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In this note we present a new proof of Sobolev's inequality under a uniformlower bound of the Ricci curvature. This result was initially obtained in 1983by Ilias. Our goal is to present a very short proof, to give a review of thefamous inequality and to explain how our method, relying on a gradient-flowinterpretation, is simple and robust. In particular, we elucidate computationsused in numerous previous works, starting with Bidaut-V{\'e}ron and V{\'e}ron's1991 classical work.

Sobolev's inequality under a curvature-dimension condition