Parallelizable manifold

In mathematics, a differentiable manifold ${\displaystyle M}$ of dimension n is called parallelizable^[1] if there exist smooth vector fields $${\displaystyle \{V_{1},\ldots ,V_{n}\}}$$ on the manifold, such that at every point ${\displaystyle p}$ of ${\displaystyle M}$ the tangent vectors $${\displaystyle \{V_{1}(p),\ldots ,V_{n}(p)\}}$$ provide a basis of the tangent space at ${\displaystyle p}$. Equivalently, the tangent bundle is a trivial bundle,^[2] 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 ${\displaystyle M}$.