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We investigate Prékopa-Leindler type inequalities on a Riemannian manifold $M$ equipped with a measure with density $e^{-V}$ where the potential $V$ and the Ricci curvature satisfy ${Hess}_{x}V+{Ric}_x\ge \lambda I$ for all $x\in M$, with some $\lambda \in ℝ$. As in our earlier work [], the argument uses optimal mass transport on $M$, but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to...