# 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 ${\displaystyle \pi :E\to X}$  is the vector bundle ${\displaystyle \pi ^{*}:E^{*}\to X}$  whose fibers are the dual spaces to the fibers of ${\displaystyle E}$ .

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

## Constructions and examples

Given a local trivialization of ${\displaystyle E}$  with transition functions ${\displaystyle t_{ij},}$  a local trivialization of ${\displaystyle E^{*}}$  is given by the same open cover of ${\displaystyle X}$  with transition functions ${\displaystyle t_{ij}^{*}=(t_{ij}^{T})^{-1}}$  (the inverse of the transpose). The dual bundle ${\displaystyle E^{*}}$  is then constructed using the fiber bundle construction theorem. As particular cases:

## Properties

If the base space ${\displaystyle X}$  is paracompact and Hausdorff then a real, finite-rank vector bundle ${\displaystyle E}$  and its dual ${\displaystyle E^{*}}$  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 ${\displaystyle E^{*}}$  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 ${\displaystyle \mathrm {Hom} (E_{1},E_{2})}$  of two vector bundles is canonically isomorphic to the tensor product bundle ${\displaystyle E_{1}^{*}\otimes E_{2}.}$

Given a morphism ${\displaystyle f:E_{1}\to E_{2}}$  of vector bundles over the same space, there is a morphism ${\displaystyle f^{*}:E_{2}^{*}\to E_{1}^{*}}$  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.

## References

• 今野, 宏 (2013). 微分幾何学. 〈現代数学への入門〉 (in Japanese). 東京: 東京大学出版会. ISBN 9784130629713.