Generalizing Coordinate Non Commutativity

In this paper, we establish and employ a local framework to the first order of Riemann's curvature tensor in order to develop the corresponding coordinate non commutativity into general manifolds. We also exploit a new translation of function at the level of quantum mechanics to show that the final correlation result of the generalized non commutativity is a mixture of the Canonical and Quadratic formalisms and does not consist only of the Lie algebraic formalism. The basic premise of this article is that the geometry of a four-dimensional pseudo Riemann manifold representing space time, is homeomorphic to Minkowski space time.