  Using a hyperbolic complex plane, we study the realization of the underlying
hyperbolic symmetry as an internal symmetry that enables the unification of
scalar fields of cosmological and particle physics interest. Such an
unification is achieved along the universal prescriptions used in physics,
avoiding the use of concepts as Euclideanization, non-canonical Lagrangians and
hidden structures, that have appeared in other approaches. The scalar
potentials constructed within the present scheme are bounded from below, and
the realization of the spontaneous symmetry breaking of the aforementioned
noncompact symmetry is studied. The profiles of these potentials with
exact/broken hyperbolic symmetry replicate qualitative aspects of those ones
used in inflationary models, and then a detailed com\-pa\-ri\-son is made.
Moreover, the homotopy constraints of the topology induced on the corresponding
vacuum manifolds, restricts the existence of topological defects associated
with continuous symmetries, allowing only those defects associated with
discrete symmetries; the consistency of these results is contrasted with
current observational tests from the LIGO/Virgo collaboration, and terrestrial
experiments based on a synchronized network of atomic magnetometers. At the
end, the nonre\-la\-tivistic limit of the model is identified with a hyperbolic
version of the nonlinear Schr\"odinger equation.
