We can also consider the category of all vector bundles over a fixed base space 'X'.
Any vector bundle associated to E can be given by the above construction.
Like other characteristic classes, it measures how "twisted" the vector bundle is.
More generally, one may speak of a metric in a vector bundle.
This is sometimes taken as the definition of a vector bundle.
More generally, a similar approach applies for connections in any vector bundle.
A further way in which the theory has been extended is as generalized sections of a smooth vector bundle.
Properties of certain vector bundles provide information about the underlying topological space.
The question of whether two ostensibly different vector bundles are the same can be quite hard to answer.
In particular, the vector bundles need not necessarily be complex.