Dual bundle

Constructions and examples

Given a local trivialization of ${\displaystyle E}$ with transition functions ${\displaystyle t_{ij},}$ a local trivialization of ${\displaystyle E^{\ast }}$ is given by the same open cover of ${\displaystyle X}$ with transition functions ${\displaystyle t_{ij}^{\ast }=(t_{ij}^T{)}^{-1}}$ (the inverse of the transpose). The dual bundle ${\displaystyle E^{\ast }}$ 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.

Properties

If the base space ${\displaystyle X}$ is paracompact and Hausdorff then a real, finite-rank vector bundle ${\displaystyle E}$ and its dual ${\displaystyle E^{\ast }}$ are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless ${\displaystyle E}$ 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 ${\displaystyle E^{\ast }}$ of a complex vector bundle ${\displaystyle E}$ is indeed isomorphic to the conjugate bundle ${\displaystyle \overline{E},}$ but the choice of isomorphism is non-canonical unless ${\displaystyle E}$ is equipped with a hermitian product.

The Hom bundle ${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(E_1,E_2)}$ of two vector bundles is canonically isomorphic to the tensor product bundle ${\displaystyle E_1^{\ast }\otimes E_2.}$

Given a morphism ${\displaystyle f:E_1\to E_2}$ of vector bundles over the same space, there is a morphism ${\displaystyle f^{\ast }:E_2^{\ast }\to E_1^{\ast }}$ between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map ${\displaystyle f_x:(E_1{)}_x\to (E_2{)}_x.}$ Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself.