Parallelizable manifold

In mathematics, a differentiable manifold ${\displaystyle M}$ of dimension n is called parallelizable if there exist smooth vector fields

on the manifold, such that at every point ${\displaystyle p}$ of ${\displaystyle M}$ the tangent vectors

provide a basis of the tangent space at ${\displaystyle p}$. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on ${\displaystyle M.}$

A particular choice of such a basis of vector fields on ${\displaystyle M}$ is called a parallelization (or an absolute parallelism) of ${\displaystyle M}$.