The line element is unambiguously related to the modeling of the curved spacetime whether an algebraic or a phenomenological or a geometric prescription is applied. Accordingly, the possible quantization of the line element can be suggested. For ds²=g_{mu nu} dx^mu dx^nu, the author derived a quantum-induced revision of the fundamental tensor. To this end, a minimum measurable length was imposed (discretization) and simultaneously the four–dimensional Riemann manifold is extended to the eight–dimensional Finsler manifold (additional curvature), in which the quadratic restriction on the length measure is relaxed, especially in the relativistic regime. The line element measure given by this expression is apparently still precise and accurate. This urges the author to suggest expressing g_{mu nu} and the 1-form dx^mu as quantum operators or suggesting noncommutation relations or imposing probability distribution functions. Alternatively, a full quatization to the local geodesiccharacterizing the additional curvature was also suggested.