Whether an algebraic or a geometric or a phenomenological prescription is applied, the first fundamental form is unambiguously related to the modeling of the curved spacetime. Accordingly, we assume that the possible quantization of the first fundamental form could be proposed. For precise

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Whether an algebraic or a geometric or a phenomenological prescription is applied, the first fundamental form is unambiguously related to the modeling of the curved spacetime. Accordingly, we assume that the possible quantization of the first fundamental form could be proposed. For precise accurate measurement of the first fundamental form

$d{s}^{2}={g}_{\mu \nu}d{x}^{\mu}d{x}^{\nu}$, the author derived a quantum-induced revision of the fundamental tensor. To this end, the four-dimensional Riemann manifold is extended to the eight-dimensional Finsler manifold, in which the quadratic restriction on the length measure is relaxed, especially in the relativistic regime; the minimum measurable length could be imposed ad hoc on the Finsler structure. The present script introduces an approach to quantize the fundamental tensor and first fundamental form. Based on gravitized quantum mechanics, the resulting relativistic generalized uncertainty principle (RGUP) is directly imposed on the Finsler structure,

$F({\widehat{x}}_{0}^{\mu},{\widehat{p}}_{0}^{\nu})$, which is obviously homogeneous to one degree in

${\widehat{p}}_{0}^{\mu}$. The momentum of a test particle with mass

$\overline{m}=m/{m}_{\mathtt{p}}$ with

${m}_{\mathtt{p}}$ is the Planck mass. This unambiguously results in the quantized first fundamental form

$d{\tilde{s}}^{2}=[1+(1+2\beta {\widehat{p}}_{0}^{\rho}{\widehat{p}}_{0\rho}){\overline{m}}^{2}\left(\right|\ddot{x}{|/\mathcal{A})}^{2}]{g}_{\mu \nu}d{\widehat{x}}^{\mu}d{\widehat{x}}^{\nu}$, where

$\ddot{x}$ is the proper spacelike four-acceleration,

$\mathcal{A}$ is the maximal proper acceleration, and

$\beta $ is the RGUP parameter. We conclude that an additional source of curvature associated with the mass

$\overline{m}$, whose test particle is accelerated at

$|\ddot{x}|$, apparently emerges. Thereby, quantizations of the fundamental tensor and first fundamental form are feasible.

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