Parallelizable manifold

Remarks

Equivalently, the tangent bundle $TM$ is isomorphic, as a smooth vector bundle, to the trivial bundle $M \times \R^n$; a global frame gives the isomorphism $(p, a) \mapsto \sum_i a_i V_i(p)$. Example: the circle $S^1 \subseteq \R^2$ is parallelizable. The vector field $V(x, y) = (-y, x)$ is tangent to $S^1$ and nonzero at every point, so it is a global frame for the $1$-dimensional manifold $S^1$. Every Lie group is parallelizable, by translating a basis of the tangent space at the identity. Non-example: $S^2$ is not parallelizable. A global frame would contain a nowhere-vanishing vector field, and by the hairy ball theorem no such field exists on $S^2$. A parallelizable manifold is orientable, so the Möbius band is not parallelizable either.