Polar factorization of conformal and projective maps of the sphere in the sense of optimal mass transport

Let M be a compact Riemannian manifold and let μ , d be the associated measure and distance on M. Robert McCann, generalizing results for the Euclidean case by Yann Brenier, obtained the polar factorization of Borel maps S : M → M pushing forward μ to a measure ν : each S factors uniquely a.e. into the composition S = T ◦ U , where U : M → M is volume preserving and T : M → M is the optimal map transporting μ to ν with respect to the cost function d 2 /2. In this article we study the polar factorization of conformal and projective maps of the sphere S n . For conformal maps, which may be identified with elements of O o (1, n +1), we prove that the polar factorization in the sense of optimal mass transport coincides with the algebraic polar factorization (Cartan decomposition) of this Lie group. For the projective case, where the group GL +( n + 1) is involved, we find necessary and sufficient conditions for these two factorizations to agree.