Parallelization (mathematics)


In mathematics, a parallelization of a manifold of dimension n is a set of n global linearly independent vector fields.

Formal definition

Given a manifold of dimension n, a parallelization of is a set of n 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

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.