In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces.
can be defined as the Hom bundle
that is, the vector bundle of morphisms from
to the trivial line bundle
Given a local trivialization of
with transition functions
a local trivialization of
is given by the same open cover of
with transition functions
(the inverse of the transpose).
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
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.
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.
of two vector bundles is canonically isomorphic to the tensor product bundle
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.