Abstract: Transport maps arising on a complete connected Riemannian manifold satisfy an associated highly-nonlinear partial differential equation known as the Monge-Ampère equation. We discuss the concept of Hessian metrics --- one of the geometric tools implemented by Eugenio Calabi in his investigation of the properties of solutions of the general Monge-Ampère equation for the n-dimensional Euclidean space --- and give an exposition of the speaker's results concerning modified Hessian pseudo-metrics. Our results generalize a result of Calabi to n-dimensional space forms of constant positive sectional curvature and lead to new lower Ricci curvature bounds.