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A [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714201.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714202.png" /> admitting a (global) field of frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714203.png" /> (cf. also [[Frame|Frame]]), that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714204.png" /> vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714205.png" /> that are linearly independent at each point. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714206.png" /> determines an isomorphism of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714207.png" /> onto the trivial bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714208.png" />, which sends a tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p0714209.png" /> to its coordinates with respect to the frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p07142010.png" /> and its origin. Therefore a parallelizable manifold can also be defined as a manifold having a trivial tangent bundle. Examples of parallelizable manifolds are open submanifolds of a Euclidean space, all three-dimensional manifolds, the space of an arbitrary Lie group, and the manifolds of frames of an arbitrary manifold. The sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p07142011.png" /> is a parallelizable manifold only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p07142012.png" />. A necessary and sufficient condition for the parallelizability of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071420/p07142013.png" />-dimensional manifold is the vanishing of the second Stiefel–Whitney [[Characteristic class|characteristic class]]. In the general case the vanishing of the second characteristic classes of Stiefel— Whitney, Chern and Pontryagin are necessary but not sufficient conditions for a manifold to be parallelizable.
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A [[Manifold|manifold]]  $  M $
 +
of dimension  $  n $
 +
admitting a (global) field of frames  $  e = ( e _ {1} \dots e _ {n} ) $(
 +
cf. also [[Frame|Frame]]), that is,  $  n $
 +
vector fields  $  e _ {1} \dots e _ {n} $
 +
that are linearly independent at each point. The field  $  e $
 +
determines an isomorphism of the tangent bundle  $  \tau :  TM \rightarrow M $
 +
onto the trivial bundle  $  \epsilon :  \mathbf R  ^ {n} \times M \rightarrow M $,
 +
which sends a tangent vector  $  v \in T _ {p} M $
 +
to its coordinates with respect to the frame  $  e \mid  _ {p} $
 +
and its origin. Therefore a parallelizable manifold can also be defined as a manifold having a trivial tangent bundle. Examples of parallelizable manifolds are open submanifolds of a Euclidean space, all three-dimensional manifolds, the space of an arbitrary Lie group, and the manifolds of frames of an arbitrary manifold. The sphere  $  S  ^ {n} $
 +
is a parallelizable manifold only for  $  n = 1, 3, 7 $.
 +
A necessary and sufficient condition for the parallelizability of a  $  4 $-
 +
dimensional manifold is the vanishing of the second Stiefel–Whitney [[Characteristic class|characteristic class]]. In the general case the vanishing of the second characteristic classes of Stiefel— Whitney, Chern and Pontryagin are necessary but not sufficient conditions for a manifold to be parallelizable.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gromov,  "Partial differential relations" , Springer  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gromov,  "Partial differential relations" , Springer  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>

Revision as of 08:05, 6 June 2020


A manifold $ M $ of dimension $ n $ admitting a (global) field of frames $ e = ( e _ {1} \dots e _ {n} ) $( cf. also Frame), that is, $ n $ vector fields $ e _ {1} \dots e _ {n} $ that are linearly independent at each point. The field $ e $ determines an isomorphism of the tangent bundle $ \tau : TM \rightarrow M $ onto the trivial bundle $ \epsilon : \mathbf R ^ {n} \times M \rightarrow M $, which sends a tangent vector $ v \in T _ {p} M $ to its coordinates with respect to the frame $ e \mid _ {p} $ and its origin. Therefore a parallelizable manifold can also be defined as a manifold having a trivial tangent bundle. Examples of parallelizable manifolds are open submanifolds of a Euclidean space, all three-dimensional manifolds, the space of an arbitrary Lie group, and the manifolds of frames of an arbitrary manifold. The sphere $ S ^ {n} $ is a parallelizable manifold only for $ n = 1, 3, 7 $. A necessary and sufficient condition for the parallelizability of a $ 4 $- dimensional manifold is the vanishing of the second Stiefel–Whitney characteristic class. In the general case the vanishing of the second characteristic classes of Stiefel— Whitney, Chern and Pontryagin are necessary but not sufficient conditions for a manifold to be parallelizable.

Comments

References

[a1] M. Gromov, "Partial differential relations" , Springer (1986) (Translated from Russian)
[a2] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
How to Cite This Entry:
Parallelizable manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallelizable_manifold&oldid=17256
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article