Let \(M\) be a \(n\)-manifold . A tuple of vector fields \((E_1, \ldots, E_n)\) on an open subset \(U\subset M\) is called local frame if \((E_1{\mid}_p, \ldots, E_n{\mid}_p)\) is a basis of \(T_pM\) for every \(p\subset U\). It is called global frame if \(U=M\).
It is called smooth local (or global) frame if each vector field \(E_i\) is smooth .