Ricci tensor

Definition.

The Ricci curvature tensor is a rank $2$ , symmetric tensor thatarises naturally in pseudo-Riemannian geometry. Let $(M,g_{ij})$ be asmooth, $n$ -dimensional pseudo-Riemannian manifold, and let $R^{i}{}_{jkl}$ denote the corresponding Riemann curvature tensor. TheRicci tensor $R_{ij}$ is commonly defined as the following contractionof the full curvature tensor:

R_{ij}=R^{k}{}_{ikj}.

The index symmetry of $R_{ij}$ , so defined, follows from the symmetryproperties of the Riemann curvature. To wit,

R_{ij}=R^{k}{}_{ikj}=R_{ki}{}^{k}{}_{j}=R^{k}{}_{jki}=R_{ji}.

It is also convenient to regard the Ricci tensor as a symmetric bilinearform. To that end for vector-fields $X, Y$ we will write

\\operatorname{Ric}(X,Y)=X^{i}Y^{j}R_{ij}.

Related objects.

Contracting the Ricci tensor, we obtain an important scalar invariant

R=R^{i}{}_{i},

called the scalar curvature, and sometimes also calledthe Ricci scalar. Closely related to the Ricci tensor is the tensor

G_{ij}=R_{ij}-\\frac{1}{2}R\\,g_{ij},

called the Einsteintensor. The Einstein tensor is also known as the trace-reversed Riccitensor owing to the fact that

G^{i}{}_{i}=-R.

Another related tensor is

S_{ij}=R_{ij}-\\frac{1}{n}R\\,g_{ij}.

This is calledthetrace-free Ricci tensor, owing to the fact that the above definitionimplies that

S^{i}{}_{i}=0.

Geometric interpretation.

In Riemannian geometry, the Ricci tensor represents the average valueof the sectional curvature along a particular direction.Let

K_{x}(u,v)=\\frac{R_{x}(u,v,v,u)}{g_{x}(u,u)g_{x}(v,v)-g_{x}(u,v)^{2}}

denote the sectional curvature of $M$ along the plane spanned byvectors $u,v\\in T_{x}M$ . Fix a point $x\\in M$ and a tangent vector $v\\in T_{x}M$ , and let

S_{x}(v)=\\{u\\in T_{x}M\\colon g_{x}(u,u)=1,\\;g_{x}(u,v)=0\\}

denote the $n-2$ dimensionalsphere of those unit vectors at $x$ that are perpendicular to $v$ .Let $\\mu_{x}$ denote the natural $(n-2)$ -dimensional volume measure on $T_{x}M$ , normalized so that

\\int_{S_{x}(v)}\\mu_{x}=1.

In this way, the quantity

\\int_{S_{x}(v)}\\!\\!K_{x}(\\cdot,v)\\mu_{x},

describes the average value of the sectional curvature for all planesin $T_{x}M$ that contain $v$ . It is possible to show that

\\operatorname{Ric}_{x}(v,v)=(1-n)\\int_{S_{x}(v)}\\!\\!K_{x}(\\cdot,v)\\mu_{x},

thereby giving us the desired geometric interpretation.

Decomposition of the curvature tensor.

For $n\\geq 3$ , the Ricci tensor can be characterized in terms of thedecomposition of the full curvature tensor into three covariantlydefined summands, namely

	$\\displaystyle F_{ijkl}$	$\\displaystyle=\\tfrac{1}{n-2}\\left(S_{jl}\\,g_{ik}+S_{ik}\\,g_{jl}-S_{il}\\,g_{jk}%-S_{jk}\\,g_{il}\\right),$
	$\\displaystyle E_{ijkl}$	$\\displaystyle=\\tfrac{1}{n(n-1)}R\\left(g_{jl}\\,g_{ik}-g_{il}\\,g_{jk}\\right),$
	$\\displaystyle W_{ijkl}$	$\\displaystyle=R_{ijkl}-F_{ijkl}-E_{ijkl}.$

The $W_{ijkl}$ is called the Weyl curvature tensor. It isthe conformally invariant, trace-free part of the curvature tensor.Indeed, with the above definitions, we have

W^{k}{}_{ikj}=0.

The $E_{ijkl}$ and $F_{ijkl}$ correspond to thetrace-free part of the Ricci curvature tensor, and to the Ricciscalar. Indeed, we can recover $S_{ij}$ and $R$ from $E_{ijkl}$ and $F_{ijkl}$ as follows:

	$\\displaystyle S_{ij}$	$\\displaystyle=F^{k}{}_{ikj},$
	$\\displaystyle E^{ij}{}_{ij}$	$\\displaystyle=R.$

Relativity.

The Ricci tensor also plays an important rolein the theory of general relativity. In this keystone application, $M$ is a 4-dimensional pseudo-Riemannian manifold with signature $(3,1)$ . The Einstein field equations assert that the energy-momentumtensor is proportional to the Einstein tensor. In particular, theequation

R_{ij}=0

is the field equation for a vacuum space-time. In geometry, apseudo-Riemannian manifold that satisfies this equation is calledRicci-flat. It is possible to prove that a manifold is Ricci flat ifand only if locally, the manifold, is conformally equivalent to flat space.