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 [math]\displaystyle{ \pi: E \to X }[/math] is the vector bundle [math]\displaystyle{ \pi^*: E^* \to X }[/math] whose fibers are the dual spaces to the fibers of [math]\displaystyle{ E }[/math].

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

Constructions and examples

Given a local trivialization of [math]\displaystyle{ E }[/math] with transition functions [math]\displaystyle{ t_{ij}, }[/math] a local trivialization of [math]\displaystyle{ E^* }[/math] is given by the same open cover of [math]\displaystyle{ X }[/math] with transition functions [math]\displaystyle{ t_{ij}^* = (t_{ij}^T)^{-1} }[/math] (the inverse of the transpose). The dual bundle [math]\displaystyle{ E^* }[/math] 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 [math]\displaystyle{ X }[/math] is paracompact and Hausdorff then a real, finite-rank vector bundle [math]\displaystyle{ E }[/math] and its dual [math]\displaystyle{ E^* }[/math] are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless [math]\displaystyle{ E }[/math] 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 [math]\displaystyle{ E^* }[/math] of a complex vector bundle [math]\displaystyle{ E }[/math] is indeed isomorphic to the conjugate bundle [math]\displaystyle{ \overline{E}, }[/math] but the choice of isomorphism is non-canonical unless [math]\displaystyle{ E }[/math] is equipped with a hermitian product.

The Hom bundle [math]\displaystyle{ \mathrm{Hom}(E_1,E_2) }[/math] of two vector bundles is canonically isomorphic to the tensor product bundle [math]\displaystyle{ E_1^* \otimes E_2. }[/math]

Given a morphism [math]\displaystyle{ f : E_1 \to E_2 }[/math] of vector bundles over the same space, there is a morphism [math]\displaystyle{ f^*: E_2^* \to E_1^* }[/math] between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map [math]\displaystyle{ f_x: (E_1)_x \to (E_2)_x. }[/math] 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) (in ja). 微分幾何学. 〈現代数学への入門〉. 東京: 東京大学出版会. ISBN 9784130629713.