In recent years it has been recognized that the hyperbolic numbers (an extension of complex numbers, defined as z = x + h y; h2 = 1 x, y ∈R, h ∉ R) can be associated to space-time geometry as stated by the Lorentz transformations of special relativity. In this paper we show that as the complex numbers had allowed the most complete and conclusive mathematical formalization of the constant-curvature surfaces in the Euclidean space, in the same way the hyperbolic numbers allow a representation of constant-curvature surfaces with non-definite line elements (Lorentz surfaces). The results are obtained just as a consequence of the space-time symmetry stated by the Lorentz group, but, from a physical point of view, they give the right link between fields and curvature as postulated by general relativity. This mathematical formalization can open new ways for application in the studies of field theories. © Società Italiana di Fisica.

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