Dual bundle
Encyclopedia
In mathematics
, the dual bundle of a vector bundle
π : E → X is a vector bundle π* : E* → X whose fibers are the dual space
s to the fibers of E. The dual bundle can be constructed using the associated bundle
construction by taking the dual representation
of the structure group.
Specifically, given a local trivialization of E with transition functions
tij, a local trivialization of E* is given by the same open cover of X with transition functions tij* = (tijT)−1 (the inverse of the transpose
). The dual bundle E* is then constructed using the fiber bundle construction theorem
.
For example, the dual to the tangent bundle
of a differentiable manifold
is the cotangent bundle
.
If the base space X is paracompact and Hausdorff
then a finite-rank vector bundle E and its dual E* are isomorphic as vector bundles. However, just as for vector space
s, there is no canonical choice of isomorphism unless E is equipped with an inner product.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the dual bundle of a vector bundle
Vector bundle
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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
π : E → X is a vector bundle π* : E* → X whose fibers are the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
s to the fibers of E. The dual bundle can be constructed using the associated bundle
Associated bundle
In mathematics, the theory of fiber bundles with a structure group G 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 with a group action of G...
construction by taking the dual representation
Dual representation
In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation is defined over the dual vector space as follows:...
of the structure group.
Specifically, given a local trivialization of E with transition functions
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
tij, a local trivialization of E* is given by the same open cover of X with transition functions tij* = (tijT)−1 (the inverse of the transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
). The dual bundle E* is then constructed using the fiber bundle construction theorem
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 isomorphic...
.
For example, the dual to the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
of a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
is the cotangent bundle
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...
.
If the base space X is paracompact and Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
then a finite-rank vector bundle E and its dual E* are isomorphic as vector bundles. However, just as for vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, there is no canonical choice of isomorphism unless E is equipped with an inner product.