Vector Bundle - Operations On Vector Bundles

Operations On Vector Bundles

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 (named for Hassler Whitney) or direct sum bundle of E and F is a vector bundle over X whose fiber over x is the direct sum of the vector spaces E_x and F_x.

• The tensor product bundle 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.

• The dual vector bundle E∗ is the Hom bundle Hom(E,R×X) of bundle homomorphisms of E and the trivial bundle R×X. There is a canonical vector bundle isomorphism

Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the category of vector spaces can also be performed on the category of vector bundles in a functorial manner. This is made precise in the language of smooth functors. 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. Hence, Whitney summing can be defined as the pullback bundle of the diagonal map from X to X x X where the bundle over X x X is E x F.