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 $$B$$ is a space $$E$$ together with a continuous map $$p:E\rightarrow B$$ with the following properties:

(1) For each $$b\in B$$, $$p^{-1}(b)$$ is isomorphic to $$\mathbf{R}^n$$

(2) $$B$$ is covered by open sets $$U_i$$ such that there exist homeomorphisms  $$h_i:p^{-1}(U_i)\rightarrow U_i\times \mathbf{R}^n$$ and $$h_i\circ h_j^{-1} U_i\cap U_j\times \mathbf{R}^n\rightarrow U_i\cap U_j\times \mathbf{R}^n$$ is  the identity on the first factor and a linear isomorphism on the second.

Replacing $$\mathbf{R}$$ with $$\mathbf{C}$$, we get the definition of a complex vector bundle.

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

One can define a smooth vector bundle as following:

$$E$$ and $$B$$ 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.