  This article addresses regularity of optimal transport maps for cost="squared
distance" on Riemannian manifolds that are products of arbitrarily many round
spheres with arbitrary sizes and dimensions. Such manifolds are known to be
non-negatively cross-curved [KM2]. Under boundedness and non-vanishing
assumptions on the transfered source and target densities we show that optimal
maps stay away from the cut-locus (where the cost exhibits singularity), and
obtain injectivity and continuity of optimal maps. This together with the
result of Liu, Trudinger and Wang [LTW] also implies higher regularity
(C^{1,\alpha}/C^\infty) of optimal maps for more smooth (C^\alpha /C^\infty))
densities. These are the first global regularity results which we are aware of
concerning optimal maps on non-flat Riemannian manifolds which possess some
vanishing sectional curvatures. Moreover, such product manifolds have potential
relevance in statistics (see [S]) and in statistical mechanics (where the state
of a system consisting of many spins is classically modeled by a point in the
phase space obtained by taking many products of spheres). For the proof we
apply and extend the method developed in [FKM1], where we showed injectivity
and continuity of optimal maps on domains in R^n for smooth non-negatively
cross-curved cost. The major obstacle in the present paper is to deal with the
non-trivial cut-locus and the presence of flat directions.

Analysis of PDEs;Differential Geometry