mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the dual bundle is an operation on

vector bundles In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

extending the operation of duality for

vector spaces In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and sc ...

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 cons ...

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 of the

transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...

). The dual bundle

E^*

is then constructed using the fiber bundle construction theorem. As particular cases: * The dual bundle of an

associated bundle Associated may refer to: *Associated, former name of Avon, Contra Costa County, California *Associated Hebrew Schools of Toronto, a school in Canada *Associated Newspapers, former name of DMG Media, a British publishing company See also *Associatio ...

is the bundle associated to the

dual representation In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows: : is the transpose of , that is, = for all . The dual representation ...

of the

structure group In mathematics, and particularly topology, a fiber bundle ( ''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 p ...

. * The dual bundle of the

tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...

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 ...

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 m ...

Properties

If the base space ''

X

'' is

paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...

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 or morphism 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 the ...

as vector bundles. However, just as for

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

s, there is no

natural Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part ...

choice of isomorphism unless ''

E

'' is equipped with an

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

. This is not true in the case of complex vector bundles: for example, the tautological line bundle over the

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...

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

of each linear map ''

f_x: (E_1)_x \to (E_2)_x.

'' Accordingly, the dual bundle operation defines a

contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

from the category of vector bundles and their morphisms to itself.

References

* {{DEFAULTSORT:Dual Bundle Vector bundles Geometry