2024/07/22 by Li, Xiang-Dong
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.15576
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∈ ℝ and m∈ [n, ∞] and some entropy differential inequalities along the geodesics on the Wasserstein space over a Riemannian manifold. The rigidity models of the entropy differential inequalities are the K-Einstein manifolds and the (K, m)-Einstein manifolds with Hessian solitons. Moreover, we prove the monotonicity and rigidity theorem of the W-entropy along the geodesics on the Wasserstein space over Riemannian manifolds with CD(0, m)-condition. Finally, we compare our work with the synthetic geometry developed by Lott-Villani and Sturm.