  Information geometry is concerned with the application of differential
geometry concepts in the study of the parametric spaces of statistical models.
When the random variables are independent and identically distributed, the
underlying parametric space exhibit constant curvature, which makes the
geometry hyperbolic (negative) or spherical (positive). In this paper, we
derive closed-form expressions for the components of the first and second
fundamental forms regarding pairwise isotropic Gaussian-Markov random field
manifolds, allowing the computation of the Gaussian, mean and principal
curvatures. Computational simulations using Markov Chain Monte Carlo dynamics
indicate that a change in the sign of the Gaussian curvature is related to the
emergence of phase transitions in the field. Moreover, the curvatures are
highly asymmetrical for positive and negative displacements in the inverse
temperature parameter, suggesting the existence of irreversible geometric
properties in the parametric space along the dynamics. Furthermore, these
asymmetric changes in the curvature of the space induces an intrinsic notion of
time in the evolution of the random field.
