In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces.

Definition

The dual bundle of a vector bundle is the vector bundle whose fibers are the dual spaces to the fibers of E. Equivalently, E^* can be defined as the Hom bundle ' that is, the vector bundle of morphisms from E to the trivial line bundle '

Constructions and examples

Given a local trivialization of E with transition functions t_{ij}, a local trivialization of E^* is given by the same open cover of X with transition functions (the inverse of the transpose). The dual bundle E^* is then constructed using the fiber bundle construction theorem. As particular cases:

Properties

If the base space X is paracompact and Hausdorff then a real, finite-rank vector bundle E and its dual E^* are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless E is equipped with an inner product. This is not true in the case of complex vector bundles: for example, the tautological line bundle over the Riemann sphere is not isomorphic to its dual. The dual E^* of a complex vector bundle E is indeed isomorphic to the conjugate bundle but the choice of isomorphism is non-canonical unless E is equipped with a hermitian product. The Hom bundle ' of two vector bundles is canonically isomorphic to the tensor product bundle ' Given a morphism ' of vector bundles over the same space, there is a morphism ' between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself.