Fibred spaces with projectable Riemannian metric

Metric measure spaces with Riemannian Ricci curvature bounded from below

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, i...

Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure

In prior work [4] of the first two authors with Savaré, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X, d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In [4] the RCD(K,∞) property is defined in thr...

Monge Problem in Metric Measure Spaces with Riemannian Curvature-dimension Condition

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X, d,m) enjoying the Riemannian curvature-dimension condition RCD∗(K,N), with N < ∞. For the first marginal measure, we assume that μ0 ≪ m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge-Kantorovich problem, attain the same minimal value. Moreover we ...

On metric spaces induced by fuzzy metric spaces

For a class of fuzzy metric spaces (in the sense of George and Veeramani) with an H-type t-norm,  we present a method to construct a metric on a  fuzzy metric space. The induced metric space shares many important properties with the given fuzzy metric space.  Specifically, they generate the same topology, and have the same completeness. Our results can give the constructive proofs to some probl...

An overview on calculus and heat flow in metric measure spaces and spaces with Riemannian curvature bounded from below