Ricci curvature
/ REE-chee /
Suppose you do not want the curvature in one specific 2-plane, but a kind of average of all the curvatures in the directions around a given one. That averaged quantity is the Ricci curvature. Intuitively, it answers a volume question: if you shoot out a small cone of geodesics in a chosen direction, does the bundle squeeze together (volume shrinks faster than in flat space) or spread apart? Positive Ricci curvature means geodesics focus and small balls have less volume than flat space predicts; negative means they defocus.
Precisely, the Ricci tensor is obtained from the Riemann tensor by contraction — summing over one pair of indices, which is the tensor version of averaging. In a chart, Ric_jk = sum_i R^i_ijk. The Ricci curvature in a unit direction v equals the sum of the sectional curvatures of all the 2-planes containing v (over an orthonormal frame), so it is genuinely an average of sectional curvatures around v, multiplied out. Because it averages, Ricci carries less information than the full Riemann tensor in dimension 4 and above — many different curved spaces share the same Ricci tensor — but in dimension 3 it actually determines the whole Riemann tensor, and in dimension 2 everything collapses to the single Gaussian curvature.
Ricci curvature is the star of general relativity. Einstein's field equations equate a curvature expression built from the Ricci tensor (the Einstein tensor) to the distribution of mass and energy: matter tells the Ricci curvature how to bend, and that bending is gravity. It also drives Ricci flow, the geometric heat equation Hamilton and Perelman used to prove the Poincare conjecture, where a metric evolves by smoothing out its own Ricci curvature. The honest caveat: Ricci curvature controls how volumes of geodesic bundles change, but it does not see every detail of shape — the leftover information, invisible to Ricci, is the Weyl part of the Riemann tensor.
In empty space far from any matter, Einstein's equations reduce to Ric = 0, called Ricci-flat. The flat plane and flat spacetime are Ricci-flat, but so is the curved spacetime outside a star — the Schwarzschild geometry has Ric = 0 everywhere outside the mass, yet is genuinely curved, because the leftover Weyl curvature is what bends light and orbits.
Ricci-flat does not mean flat: Schwarzschild spacetime has Ric = 0 outside the star yet is curved by its Weyl part.
Ricci curvature is an average and so loses information above dimension three; Ric = 0 (Ricci-flat) is far weaker than R = 0 (truly flat). General relativity's vacuum equations are Ric = 0, which is why nontrivial gravity can exist in empty space.