Dual bundle

In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. 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 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). The dual bundle is then constructed using the fiber bundle construction theorem. As particular cases: The dual bundle of an associated bundle is the bundle associated to the dual representation of the structure group. The dual bundle of the tangent bundle of a differentiable manifold is its cotangent bundle. 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 (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.