Knowlpedia

Definition

Ricci curvature

The symmetric covariant two-tensor obtained by tracing the Riemann curvature tensor.

Definition

Let (M,g)(M,g) be an nn-dimensional , and use the curvature convention R(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]ZR(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z from the . The Ricci curvature is the covariant 22-tensor

Ric⁡(X,Y)=tr⁡(Z↦R(Z,X)Y).\operatorname{Ric}(X,Y) =\operatorname{tr}\bigl(Z\mapsto R(Z,X)Y\bigr).

Equivalently, for any local orthonormal frame e1,…,ene_1,\ldots,e_n,

Ric⁡(X,Y)=∑i=1ng(R(ei,X)Y,ei).\operatorname{Ric}(X,Y) =\sum_{i=1}^n g(R(e_i,X)Y,e_i).

This contraction is independent of the chosen orthonormal frame and is symmetric. It records the curvature averaged over directions orthogonal to a given tangent direction.

Geometric meaning

For a unit tangent vector XX, extend XX to an X,e2,…,enX,e_2,\ldots,e_n. Then

Ric⁡(X,X)=∑i=2nK(X,ei),\operatorname{Ric}(X,X)=\sum_{i=2}^n K(X,e_i),

so Ricci curvature averages the sectional curvatures of planes containing XX. It retains less directional information than the full Riemann tensor but controls volume distortion, geodesic focusing, and the Bochner formula. These contraction formulas follow the convention in Besse, Chapter 1.

Contractions and special metrics

Taking the metric trace of Ric⁡\operatorname{Ric} gives the scalar curvature Scal⁡\operatorname{Scal}. A metric is Einstein when Ric⁡=λg\operatorname{Ric}=\lambda g for a function λ\lambda; in dimension at least three, the contracted Bianchi identity forces λ\lambda to be constant on each . This is the contracted form of the . A metric is Ricci-flat when Ric⁡=0\operatorname{Ric}=0. Ricci-flatness does not imply that the full Riemann tensor vanishes in dimensions four and higher.

Examples and conventions

On an nn-manifold of constant sectional curvature kk,

Ric⁡=(n−1)k g.\operatorname{Ric}=(n-1)k\,g.

Thus the unit round sphere has positive Ricci curvature and has zero Ricci curvature. In dimension two, Ric⁡=Kg\operatorname{Ric}=K g, so Ricci curvature and Gaussian curvature contain the same information.

References
  1. Arthur L. Besse, Einstein Manifolds, Springer, 1987. DOI record. Relevant: Chapter 1 on curvature conventions, contractions, and Einstein metrics.
  2. John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. DOI record. Relevant: Chapter 7, “Curvature,” on Ricci and scalar curvature.