Polar factorization of maps on Riemannian manifolds

Let (M, g) be a connected compact manifold, C-3 smooth and without boundary , equipped with a Riemannian distance d(x, y). If s : M --> M is merely Bor el and never maps positive volume into zero volume, we show s = t circle u factors uniquely a.e. into the composition of a map t(x) = exp(x)[-del psi (x)] and a volume-preserving map u : M --> M, where psi : M --> R satisfies the additional property that (psi (c))(c) = psi with psi (c)(y) := inf {c( x,y) - psi (x) \ x is an element of M} and c(x, y) = d(2)(x, y)/2. Like the factorization it generalizes from Euclidean space, this nonlinear decompos ition can be linearized around the identity to yield the Hodge decompositio n of vector fields. The results are obtained by solving a Riemannian version of the Monge-Kanto rovich problem, which means minimizing the expected value of the cost c(x, y) for transporting one distribution f greater than or equal to 0 of mass i n L-1(M) onto another. Parallel results for other strictly convex cost func tions c(x, y) greater than or equal to 0 of the Riemannian distance on non- compact manifolds are briefly discussed.