In mathematics, a parallelization, p. 160 of a manifold of dimension n is a set of n global smooth linearly independent vector fields.
Formal definition Given a manifold of dimension n, a parallelization of is a set of n smooth vector fields defined on all of such that for every the set is a basis of , where denotes the fiber over of the tangent vector bundle .
A manifold is called parallelizable whenever it admits a parallelization.
Examples Every Lie group is a parallelizable manifold. The product of parallelizable manifolds is parallelizable. Every affine space, considered as manifold, is parallelizable.
Properties Proposition. A manifold is parallelizable iff there is a diffeomorphism such that the first projection of is and for each the second factor—restricted to —is a linear map .
In other words, is parallelizable if and only if is a trivial bundle. For example, suppose that is an open subset of , i.e., an open submanifold of . Then is equal to , and is clearly parallelizable., p. 15.
See also Chart (topology) Differentiable manifold Frame bundle Orthonormal frame bundle Principal bundle Connection (mathematics) G-structure Web (differential geometry)
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