Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)

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.