Dual bundle

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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 π : E → X {\displaystyle \pi :E\to X} {\displaystyle \pi :E\to X} is the vector bundle π ∗ : E ∗ → X {\displaystyle \pi ^{*}:E^{*}\to X} {\displaystyle \pi ^{*}:E^{*}\to X} whose fibers are the dual spaces to the fibers of E {\displaystyle E} {\displaystyle E}.

Equivalently, E ∗ {\displaystyle E^{*}} {\displaystyle E^{*}} can be defined as the Hom bundle H o m ( E , R × X ) , {\displaystyle \mathrm {Hom} (E,\mathbb {R} \times X),} {\displaystyle \mathrm {Hom} (E,\mathbb {R} \times X),} that is, the vector bundle of morphisms from E {\displaystyle E} {\displaystyle E} to the trivial line bundle R × X → X . {\displaystyle \mathbb {R} \times X\to X.} {\displaystyle \mathbb {R} \times X\to X.}

Constructions and examples

Given a local trivialization of E {\displaystyle E} {\displaystyle E} with transition functions t i j , {\displaystyle t_{ij},} {\displaystyle t_{ij},} a local trivialization of E ∗ {\displaystyle E^{*}} {\displaystyle E^{*}} is given by the same open cover of X {\displaystyle X} {\displaystyle X} with transition functions t i j ∗ = ( t i j T ) − 1 {\displaystyle t_{ij}^{*}=(t_{ij}^{T})^{-1}} {\displaystyle t_{ij}^{*}=(t_{ij}^{T})^{-1}} (the inverse of the transpose). The dual bundle E ∗ {\displaystyle E^{*}} {\displaystyle E^{*}} is then constructed using the fiber bundle construction theorem. As particular cases:

Properties

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

The Hom bundle H o m ( E 1 , E 2 ) {\displaystyle \mathrm {Hom} (E_{1},E_{2})} {\displaystyle \mathrm {Hom} (E_{1},E_{2})} of two vector bundles is canonically isomorphic to the tensor product bundle E 1 ∗ ⊗ E 2 . {\displaystyle E_{1}^{*}\otimes E_{2}.} {\displaystyle E_{1}^{*}\otimes E_{2}.}

Given a morphism f : E 1 → E 2 {\displaystyle f:E_{1}\to E_{2}} {\displaystyle f:E_{1}\to E_{2}} of vector bundles over the same space, there is a morphism f ∗ : E 2 ∗ → E 1 ∗ {\displaystyle f^{*}:E_{2}^{*}\to E_{1}^{*}} {\displaystyle f^{*}:E_{2}^{*}\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 → ( E 2 ) x . {\displaystyle f_{x}:(E_{1})_{x}\to (E_{2})_{x}.} {\displaystyle f_{x}:(E_{1})_{x}\to (E_{2})_{x}.} 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). 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.