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 ${\pi }^{\ast }:E^{\ast }\to X$ whose fibers are the dual spaces to the fibers of $E$.

Equivalently, $E^{\ast }$ can be defined as the Hom bundle $\mathrm{H}\mathrm{o}\mathrm{m}(E,ℝ\times X),$ 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 $E^{\ast }$ is given by the same open cover of $X$ with transition functions $t_{ij}^{\ast }=(t_{ij}^T{)}^{-1}$ (the inverse of the transpose). The dual bundle $E^{\ast }$ 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 $X$ is paracompact and Hausdorff then a real, finite-rank vector bundle $E$ and its dual $E^{\ast }$ 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 $E^{\ast }$ of a complex vector bundle $E$ is indeed isomorphic to the conjugate bundle $\bar{E},$ but the choice of isomorphism is non-canonical unless $E$ is equipped with a hermitian product.

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

Given a morphism $f:E_1\to E_2$ of vector bundles over the same space, there is a morphism $f^{\ast }:E_2^{\ast }\to E_1^{\ast }$ between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map $f_x:(E_1{)}_x\to (E_2{)}_x.$ 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 日本語). 東京: 東京大学出版会. ISBN 9784130629713.