[en] We demonstrate that if the relaxation of a non-equilibrium system towards a steady-state satisfies the shortest path principle, then a covariant form of the Glansdorff-Prigogine Universal Criterion of Evolution is also satisfied. We further prove that the Glansdorff-Prigogine quantity is locally minimized when the evolution traces out a geodesic in the space of thermodynamic configurations. Physically the minimization of this term is the Minimum Rate of Dissipation Principle, which states that a thermodynamic system evolves towards a steady-state with the least possible dissipation and therefore relaxes along a geodesic