Non-commutativity on another Minkowski space-time: Vierbein formalism and Higgs approach

We extend the non-commutative coordinates relationship into other than the Minkowski space-time. We clarify the non-commutativity dependency to the geometrical structure. As well as, we find an inverse map between Riemann's normal and global coordinates. Furthermore, we show that behavior of the corresponding coordinates non-commutativity like as a tensor. All results summarized for the Schwarzschild metric. And they summarized for the space-time in the presence of weak gravitational waves.