---
tag: K83HM845A8
kind: definition
name: Ricci curvature
msc: [53B20, 53C21]
defines: 4GTNM0WHDH
uses: [GH6CXWFNDJ, JA29ETCM02, 07PTPSZPME]
sources:
  - url: https://en.wikipedia.org/wiki/Ricci_curvature
---

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] with curvature $R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$ (see [[Riemann curvature tensor|JA29ETCM02]]).
The *Ricci curvature* of $(M, g)$ is the $(0,2)$-tensor field $\operatorname{Ric}$ defined, for $p \in M$ and $u, v \in T_pM$, by
$$\operatorname{Ric}(u, v) = \tr\bigl( w \mapsto R(w, u)v \bigr),$$
the [[trace|07PTPSZPME]] of the linear map $T_pM \to T_pM$, $w \mapsto R(w, u)v$.

## Remarks

If $e_1, \ldots, e_n$ is a $g$-orthonormal basis of $T_pM$ in the Riemannian case, then $\operatorname{Ric}(u, v) = \sum_i g(R(e_i, u)v, e_i)$, and for a unit vector $v = e_1$ this gives $\operatorname{Ric}(v, v) = \sum_{i \ge 2} K(e_i, v)$, a sum of [[sectional curvatures|1X42JAA349]].
In coordinates, $R_{ij} = \sum_k R^k{}_{kij}$ where $R(\partial_k, \partial_i)\partial_j = \sum_l R^l{}_{kij} \partial_l$.
The Ricci tensor is symmetric, by the first Bianchi identity.

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

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

A metric with $\operatorname{Ric} = \lambda g$ for a constant $\lambda$ is an [[Einstein metric|41S2EZ0KR3]].