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

## TL;DR

This paper explores the polar factorization of conformal and projective maps on the sphere, showing how optimal mass transport relates to algebraic decompositions in Lie groups, with specific conditions for their equivalence.

## Contribution

It establishes the connection between optimal mass transport polar factorization and algebraic group decompositions for sphere maps, extending previous Euclidean results.

## Key findings

- Polar factorization of conformal maps matches Cartan decomposition.
- Necessary and sufficient conditions for projective maps to have coinciding factorizations.
- Extension of McCann's polar factorization to sphere conformal and projective maps.

## Abstract

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.