Can the definition of an holonomic basis be done point by point?
Namely, instead of taking the commutator of two vector fields (local basis) $[e_i,e_j]=0$, is it equivalent to state that the commutator of two vectors must vanish at every point of the manifold $[e^p_i,e^p_j]=0$ (where $\{e^p_i\}$ is an ordered basis of the tangent space $T_pM$)?