en>Sweetestbilly at 16:09, 26 January 2014

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In [[mathematics]], a '''parallelization''' <ref>{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I. | title = Tensor Analysis on Manifolds | year=1968|page=160}}</ref> of a [[manifold]] <math>M\,</math> of dimension ''n'' is a set of ''n'' global [[linearly independent]] [[vector field]]s.

==Formal definition==
Given a manifold <math>M\,</math> of dimension ''n'', a '''parallelization''' of <math>M\,</math> is a set <math>\{X_1, \dots,X_n\}</math> of ''n'' vector fields defined on ''all'' of <math>M\,</math> such that for every <math>p\in M\,</math> the set <math>\{X_1(p), \dots,X_n(p)\}</math> is a [[Basis_(mathematics)|basis]] of <math>T_pM\,</math>, where <math>T_pM\,</math> denotes the fiber over <math>p\,</math> of the [[tangent vector bundle]] <math>TM\,</math>.

A manifold is called '''parallelizable''' whenever admits a '''parallelization'''.

==Examples==
*Every [[Lie group]] is a [[parallelizable manifold]].
*The product of parallelizable [[manifold]]s is parallelizable.

==Properties==
'''Proposition'''. A manifold <math>M\,</math> is parallelizable iff there is a diffeomorphism <math>\phi \colon TM \longrightarrow M\times {\mathbb R^n}\,</math> such that the first projection of <math>\phi\,</math> is <math>\tau_{M}\colon TM \longrightarrow M\,</math> and for each <math>p\in M\,</math> the second factor—restricted to <math>T_pM\,</math>—is a linear map <math>\phi_{p} \colon T_pM \rightarrow {\mathbb R^n}\,</math>.

In other words, <math>M\,</math> is parallelizable if and only if <math>\tau_{M}\colon TM \longrightarrow M\,</math> is a trivial [[Bundle (mathematics)|bundle]]. For example suppose that <math>M\,</math> is an [[open subset]] of <math>{\mathbb R^n}\,</math>, i.e., an open submanifold of <math>{\mathbb R^n}\,</math>. Then <math>TM\,</math> is equal to <math>M\times {\mathbb R^n}\,</math>, and <math>M\,</math> is clearly parallelizable.<ref>{{harvtxt|Milnor|Stasheff|1974}}, p. 15.</ref>

==See also==
*[[Chart (topology)]]
*[[Differentiable manifold]]
* [[Frame bundle]]
* [[Orthonormal frame bundle]]
* [[Principal bundle]]
* [[Connection (mathematics)]]
* [[G-structure]]
* [[Web (differential geometry)]]

==Notes==
{{Reflist}}

== References ==
* {{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I. | title = Tensor Analysis on Manifolds| publisher=The Macmillan Company | year=1968|edition=First Dover 1980|isbn=0-486-64039-6}}
* {{citation | last1=Milnor|first1=J.W.|last2=Stasheff|first2=J.D.|author2-link=Jim Stasheff | title = Characteristic Classes| publisher=Princeton University Press | year=1974}}

[[Category:Differential geometry]]
[[Category:Fiber bundles]]
[[Category:Vector bundles]]

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en>Sweetestbilly at 16:09, 26 January 2014