Ricci curvature

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In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, measures how a curved space locally differs from flat space by tracking how nearby geodesics spread apart or converge. Formally, it is a symmetric rank-two tensor obtained by taking a trace of the Riemann curvature tensor of a Riemannian or pseudo-Riemannian metric. In Riemannian geometry, the Ricci curvature in a given tangent direction may be interpreted as an average of the sectional curvatures of planes containing that direction. A further trace of the Ricci curvature yields the scalar curvature. In general relativity, the Ricci curvature tensor enters the Einstein field equations through the Einstein tensor, formed from the Ricci tensor, the scalar curvature, and the metric. The Ricci tensor is also fundamental in Riemannian geometry and geometric analysis. Bounds on Ricci curvature imply strong global geometric and topological consequences, as in Myers's theorem and related comparison theorems...

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