Dual bundle


In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces.

Definition edit

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  

Constructions and examples edit

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:

Properties edit

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.

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.

The Hom bundle   of two vector bundles is canonically isomorphic to the tensor product bundle  

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.

References edit

  • 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.