Parallelization (mathematics)

In mathematics, a parallelization of a manifold formula_1 of dimension "n" is a set of "n" global linearly independent vector fields.
Formal definition.
Given a manifold formula_1 of dimension "n", a parallelization of formula_1 is a set formula_4 of "n" vector fields defined on "all" of formula_1 such that for every formula_6 the set formula_7 is a basis of formula_8, where formula_8 denotes the fiber over formula_10 of the tangent vector bundle formula_11.
A manifold is called parallelizable whenever it admits a parallelization.
Properties.
Proposition. A manifold formula_1 is parallelizable iff there is a diffeomorphism formula_13 such that the first projection of formula_14 is formula_15 and for each formula_6 the second factor—restricted to formula_8—is a linear map formula_18.
In other words, formula_1 is parallelizable if and only if formula_15 is a trivial bundle. For example, suppose that formula_1 is an open subset of formula_22, i.e., an open submanifold of formula_22. Then formula_11 is equal to formula_25, and formula_1 is clearly parallelizable.