Dual bundle

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

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

Constructions and examples

Given a local trivialization of ${\displaystyle E}$ with transition functions ${\displaystyle t_{ij},}$ a local trivialization of ${\displaystyle E^{*}}$ is given by the same open cover of ${\displaystyle X}$ with transition functions ${\displaystyle t_{ij}^{*}=(t_{ij}^{T})^{-1}}$ (the inverse of the transpose). The dual bundle ${\displaystyle E^{*}}$ 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.

Properties

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

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

Given a morphism ${\displaystyle f:E_{1}\to E_{2}}$ of vector bundles over the same space, there is a morphism ${\displaystyle f^{*}:E_{2}^{*}\to E_{1}^{*}}$ between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map ${\displaystyle f_{x}:(E_{1})_{x}\to (E_{2})_{x}.}$ 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.