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

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