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**(**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_{x}X 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 S^{1} x **R**, regarded as a vector bundle whose base space is the circle S^{1} 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.