View source for Parallelizable manifold

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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====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>