Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures

Abstract

The Lott-Sturm-Villani theory of CD(K, N) metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense via optimal transport, though extremely successful, has been shown not to directly apply to sub-Riemannian geometries. Nonetheless, still using optimal transport tools, some entropy inequalities have been proved to hold in the case of the Heisenberg group and more in general in sub-Riemannian manifolds.
In this talk we survey the known results and motivate a new approach we propose aiming to unify Riemannian and sub-Riemannian synthetic Ricci lower bounds, introducing suitable curvature dimension conditions. A main novelty is that we consider metric measure spaces endowed with a gauge function, and we allow general distortion coefficients. 
The talk is based on a joint work with Andrea Mondino (Oxford) and Luca Rizzi (SISSA).

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