Synthetic Ricci curvature

Synthetic Ricci curvature is the study of Ricci curvature lower bounds in metric measure spaces that lack the smooth structure of Riemannian manifolds. Developed independently by John Lott and Cédric Villani on one hand and by Karl-Theodor Sturm on the other, the theory uses the language of optimal transport to define what it means for a metric measure space to have Ricci curvature bounded below — a condition that makes no reference to derivatives, tangent spaces, or Christoffel symbols.

The central insight is that Ricci curvature lower bounds in Riemannian geometry are equivalent to convexity properties of certain entropy functionals along geodesics in the space of probability measures equipped with the Wasserstein metric. This equivalence, first observed in the work of Otto and Villani on the geometry of dissipative evolution equations, allows the definition of Ricci curvature bounds to be extended to arbitrary metric measure spaces. A space satisfies the curvature-dimension condition CD(K,N) if the entropy functional is K-convex along Wasserstein geodesics and satisfies a dimensional refinement corresponding to dimension N.

The theory has profound consequences. The Bakry-Émery curvature-dimension condition, the Brunn-Minkowski inequality, the Bishop-Gromov volume comparison theorem, and the Lichnerowicz eigenvalue bound — all classical results of Riemannian geometry — can be proved in the synthetic setting. This demonstrates that these results are not consequences of smoothness but of the curvature-dimension condition itself. The implications extend to metric geometry, probability theory, and the study of Ricci flow through singularities.