Parallelizable manifold

In mathematics, a differentiable manifold formula_1 of dimension "n" is called parallelizable if there exist smooth vector fields
on the manifold, such that at every point formula_3 of formula_1 the tangent vectors
provide a basis of the tangent space at formula_3. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on formula_7
A particular choice of such a basis of vector fields on formula_1 is called a parallelization (or an absolute parallelism) of formula_1.