# 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 $\mathbb {R} \times X\to X.$

## 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:

## 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.