  Conformal symmetry and color confinement in the infrared regime of QCD are
interpreted by means of a conjectured deSitter $dS_4$ geometry of the internal
space-time of hadrons, an assumption inspired by the hypothesis on deSitter
special relativity. Within such a scenario, the interactions involving the
virtual gluon and constituent quark degrees of freedom of hadrons are deduced
from the Green functions of Laplace operators on the $dS_4$ geodesics. Then the
conformal symmetry of QCD emerges as a direct consequence of the conformal
symmetry of the $dS_4$ space-time, while the color confinement, understood as
colorlessness of hadrons, appears as a consequence of the inevitable charge
neutrality of the unique closed space-like manifold, the three dimensional
hyper-sphere $S^3$, on whose geodesics the hadron's constituents are
conjectured to reside when near rest frame.
  Mesons are now modelled as quarkish color-anticolor dipoles, whose free
quantum motions on the aforementioned $S^3$ geodesic are perturbed by a
potential generated by a gluon--anti-gluon color dipole. The potential
predicted presents itself as the color charge analogue to the "curved" Coulomb
potential, i.e. to the electric potential that defines a consistent
electrostatic theory on a hyper-spherical surface. The advantage of this method
is that it allows to establish a direct relationship of the potential
parameters to the fundamental constants of QCD. We apply the model to the
description of the spectra of the $a_1$ and $f_1$ mesons, and the pion electric
charge form factor, finding fair agreement with data.
