Dual bundle

Definition

The dual bundle of a vector bundle \pi: E \to X is the vector bundle \pi^: E^ \to X whose fibers are the dual spaces to the fibers of E.

Equivalently, E^* can be defined as the Hom bundle \mathrm{Hom}(E,\mathbb{R} \times X), that is, the vector bundle of morphisms from E to the trivial line bundle \R \times X \to X.

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 t_{ij}^* = (t_{ij}^T)^{-1} (the inverse of the transpose). The dual bundle E^* 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 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 \overline{E}, but the choice of isomorphism is non-canonical unless E is equipped with a hermitian product.

The Hom bundle \mathrm{Hom}(E_1,E_2) of two vector bundles is canonically isomorphic to the tensor product bundle E_1^ \otimes E_2.*

Given a morphism f : E_1 \to E_2 of vector bundles over the same space, there is a morphism f^: E_2^* \to E_1^** between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map 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.

References