  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.
