Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {rm exp}_x[-nablapsi(x)] and a volume-preserving map u : M ? M u : M to M , where y: M ? bold R psi : M to {bold R} satisfies the additional property that (y^c)^c = y (psi^c)^c = psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} psi^c(y) :={rm inf}{c(x,y) - psi(x),vert,x in M} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.