In
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 ...
, a parallelization of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
of dimension ''n'' is a set of ''n'' global
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
vector fields.
Formal definition
Given a manifold
of dimension ''n'', a parallelization of
is a set
of ''n'' smooth vector fields defined on ''all'' of
such that for every
the set
is a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of
, where
denotes the fiber over
of the
tangent vector bundle .
A manifold is called parallelizable whenever it admits a parallelization.
Examples
*Every
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
is a
parallelizable manifold
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields
\
on the manifold, such that at every point p of M the tangent vectors
\
provide a basis of the tangent space at p. Equi ...
.
*The product of parallelizable
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s is parallelizable.
*Every
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, considered as manifold, is parallelizable.
Properties
Proposition. A manifold
is parallelizable iff there is a diffeomorphism
such that the first projection of
is
and for each
the second factor—restricted to
—is a linear map
.
In other words,
is parallelizable if and only if
is a trivial
bundle
Bundle or Bundling may refer to:
* Bundling (packaging), the process of using straps to bundle up items
Biology
* Bundle of His, a collection of heart muscle cells specialized for electrical conduction
* Bundle of Kent, an extra conduction pat ...
. For example, suppose that
is an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of
, i.e., an open submanifold of
. Then
is equal to
, and
is clearly parallelizable.
[, p. 15.]
See also
*
Chart (topology)
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an ...
*
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 ...
*
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
*
Orthonormal frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
*
Principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
*
Connection (mathematics) In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are var ...
*
G-structure
In differential geometry, a ''G''-structure on an ''n''- manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes var ...
*
Web (differential geometry) In mathematics, a web permits an intrinsic characterization in terms of Riemannian geometry of the additive separation of variables in the Hamilton–Jacobi equation.
Formal definition
An orthogonal web on a Riemannian manifold ''(M,g)'' is a set \ ...
Notes
References
*
* {{citation , last1=Milnor, first1=J.W., last2=Stasheff, first2=J.D., author2-link=Jim Stasheff , title = Characteristic Classes, publisher=Princeton University Press , year=1974
Differential geometry
Fiber bundles
Vector bundles