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

Here GL(Rm) is the general linear group of the vector space Rm, that is, the group of invertible m x m matrices. It is actually enough to require that the values gUV(x) of the transition map be linear, because requiring the bundle charts to be diffeomorphisms forces them to be invertible. The vector space Rm is called the fiber of the bundle p. Replacing GL(Rm) by another topological group G (or Lie group for the smooth category), and the fiber Rm 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 TxX and cotangent space Tx*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.

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