Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.

The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x) = V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X × V over X. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles: for example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if and only if its tangent bundle is trivial.