mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the dual bundle is an operation on vector bundles extending the operation of

duality Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** Dual ...

for vector spaces.

Definition

The dual bundle of a vector bundle

\pi: E \to X

is the vector bundle

\pi^*: E^* \to X

whose fibers are the

dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...

s to the fibers of

E

. Equivalently,

E^*

can be defined as the Hom bundle ''

\mathrm(E,\mathbb \times X),

'' that is, the vector bundle of morphisms from ''

E

'' to the trivial line bundle ''

\R \times X \to X.

Constructions and examples

Given a local trivialization of ''

E

'' with transition functions

t_,

a local trivialization of

E^*

is given by the same open cover of ''

X

'' with transition functions

t_^* = (t_^T)^

(the

inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...

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 In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...

. * The dual bundle of the tangent bundle of a

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

is its

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...

Properties

If the base space ''

X

'' is paracompact and Hausdorff then a real, finite-rank vector bundle ''

E

'' and its dual

E^*

are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

as vector bundles. However, just as for vector spaces, 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 In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k-dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector ...

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 ''

\overline,

'' but the choice of isomorphism is non-canonical unless ''

E

'' is equipped with a hermitian product. The Hom bundle ''

\mathrm(E_1,E_2)

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

E_1^* \otimes E_2.

'' Given a morphism ''

f : E_1 \to E_2

'' of vector bundles over the same space, there is a morphism ''

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 \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

* {{DEFAULTSORT:Dual Bundle Vector bundles Geometry