User:Silly rabbit/Operations on vector bundles

In the mathematical fields of topology and differential geometry, many constructions from vector spaces can be defined for vector bundles as well by performing the operation on the fibers and then gluing together the fibers.

Most operations on vector spaces can be extended to vector bundles by performing the vector space operation fiberwise.

For example, if E is a vector bundle over X, then the there is a bundle E^* over X, called the dual bundle, whose fiber at x∈X is the dual vector space (E_x)^*. Formally E^* can be defined as the set of pairs (x,φ), where x∈X and φ∈(E_x)^*. The dual bundle is locally trivial because the dual space of the inverse of a local trivialization of E is a local trivialization of E^*: the key point here, is that the operation of taking the dual vector space is functorial.

There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles E, F on X (over the given field). A few examples follow.

• The Whitney sum or direct sum bundle of E and F is a vector bundle $E\oplus F$ over X whose fiber over x is the direct sum $E_{x}\oplus F_{x}$ of the vector spaces E_x and F_x.

• The tensor product bundle $E\otimes F$ is defined in a similar way, using fiberwise tensor product of vector spaces.

• The Hom-bundle Hom(E,F) is a vector bundle whose fiber at x is the space of linear maps from E_x to F_x (which is often denoted Hom(E_x,F_x) or L(E_x,F_x)). The Hom-bundle is so-called (and useful) because there is a bijection between vector bundle homomorphisms from E to F over X and sections of Hom(E,F) over X.

An operation of a different nature is the pullback bundle construction. Given a vector bundle E → Y and a continuous map f : X → Y one can "pull back" E to a vector bundle f*E over X. The fiber over a point x ∈ X is essentially just the fiber over f(x) ∈ Y.