  In their seminal work, Cordero-Erausquin, Nazaret and Villani [Adv. Math.,
2004] proved sharp Sobolev inequalities in Euclidean spaces via Optimal Mass
Transportation, raising the question whether their approach is powerful enough
to produce sharp Sobolev inequalities also on Riemannian manifolds. In this
paper we affirmatively answer their question for Riemannian manifolds with
non-negative Ricci curvature; namely, by using Optimal Mass Transportation with
quadratic distance cost, sharp $L^p$-Sobolev and $L^p$-logarithmic Sobolev
inequalities (both for $p>1$ and $p=1$) are established, where the optimal
constants contain the asymptotic volume growth arising from precise asymptotic
properties of the Talentian and Gaussian bubbles. As a byproduct, we give an
alternative, elementary proof to the main result of do Carmo and Xia [Compos.
Math., 2004] (and subsequent results) concerning the quantitative volume
non-collapsing estimates on Riemannian manifolds with non-negative Ricci
curvature that support certain Sobolev inequalities.
