A vector bundle is special class of fiber bundle in which the fiber is a vector space . Technically, a little more is required; namely, if is a bundle with fiber , to be a vector bundle, all of the fibers for need to have a coherent vector space structure. One way to say this is that the "trivializations" , are fiber-for-fiber vector space isomorphisms.
One use for vector bundles is a generalization of vector functions. For instance, the tangent vectors of an -dimensional manifold are isomorphic to at a point in a coordinate chart. But the isomorphism with depends on the choice of coordinate chart. Nearby , the vector fields look like functions. To define vector fields on the whole manifold requires the tangent bundle, which is a special case of a vector bundle.
Near every point in a vector bundle, there is a trivialization. The structure of the vector bundle, as in all bundles, is that it is locally trivial. In the case of a vector bundle, the transition functions between the trivializations take values in linear invertible transformations of the fiber.
Since the element zero in is fixed by any linear transformation, the zero section always exists. By "nontrivial section," it is meant that it is not the zero section.
There are several adjectives that can specify properties of a vector bundle. A complex vector bundle has a fiber that is a complex vector space. A real vector bundle has a fiber that is a real vector space, which is the default kind of vector bundle. A line bundle has a fiber that is one dimensional.
A continuous vector bundle is a manifold with a continuous projection map . A smooth vector bundle is a smooth manifold with a smooth projection . Finally, a holomorphic vector bundle is a complex manifold with a holomorphic projection . In this last case, the fiber must be a complex vector space. So there could be a smooth complex vector bundle, but not a holomorphic real vector bundle.