Dominating manifolds by radial spaces

This paper is devoted to a kind of rearrangement of functions on CD(k,n)-spaces, which satisfy a Polya-Szegö type inequality. We use this rearrangement to prove the validity of a Moser-Trudinger type inequality on a wide class of metric measure spaces satisfying a CD(k,n)-curvature-dimension inequality. As a consequence, we give a characterization, among manifolds with lower bounded Ricci curvature, of those admitting a Moser-Trudinger type inequality.