Editing Vector bundle (section)

==Additional structures and generalizations==
Vector bundles are often given more structure. For instance, vector bundles may be equipped with a [[metric (vector bundle)|vector bundle metric]]. Usually this metric is required to be [[definite bilinear form|positive definite]], in which case each fibre of ''E'' becomes a [[Euclidean space]]. A vector bundle with a [[Linear complex structure|complex structure]] corresponds to a [[complex vector bundle]], which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings be [[Complex linear structure|complex-linear]] in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resulting [[reduction of the structure group of a bundle]]. Vector bundles over more general [[topological field]]s may also be used.

If instead of a finite-dimensional vector space, the fiber ''F'' is taken to be a [[Banach space]] then a '''[[Banach bundle]]''' is obtained.{{sfn|Lang|1995}} Specifically, one must require that the local trivializations are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions
:<math>g_{UV} \colon U\cap V \to \operatorname{GL}(F)</math>
are continuous mappings of [[Banach manifold]]s.  In the corresponding theory for C<sup>''p''</sup> bundles, all mappings are required to be C<sup>''p''</sup>.

Vector bundles are special [[fiber bundle]]s, those whose fibers are vector spaces and whose cocycle respects the vector space structure.  More general fiber bundles can be constructed in which the fiber may have other structures; for example [[sphere bundle]]s are fibered by spheres.