  For a complete connected Riemannian manifold $M$ let $V\in C^2(M)$ be such
that $\mu(d x)={\rm e}^{-V(x)} \mbox{vol}(d x)$ is a probability measure on
$M$. Taking $\mu$ as reference measure, we derive inequalities for probability
measures on $M$ linking relative entropy, Fisher information, Stein discrepancy
and Wasserstein distance. These inequalities strengthen in particular the
famous log-Sobolev and transportation-cost inequality and extend the so-called
Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux,
Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space
to the setting of Riemannian manifolds.
