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Let M be a compact Riemannian manifold and let $\mu$,d be the associatedmeasure and distance on M. Robert McCann obtained, generalizing results for theEuclidean case by Yann Brenier, the polar factorization of Borel maps S : M ->M pushing forward $\mu$ to a measure $\nu$: each S factors uniquely a.e. intothe composition S = T \circ U, where U : M -> M is volume preserving and T : M-> M is the optimal map transporting $\mu$ to $\nu$ with respect to the costfunction d^2/2.  In this article we study the polar factorization of conformal and projectivemaps of the sphere S^n. For conformal maps, which may be identified withelements of the identity component of O(1,n+1), we prove that the polarfactorization in the sense of optimal mass transport coincides with thealgebraic polar factorization (Cartan decomposition) of this Lie group. For theprojective case, where the group GL_+(n+1) is involved, we find necessary andsufficient conditions for these two factorizations to agree.

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