  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.
