A differentiable manifold $M$ of dimension $n$ is called parallelizable if there exist smooth vector fields
$$ \{V_1, \ldots,V_n\} $$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.
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Author of the notes: Antonio J. Pan-Collantes
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