Dual bundle


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.
Equivalently, can be defined as the Hom bundle ' that is, the vector bundle of morphisms from ' to the trivial line bundle ''''

Constructions and examples

Given a local trivialization of ' with transition functions a local trivialization of is given by the same open cover of ' with transition functions . The dual bundle is then constructed using the fiber bundle construction theorem. As particular cases:
If the base space ' is paracompact and Hausdorff then a real, finite-rank vector bundle ' and its dual are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless ' 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 of a complex vector bundle ' is indeed isomorphic to the conjugate bundle ' but the choice of isomorphism is non-canonical unless ' 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, 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.