Some inequalities on Riemannian manifolds linking Entropy,Fisher information, Stein discrepancy and Wasserstein distance

For a complete connected Riemannian manifold M let V∈C2(M) be such that μ(dx)=e−V(x)vol(dx) is probability measure on M. Taking μ as reference measure, we derive inequalities for measures linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These strengthen in particular the famous log-Sobolev transportation-cost inequality extend so-called Entropy/Stein-discrepancy/Information (HSI) established by Ledoux, Nourdin Peccati (2015) standard Gaussian Euclidean space to setting of manifolds.