View source for Parallelizable manifold

<!--
p0714201.png
$#A+1 = 13 n = 0
$#C+1 = 13 : ~/encyclopedia/old_files/data/P071/P.0701420 Parallelizable manifold
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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====

====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>