Sectional and intermediate Ricci curvature lower bounds via optimal transport

© 2018 The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth n-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called p-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on p-dimensional planes, 1≤p≤n. Such characterization roughly consists on a convexity condition of the p-Renyi entropy along L2-Wasserstein geodesics, where the role of reference measure is played by the p-dimensional Hausdorff measure. As application we establish a new Brunn–Minkowski type inequality involving p-dimensional submanifolds and the p-dimensional Hausdorff measure.