Summary: Let \(M\) and \(\overline M\) be \(n\)-dimensional manifolds equipped with suitable Borel probability measures \(\rho \) and \(\overline{\rho }\). For subdomains \(M\) and \(\overline M\) of \(\mathbb R^n\), Ma, Trudinger & Wang gave sufficient conditions on a transportation cost \(c\in C^{4} (M \times \overline M)\) to guarantee smoothness of the optimal map pushing \(\rho \) forward to \(\overline{\rho }\); the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost \(c\) induces on the product space \(M\times\overline M\). We also explore some connections between optimal transportation and space-like Lagrangian submanifolds in symplectic geometry.
Using the pseudo-Riemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principle characterizing it due to Loeper, relaxing his hypotheses even for subdomains \(M\) and \(\overline M\) of \(\mathbb R^n\). This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered.