Ricci curvature
/ REE-chee /
Sectional curvature gives you a number for every 2-plane, which is a lot of data. Ricci curvature is a coarser, averaged version that answers a more physical question: in a given direction, do nearby geodesics (and hence small volumes of dust) tend to converge or spread apart? It is the curvature quantity that controls how volume behaves, and the one that appears in Einstein's equations of gravity.
The Ricci tensor Ric is a trace of the full Riemann tensor: Ric(X, Y) = trace of the map Z -> R(Z, X) Y, equivalently Ric_ij = R^k_ikj (summed). It is a symmetric (0,2)-tensor, the same type as the metric. Geometrically, for a unit vector u, Ric(u, u) is (n-1) times the average of the sectional curvatures of all 2-planes containing u. This average is exactly what governs volume: in normal coordinates the volume of a small geodesic ball deviates from the Euclidean value by a term proportional to the scalar curvature, and the directional spreading of geodesics emanating from a point is controlled by Ric(u,u) along u.
Why it matters and an honest limit. In general relativity the Einstein field equations relate Ric (and scalar curvature) to the energy-momentum of matter, so Ricci curvature literally IS gravity in that theory. A metric with Ric = lambda g is called Einstein. But Ricci is a trace, so it forgets information: in dimensions 4 and higher there are nonzero Riemann tensors with Ric = 0 (Ricci-flat but NOT flat — the Weyl curvature survives), and a Ricci bound is strictly weaker than a sectional-curvature bound. In dimension 3 alone Ric does determine the full Riemann tensor, a low-dimensional coincidence.
On the round n-sphere of radius a, Ric = ((n-1)/a^2) g, so the sphere is Einstein with lambda = (n-1)/a^2 > 0; positive Ricci means a thin pencil of geodesics fired from a point reconverges, matching the sphere's tendency to refocus light at the antipode. Flat R^n has Ric = 0; hyperbolic space has Ric = -((n-1)/a^2) g, and there geodesics splay apart.
Positive Ricci refocuses geodesics (sphere); zero is flat; negative spreads them (hyperbolic).
Ricci-flat (Ric = 0) is NOT the same as flat in dimension >= 4: the trace-free Weyl part can be nonzero (Ricci-flat Kahler / Calabi-Yau manifolds are the famous examples). Only in dimension 3 does Ric determine the full curvature tensor.