Английская Википедия: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 <math>\pi: E \to X</math> is the vector bundle <math>\pi^*: E^* \to X</math> whose fibers are the dual spaces to the fibers of <math>E</math>.
Equivalently, <math>E^*</math> can be defined as the Hom bundle <math>\mathrm{Hom}(E,\mathbb{R} \times X),</math> that is, the vector bundle of morphisms from <math>E</math> to the trivial line bundle <math>\R \times X \to X.</math>
Constructions and examples
Given a local trivialization of <math>E</math> with transition functions <math>t_{ij},</math> a local trivialization of <math>E^*</math> is given by the same open cover of <math>X</math> with transition functions <math>t_{ij}^* = (t_{ij}^T)^{-1}</math> (the inverse of the transpose). The dual bundle <math>E^*</math> 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 <math>X</math> is paracompact and Hausdorff then a real, finite-rank vector bundle <math>E</math> and its dual <math>E^*</math> are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless <math>E</math> 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 <math>E^*</math> of a complex vector bundle <math>E</math> is indeed isomorphic to the conjugate bundle <math>\overline{E},</math> but the choice of isomorphism is non-canonical unless <math>E</math> is equipped with a hermitian product.
The Hom bundle <math>\mathrm{Hom}(E_1,E_2)</math> of two vector bundles is canonically isomorphic to the tensor product bundle <math>E_1^* \otimes E_2.</math>
Given a morphism <math>f : E_1 \to E_2</math> of vector bundles over the same space, there is a morphism <math>f^*: E_2^* \to E_1^*</math> between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map <math>f_x: (E_1)_x \to (E_2)_x.</math> Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself.
References
