At relativistic energies and finite magnetic fields, the noncommutative relation of distance and momentum, the Heisenberg uncertainty principle, the fundamental theory of quantum mechanics, is conjectured to get modifications. Results from various rigorous approaches to quantum gravity, such as string theory, loop quantum gravity and doubly special relativity support the generalization of the noncommutative relation of the distance and momentum operators and the emergence of a minimal measurable length. With the relativistic four-dimensional generalized uncertainty principle (RGUP) in curved spacetime and Born reciprocity principle, the distance–momentum duality symmetry, we suggest to generalize Riemann to Finsler geometry. The Finsler structure allows the direct implementation of RGUP with its quantum-mechanical nature on a free particle with mass [Formula: see text], so that the Finsler structure [Formula: see text] can be expressed as [Formula: see text], from which the quantized fundamental tensor can be deduced. We present a systematic analytic and numerical evaluation of the additional geometric structures and connections which exclusively emerged from the proposed quantization approach on three-sphere. When limiting the discussion on the Einstein tensor, we find that the emerged curvatures, i.e. additional sources of gravitation, are dominant almost everywhere on the three-sphere. The nature of those curvatures is radically distinct from the ones of the classical Einstein tensor. For instance, the additional curvatures are no longer smooth or continuous.

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The main findings of the generalized uncertainty principle (GUP), the phenomenological approach, for instance, the emergence of a minimal measurable length uncertainty, are obtained in various versions from theories of quantum gravity, such as string theory, loop quantum gravity, doubly special relativity and black hole physics. GUP counts for impacts of relativistic energies and finite gravitational fields on the fundamental theories of quantum mechanics (QM), the noncommutation and measurement uncertainty. Utilizing GUP in reconciling principles of general relativity (GR) and QM, thereby enables to draw convincing conclusions about quantum gravity. To resolve the shortcuts reported with the nonrelativistic three-dimensional GUP, namely, violation of Lorentz covariance, dependence on frame of reference, and violation of the linear additional law of momenta, we introduce relativistic four-dimensional generalized uncertainty principle (RGUP) to curved spacetime. To unify GR and QM, we apply the Born reciprocity principle (BRP), distance-momentum duality symmetry and RGUP to estimate the fundamental tensor in discretized curved spacetime. To this end, we generalize Riemann geometry. The Finsler geometry, which is characterized by manifold and Finsler structure, allows to directly apply RGUP to the Finsler structure of a free particle so that [Formula: see text] can be expressed as [Formula: see text], from which the metric tensor in discretized Riemann spacetime could be deduced. We conclude that [Formula: see text] is homogeneous with degree [Formula: see text] in [Formula: see text], while [Formula: see text] is [Formula: see text]-homogeneous resulting in [Formula: see text]. Despite, the astonishing similarity with the conformal transformation, know as Weyl tensor, this study suggests that principles of QMs could be unambiguously imposed on the resulting fundamental tensor. Also, we conclude that the features of Finsler geometry assumed in this study are likely the ones of the duel Hamilton geometry.