 Parallelizable manifold

A parallelizable manifold is a smooth $n$-manifold whose tangent bundle is trivial, equivalently one admitting $n$ smooth vector fields that form a basis of every tangent space. Every Lie group is parallelizable because a basis at the identity extends to a global frame by left translation.

A **parallelizable manifold** is a differentiable manifold that has a global frame of vector fields. This means there exists a set of smooth vector fields that span the tangent space at every point of the manifold, and these vector fields can be chosen to vary smoothly. In more formal terms, a manifold \( M \) is said to be parallelizable if there exists a smooth bundle of vector fields \( \{V_1, V_2, ...