  We present an argument which intends to explore a potential geometric origin
of a class of non-linear Fokker-Planck equations related to the mesoscopic
behavior of systems conjecturally described by the $q$-entropy. We argue that
the appearance of the non-linear term(s) in such equations can be ascribed to
the fact that the effective mesoscopic metric describing the behavior of the
underlying system may not be the originally chosen one, but a conformal
deformation of it. Motivated by Liouville's theorem, we highlight the role
played by the scalar curvature of conformally related metrics in establishing
such a non-linear Fokker-Planck equation.
