Abstract

The variant A3w of Ma, Trudinger and Wang's condition for regularity of optimal transportation maps is implied by the non-negativity of a pseudo-Riemannian curvature—which we call cross-curvature—induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature non-negativity is preserved for products of two manifolds; (2) both A3w and cross-curvature non-negativity are inherited by Riemannian submersions, as is domain convexity for the exponential maps; and (3) the n-dimensional round sphere satisfies cross-curvature non-negativity. From these results, a large new class of Riemannian manifolds satisfying cross-curvature non-negativity (thus A3w) is obtained, including many whose sectional curvature is far from constant. All known obstructions to the regularity of optimal maps are absent from these manifolds, making them a class for which it is natural to conjecture that regularity holds. This conjecture is confirmed for certain Riemannian submersions of the sphere such as the complex projective spaces ℂℙⁿ.