# Ricci tensor

## Definition.

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

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

The index symmetry of $R_{ij}$, so defined, follows from the symmetry properties 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 bilinear form. 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 called the Ricci scalar. Closely related to the Ricci tensor is the tensor

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

called the Einstein tensor. The Einstein tensor is also known as the trace-reversed Ricci tensor 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 called the trace-free Ricci tensor, owing to the fact that the above definition implies that

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

## Geometric interpretation.

In Riemannian geometry, the Ricci tensor represents the average value of 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 by vectors $u,v\in T_{x}M$. 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 $n-2$ dimensional sphere 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 planes in $T_{x}M$ 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 the decomposition of the full curvature tensor into three covariantly defined 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 is the 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 the trace-free part of the Ricci curvature tensor, and to the Ricci scalar. 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 role in 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-momentum tensor is proportional to the Einstein tensor. In particular, the equation

 $R_{ij}=0$

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

Title Ricci tensor RicciTensor 2013-03-22 15:02:38 2013-03-22 15:02:38 rmilson (146) rmilson (146) 9 rmilson (146) Definition msc 83C05 scalar curvature Einstein tensor ricci scalar Weyl tensor Weyl curvature tensor