KNOWPIA
WELCOME TO KNOWPIA

In mathematics, the **dual bundle** is an operation on vector bundles extending the operation of duality for vector spaces.

The **dual bundle** of a vector bundle is the vector bundle whose fibers are the dual spaces to the fibers of .

Equivalently, can be defined as the Hom bundle * * that is, the vector bundle of morphisms from * * to the trivial line bundle * *

Given a local trivialization of * * with transition functions a local trivialization of is given by the same open cover of * * with transition functions (the inverse of the transpose). The dual bundle 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.

If the base space * * is paracompact and Hausdorff then a real, finite-rank vector bundle * * and its dual are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless * * 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 of a complex vector bundle * * is indeed isomorphic to the conjugate bundle * * but the choice of isomorphism is non-canonical unless * * is equipped with a hermitian product.

The Hom bundle * * of two vector bundles is canonically isomorphic to the tensor product bundle * *

Given a morphism * * of vector bundles over the same space, there is a morphism * * between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map * * Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself.

- 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.