Affine connections in modified general relativity with Quantum-Deformed metric tensor

Through a consistent extension of the minimal length, derived from the noncommutative Heisenberg algebra and the relativistic generalized uncertainty principle (RGUP) that integrates finite gravitational fields into quantum mechanics (QM), quantum-induced revisions are imposed on the spacetime metric along with quantum modifications to general relativity (GR). The proposed quantization scheme is conjectured to remain compatible with conventional GR by introducing additional curvatures on relativistic eight-dimensional tangent bundle, Finsler geometry that generalizes the Riemann geometry. We analyze the symmetry properties of both quantum-deformed metric and affine connections, and examine how parallel transport acquires dependence on second-order derivatives of the tangent covectors, normalized to quantities induced by quantum and gravitational effects. Our analysis shows that the quantum-induced contributions factorize linearly, such that their removal restores the classical geometric structures exactly. We conclude that the quantum-induced corrections endow both quantities with distinct quantum characteristics, particularly in the relativistic regime. At the same time, their fully quantum nature implies that the fundamental metric and the associated 1-form variations must be described by probability distributions, noncommutative structures, or quantum operators.