Dual bundle explained

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 Hom (E, R x X), that is, the vector bundle of morphisms from E to the trivial line bundle \R x 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 ^{T)} _{ij} ^{-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

Hom (E _{1,E} _{2)} of two vector bundles is canonically isomorphic to the tensor product bundle E ^{*} _{1} ⊗ 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

• Book: 今野, 宏. ja. 微分幾何学. 東京大学出版会. 2013. 東京. 〈現代数学への入門〉. 9784130629713.