  Let M be a compact Riemannian manifold and let $\mu$,d be the associated
measure and distance on M. Robert McCann obtained, generalizing results for the
Euclidean 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. into
the 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 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 the identity component of 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.
