Topology/Vector Bundles

A vector bundle is, broadly speaking, a family of vector spaces which is continuously indexed by a topological space. An important example is the tangent bundle of a manifold.

The formal definition is the following:

Definition (Vector bundle) A real vector bundle on a topological space is a space together with a continuous map with the following properties:

(1) For each , is isomorphic to

(2) is covered by open sets such that there exist homeomorphisms and is the identity on the first factor and a linear isomorphism on the second.

Replacing with , we get the definition of a complex vector bundle.

We call the total space of the vector bundle and , the base space.

One can define a smooth vector bundle as following:

and have to be smooth manifolds and every maps appearing in the previous definition have to be smooth.

As we have stated before, the tangent bundle of a smooth manifold is a smooth vector bundle.