Let \(M\) be a \(n\)-manifold . A tuple of covector fields \((\epsilon_1, \ldots, \epsilon_n)\) on an open subset \(U\subset M\) is called local coframe if \((\epsilon_{\mid}_p, \ldots, \epsilon_n{\mid}_p\) is a basis of \(T_p^*M\) for every \(p\subset U\). It is called global coframe if \(U=M\).
It is called smooth local (or global) frame if each covector field \(\epsilon_i\) is smooth .