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 $$\pi: E \to X$$ 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.