorphisms forces them to be invertible. The vector space R^m is called the fiber of the bundle p. Replacing GL(R^m) by another topological group G (or Lie group for the smooth category), and the fiber R^m by a Hausdorff space F on which G acts effectively (respectively, a homogeneous space F of G), yields the definition of a general fiber bundle.

The paradigmatic examples of a vector bundle are the tangent and cotangent bundles TX and T*X of a smooth manifold X. These are the bundles whose fibers at x in X are the tangent space T_xX and cotangent space T_x*X respectively. In this case the bundle charts and transition maps are induced by the derivatives of the coordinate charts on X (or their adjoints, for the cotangent bundle). (If that remark is obvious, then you understand the definition.)

A simpler geometric example is the cylinder S¹ x R, regarded as a vector bundle whose base space is the circle S¹ and whose dimension is 1. This is a trivial bundle since it is actually globally diffeomorphic to the Cartesian product of the base and fiber. If you give the bundle a half twist, so that it looks like a Moebius strip, then you get a bundle which is no longer trivial. In fact these are the only isomorphism classes of line bundles over the circle. The study of isomorphism classes of vector bundles as a topological invariant of the base space is the beginning of the subject of K-theory.

If you know some traditional multivariable calculus and want to learn about these ideas the best place to start is probably the first volume of A comprehensive introduction to differential geometry by Michael Spivak. If you already know some differential topology and want to learn more neat stuff about bundles I can recommend Differential forms in algebraic topology by Raoul Bott and Loring Tu, and Characteristic classes by John Milnor and Christopher Stasheff.