Additional Structures and Generalizations
Vector bundles are often given more structure. For instance, vector bundles may be equipped with a vector bundle metric. Usually this metric is required to be positive definite, in which case each fibre of E becomes a Euclidean space. A vector bundle with a 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 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 fields may also be used.
If instead of a finite-dimensional vector space, if the fiber F is taken to be a Banach space then a Banach bundle is obtained. 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
are continuous mappings of Banach manifolds. In the corresponding theory for Cp bundles, all mappings are required to be Cp.
Vector bundles are special fiber bundles, 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 bundles are fibered by spheres.
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