  In recent years it has been recognized that the hyperbolic numbers (an
extension of complex numbers, defined as z=x+h*y with h*h=1 and x,y real
numbers) 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.
