Parallelizable manifold

A differentiable manifold $M$ of dimension $n$ is called parallelizable if there exist smooth vector fields

{V_{1}, \dots, V_{n}}

on the manifold, such that at every point $p$ of $M$ the tangent vectors

{V_{1} (p), \dots, V_{n} (p)}

provide a basis of the tangent space at $p$ . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames (the frame bundle) has a global section on $M$ .

A particular choice of such a basis of vector fields on $M$ is called a parallelization (or an absolute parallelism) of $M$ . I think that this choice gives rise to a parallel transport and therefore to a covariant derivative operator (that I guess is flat since the holonomy is null)

Important examples: the Lie groups, and some of their homogeneous spaces (see this paper).