The key condition A3w of Ma, Trudinger and Wang for regularity of optimal
transportation maps is implied by the nonnegativity 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 nonnegativity is preserved for products of two manifolds; (2)
both A3w and cross-curvature nonnegativity are inherited by Riemannian
submersions; and (3) the $n$-dimensional round sphere satisfies cross-curvature
nonnegativity. From these results, a large new class of Riemannian manifolds
satisfying cross-curvature nonnegativity (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 complex projective space CP^n.

该论文研究了优化传输映射正则性的关键条件 A3w。通过运输成本产生的伪 - 黎曼曲率 - 我们称之为交叉曲率 - 的非负性，暗示了 Ma、Trudinger 和 Wang 的 A3w。得出了满足交叉曲率非负性的一大类 Riemann 流形。这些流形缺少已知的所有妨碍最优映射正则性的障碍，因此自然而然地可以推测正则性成立。这个猜想已经在复投射空间 CP ^ n 中得到证实。