Parallelizable manifold

Summary

A smooth manifold is parallelizable if it carries a global frame. This is a family of vector fields that form a basis of the tangent space at every point. Equivalently, its tangent bundle is trivial. Lie groups are parallelizable, while the $2$-sphere is not, by the hairy ball theorem. Among the spheres, exactly $S^0$, $S^1$, $S^3$ and $S^7$ are parallelizable, a fact tied to the real division algebras.