Math 2.0

Differential geometry Definition

Ricci curvature

Builds on Pseudo-Riemannian manifoldRiemann curvature tensorTrace of a matrix

Definition

Let (M,g)(M, g) be a pseudo-Riemannian manifold with curvature R(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]ZR(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z (see Riemann curvature tensor). The Ricci curvature of (M,g)(M, g) is the (0,2)(0,2)-tensor field Ric⁡\operatorname{Ric} defined, for p∈Mp \in M and u,v∈TpMu, v \in T_pM, by Ric⁡(u,v)=tr⁡(w↦R(w,u)v),\operatorname{Ric}(u, v) = \tr\bigl( w \mapsto R(w, u)v \bigr), the trace of the linear map TpM→TpMT_pM \to T_pM, w↦R(w,u)vw \mapsto R(w, u)v.

Remarks

If e1,…,ene_1, \ldots, e_n is a gg-orthonormal basis of TpMT_pM in the Riemannian case, then Ric⁡(u,v)=∑ig(R(ei,u)v,ei)\operatorname{Ric}(u, v) = \sum_i g(R(e_i, u)v, e_i), and for a unit vector v=e1v = e_1 this gives Ric⁡(v,v)=∑i≥2K(ei,v)\operatorname{Ric}(v, v) = \sum_{i \ge 2} K(e_i, v), a sum of sectional curvatures. In coordinates, Rij=∑kRkkijR_{ij} = \sum_k R^k{}_{kij} where R(∂k,∂i)∂j=∑lRlkij∂lR(\partial_k, \partial_i)\partial_j = \sum_l R^l{}_{kij} \partial_l. The Ricci tensor is symmetric, by the first Bianchi identity.

Example: on the unit sphere SnS^n, R(w,u)v=g(u,v)w−g(w,v)uR(w, u)v = g(u, v)w - g(w, v)u, whose trace in ww is n g(u,v)−g(u,v)n \, g(u, v) - g(u, v), so Ric⁡=(n−1)g\operatorname{Ric} = (n - 1)g.

Non-example: Ric⁡(u,v)=tr⁡(w↦R(u,v)w)\operatorname{Ric}(u, v) = \tr(w \mapsto R(u, v)w) is a different trace, and it is identically zero because R(u,v)R(u, v) is skew-adjoint; it is not the Ricci curvature.

A metric with Ric⁡=λg\operatorname{Ric} = \lambda g for a constant λ\lambda is an Einstein metric.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionPseudo-Riemannian manifold, concept, no evidence levelPseudo-Riemannian man…conceptRiemann curvature tensor, concept, no evidence levelRiemann curvature ten…conceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionRicci curvature, definition, no evidence levelRicci curvaturedefinition
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  1. Matrix definition
  2. Pseudo-Riemannian manifold concept
  3. Riemann curvature tensor concept
  4. Square matrix definition
  5. Trace of a matrix definition
  6. Ricci curvature definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Ricci_curvature
About this page
TagK83HM845A8
URIm2://K83HM845A8
Kinddefinition
WritingW0 stub
MSC53B20, 53C21
DefinesRicci curvature
UsesPseudo-Riemannian manifold, Riemann curvature tensor, Trace of a matrix
Changed2 November 2026

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