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 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'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

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 In mathematics, a transition function may refer to: * a transition map between two charts of an atlas of a manifold or other topological space * the function that defines the transitions of a state transition system in computing, which may refer mo ...

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 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 notations). The tr ...

). The dual bundle

E^*

is then constructed using the

fiber bundle construction theorem In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomo ...

. As particular cases: * The dual bundle of an

associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...

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 i ...

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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

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 In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...

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

s, there is no

natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...

choice of isomorphism unless ''

E

'' is equipped with an

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

. This is not true in the case of

complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...

s: 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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...

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 In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...

. The Hom bundle ''

\mathrm(E_1,E_2)

'' of two vector bundles is canonically isomorphic to the

tensor product bundle In differential geometry, the tensor product of vector bundles ''E'', ''F'' (over same space X) is a vector bundle, denoted by ''E'' ⊗ ''F'', whose fiber over a point x \in X is the tensor product of vector spaces ''E'x'' ⊗ ''F'x''.To co ...

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 ma ...

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

References

* {{DEFAULTSORT:Dual Bundle Vector bundles Geometry