When considering the combination of quantum mechanics (QM) and general relativity (GR), we can extend the fundamental theory of QM by incorporating concepts such as the non-commutative Heisenberg algebra, the generalized uncertainty principle (GUP), and the integration of gravitational fields. This extension leads us to suggest a possible deformation of the metric tensor that combines the effects of QM and GR. The deformation arises from the non-commutative algebra and the maximal space-like four-acceleration, and it corresponds to curvature in an 8-dimensional spacetime tangent bundle, which is a generalization of Riemannian spacetime. By applying this concept, we can derive a deformed metric tensor that determines how the affine connection on a Riemannian manifold is affected. In this paper, I explored the symmetric properties of the deformed metric tensor, the affine connection and found out how a parallelly transported tangent vector depends on the spacelike four-acceleration that is given in the units of Length(L) where \(L=\sqrt{\frac{\hbar\cdot G}{c^ 3}}\)

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