Ricci curvature

Summary

The Ricci curvature of a Riemannian manifold is a trace of the Riemann curvature tensor. It is a symmetric bilinear form on each tangent space. Let $v$ be a unit vector and $e_2, \ldots, e_n$ an orthonormal basis of $v^\perp$. Then $\operatorname{Ric}(v, v)$ is the sum of the sectional curvatures of the planes spanned by $v$ and $e_i$. It controls how the volume of small balls and cones of geodesics differs from the Euclidean value. It appears in Einstein's equations of general relativity, in the Ricci flow, and in theorems such as Myers's theorem and the Bishop–Gromov inequality.