{{Short description|Mathematical operation on vector bundles}} {{no footnotes|date=September 2025}} 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>\pi: E \to X</math> is the vector bundle <math>\pi^*: E^* \to X</math> whose fibers are the dual spaces to the fibers of <math>E</math>.

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

==Constructions and examples==

Given a local trivialization of ''<math>E</math>'' with transition functions <math>t_{ij},</math> a local trivialization of <math>E^*</math> is given by the same open cover of ''<math>X</math>'' with transition functions <math>t_{ij}^* = (t_{ij}^T)^{-1}</math> (the inverse of the transpose). The dual bundle <math>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>X</math>'' is paracompact and Hausdorff then a real, finite-rank vector bundle ''<math>E</math>'' and its dual <math>E^*</math> are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless ''<math>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>E^*</math> of a complex vector bundle ''<math>E</math>'' is indeed isomorphic to the conjugate bundle ''<math>\overline{E},</math>'' but the choice of isomorphism is non-canonical unless ''<math>E</math>'' is equipped with a hermitian product.

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

Given a morphism ''<math>f : E_1 \to E_2</math>'' of vector bundles over the same space, there is a morphism ''<math>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>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==

{{reflist}}

* {{cite book|last=今野|first=宏|language=ja|title=微分幾何学|publisher=東京大学出版会|year=2013|location=東京|series=〈現代数学への入門〉|isbn=9784130629713}}

{{Manifolds}}

{{DEFAULTSORT:Dual Bundle}} Category:Vector bundles Category:Geometry