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| In [[mathematics]], a differentiable manifold <math>\scriptstyle M</math> of dimension ''n'' is called '''parallelizable''' <ref>{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I.|title=Tensor Analysis on Manifolds|year=1968|page=160}}</ref> if there exist [[Smooth function|smooth]] [[vector field]]s
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| :<math>\{V_1, \dots,V_n\}</math>
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| on the manifold, such that at any point <math>\scriptstyle p</math> of <math>\scriptstyle M</math> the [[tangent vector]]s
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| :<math>\{V_i(p)\}_{i\in \{1,\dots,n\}}</math>
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| provide a [[basis of a vector space|basis]] of the [[tangent space]] at <math>\scriptstyle p</math>. Equivalently, the [[tangent bundle]] is a [[trivial bundle]],<ref>{{citation | last1=Milnor|first1=J.W.|last2=Stasheff|first2=J.D.|title=Characteristic Classes|year=1974|page=15}}</ref> so that the associated [[principal bundle]] of [[frame bundle|linear frames]] has a section on <math>\scriptstyle M</math>.
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| A particular choice of such a basis of vector fields on <math>\scriptstyle M</math> is called a '''[[Parallelization (mathematics)|parallelization]]''' (or an '''absolute parallelism''') of <math>\scriptstyle M</math>.
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| ==Examples==
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| *An example with ''n'' = 1 is the [[circle]]: we can take ''V''<sub>1</sub> to be the unit tangent vector field, say pointing in the anti-clockwise direction. The [[torus]] of dimension ''n'' is also parallelizable, as can be seen by expressing it as a [[cartesian product]] of circles. For example, take ''n'' = 2, and construct a torus from a square of [[graph paper]] with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, any [[Lie group]] ''G'' is parallelizable, since a basis for the tangent space at the [[identity element]] can be moved around by the action of the translation group of ''G'' on ''G'' (any translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in ''G'').
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| *A classical problem was to determine which of the [[sphere]]s ''S''<sup>''n''</sup> are parallelizable. The zero-dimensional case ''S''<sup>0</sup> is trivially parallelizable. The case ''S''<sup>1</sup> is the circle, which is parallelizable as has already been explained. The [[hairy ball theorem]] shows that ''S''<sup>2</sup> is not parallelizable. However ''S''<sup>3</sup> is parallelizable, since it is the Lie group [[SU(2)]]. The only other parallelizable sphere is ''S''<sup>7</sup>; this was proved in 1958, by [[Michel Kervaire]], and by [[Raoul Bott]] and [[John Milnor]], in independent work. The parallelizable spheres correspond precisely to elements of unit norm in the [[normed division algebra]]s of the real numbers, complex numbers, [[quaternion]]s, and [[octonion]]s, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires [[algebraic topology]].
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| *The product of parallelizable [[manifold]]s is parallelizable.
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| *Any [[Orientability|orientable]] [[3-manifold|three-dimensional manifold]] is parallelizable.
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| ==Remarks==
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| *The term '''''framed manifold''''' (occasionally '''''rigged manifold''''') is most usually applied to an embedded manifold with a given trivialisation of the [[normal bundle]], and also for an abstract (i.e. non-embedded) manifold with a given stable trivialisation of the [[tangent bundle]].
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| ==See also== | |
| * [[Chart (topology)]]
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| * [[Differentiable manifold]]
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| * [[Frame bundle]]
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| * [[Kervaire invariant]]
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| * [[Orthonormal frame bundle]]
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| * [[Principal bundle]]
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| * [[Connection (mathematics)]]
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| * [[G-structure]]
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| ==Notes==
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| {{Reflist}}
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| == References ==
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| * {{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}}
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| * {{citation | last1=Milnor|first1=J.W.|last2=Stasheff|first2=J.D. |author2-link=Jim Stasheff| title = Characteristic Classes| publisher=Princeton University Press | year=1974}}
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| [[Category:Differential topology]]
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| [[Category:Fiber bundles]]
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| [[Category:Vector bundles]]
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| [[Category:Manifolds]]
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