In mathematics, a parallelization of a manifold M of dimension n is a set of n global linearly independent vector fields.
Given a manifold M of dimension n, a parallelization of M is a set { X 1 , … , X n } of n vector fields defined on all of M such that for every p ∈ M the set { X 1 ( p ) , … , X n ( p ) } is a basis of T p M , where T p M denotes the fiber over p of the tangent vector bundle T M .
A manifold is called parallelizable whenever admits a parallelization.
Every Lie group is a parallelizable manifold.The product of parallelizable manifolds is parallelizable.Every affine space, considered as manifold, is parallelizable.Proposition. A manifold M is parallelizable iff there is a diffeomorphism ϕ : T M ⟶ M × R n such that the first projection of ϕ is τ M : T M ⟶ M and for each p ∈ M the second factor—restricted to T p M —is a linear map ϕ p : T p M → R n .
In other words, M is parallelizable if and only if τ M : T M ⟶ M is a trivial bundle. For example, suppose that M is an open subset of R n , i.e., an open submanifold of R n . Then T M is equal to M × R n , and M is clearly parallelizable.
