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

TL;DR

This paper introduces gauge metric measure spaces as a step towards unifying Riemannian and sub-Riemannian geometries through synthetic Ricci curvature bounds, aiming for a comprehensive geometric framework.

Contribution

It proposes the concept of gauge metric measure spaces to bridge Riemannian and sub-Riemannian geometries within a unified synthetic Ricci curvature theory.

Findings

Introduction of gauge metric measure spaces

Potential for unification of geometric frameworks

Foundation for future synthetic curvature analysis

Abstract

Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework of synthetic Ricci curvature lower bounds. With the aim of achieving such a unification program, in this paper we initiate the study of gauge metric measure spaces.