# 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

$$\mathrm{Ric}(X,Y)={X}^{i}{Y}^{j}{R}_{ij}.$$ |

## 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

$${S}_{x}(v)=\{u\in {T}_{x}M:{g}_{x}(u,u)=1,{g}_{x}(u,v)=0\}$$ |

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

$${\mathrm{Ric}}_{x}(v,v)=(1-n){\int}_{{S}_{x}(v)}{K}_{x}(\cdot ,v){\mu}_{x},$$ |

thereby giving us the desired geometric interpretation^{}.

## Decomposition of the curvature tensor.

For $n\ge 3$, the Ricci tensor can be characterized in terms of the decomposition of the full curvature tensor into three covariantly defined summands, namely

${F}_{ijkl}$ | $=\frac{1}{n-2}\left({S}_{jl}{g}_{ik}+{S}_{ik}{g}_{jl}-{S}_{il}{g}_{jk}-{S}_{jk}{g}_{il}\right),$ | ||

${E}_{ijkl}$ | $=\frac{1}{n(n-1)}R\left({g}_{jl}{g}_{ik}-{g}_{il}{g}_{jk}\right),$ | ||

${W}_{ijkl}$ | $={R}_{ijkl}-{F}_{ijkl}-{E}_{ijkl}.$ |

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:

${S}_{ij}$ | $={F}^{k}{}_{ikj},$ | ||

${E}^{ij}{}_{ij}$ | $=R.$ |

## 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 |
---|---|

Canonical name | RicciTensor |

Date of creation | 2013-03-22 15:02:38 |

Last modified on | 2013-03-22 15:02:38 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 9 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 83C05 |

Defines | scalar curvature |

Defines | Einstein tensor |

Defines | ricci scalar |

Defines | Weyl tensor |

Defines | Weyl curvature tensor |