Figure 2
Positive Ricci curvature is reflected by the characteristic that for two very close points x and y with a tangent vector v connecting xy as well as tangent vectors w (at x) and w′ (at y), in which w′ is obtained by parallel transport of w, that the two corresponding geodesics will get closer.
This can be compared to the traditional flat geometry of a Euclidean space where such distances are unaffected during the parallel transport. Equivalently, this may be formulated by the fact that the transportation distance between two small (geodesic balls) is less than the distance of their centers. Ricci curvature along the direction xy quantifies this, averaged on all directions w at x.
