This is sometimes taken as the definition of a vector bundle.

In particular, the vector bundles need not necessarily be complex.

Any vector bundle associated to E can be given by the above construction.

More generally, one may speak of a metric in a vector bundle.

More generally, a similar approach applies for connections in any vector bundle.

The question of whether two ostensibly different vector bundles are the same can be quite hard to answer.

Vector bundles over more general topological fields may also be used.

We can also consider the category of all vector bundles over a fixed base space 'X'.

Like other characteristic classes, it measures how "twisted" the vector bundle is.

A more specific type of connection form can be constructed when the vector bundle E carries a structure group.

Properties of certain vector bundles provide information about the underlying topological space.

On the other hand, a vector bundle always has a global section, namely the zero section.

A Whitney sum is an analog of the direct product for vector bundles.

Around any generator , we now have a trivial normal 3-plane vector bundle.

A further way in which the theory has been extended is as generalized sections of a smooth vector bundle.

These are called the transition functions (or the coordinate transformations) of the vector bundle.

KO-theory is a version of this construction which considers real vector bundles.

The Euler class, in turn, relates to all other characteristic classes of vector bundles.

The space of sections of this vector bundle is the induced representation.

The class of all vector bundles together with bundle morphisms forms a category.

Why is it useful to study vector bundles ?

The Lie algebroid is now the vector bundle .

Likewise, every complex vector bundle on a manifold carries a Spin structure.

More generally, let E M be a vector bundle.

There is the following canonical short exact sequence of vector bundles over :