  Random field models are mathematical structures used in the study of
stochastic complex systems. In this paper, we compute the shape operator of
Gaussian random field manifolds using the first and second fundamental forms
(Fisher information matrices). Using Markov Chain Monte Carlo techniques, we
simulate the dynamics of these random fields and compute the Gaussian curvature
of the parametric space, analyzing how this quantity changes along phase
transitions. During the simulation, we have observed an unexpected phenomenon
that we called the \emph{curvature effect}, which indicates that a highly
asymmetric geometric deformation happens in the underlying parametric space
when there are significant increase/decrease in the system's entropy. This
asymmetric pattern relates to the emergence of hysteresis, leading to an
intrinsic arrow of time along the dynamics.
