Topology/Vector Bundles

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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.