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

TL;DR

This paper explores the curvature-dimension condition on Riemannian manifolds using an information-theoretic approach, establishing new equivalences, rigidity theorems, and characterizations involving entropy inequalities and Einstein manifolds.

Contribution

It introduces new simple characterizations of the CD(K, m) condition and provides novel rigidity theorems and entropy differential inequalities on Riemannian manifolds.

Findings

Equivalence of CD(K, m) condition and entropy differential inequalities.

Rigidity models identified as K-Einstein and (K, m)-Einstein manifolds.

Monotonicity and rigidity of W-entropy along Wasserstein geodesics.

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.