Dual bundle

 ( Vector Bundles ) 

In mathematics, the dual bundle is an operation on Vector bundle extending the operation of duality for Vector space.

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Definition The dual bundle of a vector bundle $\backslash pi:\; E\; \backslash to\; X$ is the vector bundle $\backslash pi^*:\; E^*\; \backslash to\; X$ whose fibers are the to the fibers of $E$.

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

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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 $t\_\{ij\}^*\; =\; (t\_\{ij\}^T)^\{-1\}$ (the Inverse matrix 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.

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Properties If the base space $X$ is paracompact and Hausdorff space then a real, finite-rank vector bundle $E$ and its dual $E^*$ are isomorphic as vector bundles. However, just as for , 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 $\backslash overline\{E\},$ but the choice of isomorphism is non-canonical unless $E$ is equipped with a hermitian product.

The Hom bundle $\backslash mathrm\{Hom\}(E\_1,E\_2)$ of two vector bundles is canonically isomorphic to the tensor product bundle $E\_1^*\; \backslash otimes\; E\_2.$

Given a morphism $f\; :\; E\_1\; \backslash to\; E\_2$ of vector bundles over the same space, there is a morphism $f^*:\; E\_2^*\; \backslash to\; E\_1^*$ between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map $f\_x:\; (E\_1)\_x\; \backslash to\; (E\_2)\_x.$ Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself.

• 宏 今野 (2026). 9784130629713, 東京大学出版会. ISBN 9784130629713

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