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