In general relativity, Einstein used tensor calculus as mathematical descriptions of the global curvatures of spacetime, at best, these appeared positive definite. In a sense, positive definiteness is equivalent to absoluteness. This was strongly emphasized by one of the founders of tensor analysis, Tullio Levi-Civita and called his magnum opus ‘The Absolute Differential Calculus’ in 1923. The idea of positive curvature came from Riemann’s differential geometry of the closed sphere. However, other structures of non-Euclidean geometries would include Weyl’s non-metric affine geometry, Eisenhart-Veblen’s path geometry, Finsler’s generalization of Riemann-Ricci-Christoffel geometry, and as well as the hyperbolic geometries of Gauss and Lobatchevsky. It must be noted that geometries of negative curvatures are among the ones being investigated before Einstein gave his preference for positive curvature. After the experimental successes of general relativity and its modern applications in the GPS (global positioning system) tracking systems, few interests remain regarding the study of practical applications of negative curvatures. Nonetheless, a study of non-absolute or relativity of curvatures is required for complete descriptions of both global and local dynamic curvatures of spacetime. This is most important for describing the complete infinitesimal local curvature of spacetime quantization.

On the other hand, infinitesimal local curvatures have been described successfully using complex imaginary spinors. Higher dimensional spinors were used by Pauli and Dirac to describe effectively both non-relativistic and relativistic quantum mechanics. However, attempts to combine spinors and tensors, for example, Penrose’s twistors failed to give complete descriptions of spacetime quantization. Other attempts, superstring and M-theory also failed or bogged down with complex abstract representations with no hopes of eventfully giving any plausible experimental verification. Fortunately, the opened can of worms can be magically transformed into a bowl of delicious mathematically edible confetti simply by taking into consideration the connection of spacetime quantization to the topology of both negative as well as positive curvature. Where and when both positive and negative curvatures simultaneously exist, the topology ceases to be a closed sphere of zero genus but that of genus unity. Relative to either topology one appears positive and the other negative, vice versa. In addition to spacetime quantization, these can also describe mass quantization and charge quantization from first principles.