Integrably Parallelizable Manifolds

A smooth manifold ${M^n}$ is called integrably parallelizable if there exists an atlas for the smooth structure on ${M^n}$ such that all differentials in overlap between charts are equal to the identity map of the model for ${M^n}$. We show that the class of connected, integrably parallelizable, n-dimensional smooth manifolds consists precisely of the open parallelizable manifolds and manifolds diffeomorphic to the n-torus.