Curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds

Abstract

In this paper, we use the information-theoretic approach to study curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds. We prove the equivalence of the ${\rm CD}(K, m)$-condition for $K\in \mathbb{R}$ and $m\in [n, \infty]$ and a family of Shannon and R\'enyi entropy differential inequalities along the geodesics on the Wasserstein space over a Riemannian manifold. {The rigidity models of the enhanced entropy differential inequalities are the $K$-Einstein manifolds and the $(K, m)$-Einstein manifolds}. Moreover, we prove the monotonicity and rigidity theorem of the $W$-entropy associated with the Shannon entropy and the R\'enyi entropy along the geodesics on the Wasserstein space over Riemannian manifolds with CD$(0, m)$-condition. Comparing with the characterization of the the CD$(K, m)$ curvature-dimension condition in the framework of the synthetic geometry developed by Lott, Sturm and Villani, we provide more simple equivalent characterizations for the CD$(K, m)$-condition, and we provide a characterization of the Einstein and quasi-Einstein manifolds by the enhanced entropy differential equality and the enhanced entropy power differential equality. These are new in the literature.