| Version current |
Version 5 |
| \paragraph{Definition.} |
\paragraph{Definition.} |
| The \emph{Ricci curvature tensor} is a rank $2$, symmetric tensor that |
The \emph{Ricci curvature tensor} is a rank $2$, symmetric tensor that |
| arises naturally in pseudo-Riemannian geometry. Let $(M,g_{ij})$ be a |
arises naturally in pseudo-Riemannian geometry. Let $(M,g_{ij})$ be a |
| smooth, $n$-dimensional pseudo-Riemannian manifold, and let |
smooth, $n$-dimensional pseudo-Riemannian manifold, and let |
| $R^i{}_{jkl}$ denote the corresponding Riemann curvature tensor. The |
$R^i{}_{jkl}$ denote the corresponding Riemann curvature tensor. The |
| Ricci tensor $R_{ij}$ is commonly defined as the following contraction |
Ricci tensor $R_{ij}$ is commonly defined as the following contraction |
| of the full curvature tensor: |
of the full curvature tensor: |
| \[R_{ij} = R^k{}_{ikj}. |
\[R_{ij} = R^k{}_{ikj}. |
| \] |
\] |
| The index symmetry of $R_{ij}$, so defined, follows from the symmetry |
The index symmetry of $R_{ij}$, so defined, follows from the symmetry |
| properties of the Riemann curvature. To wit, |
properties of the Riemann curvature. To wit, |
| \[ R_{ij} = R^k{}_{ikj} = R_{ki}{}^k{}_j = R^k{}_{jki} = R_{ji}.\] |
\[ R_{ij} = R^k{}_{ikj} = R_{ki}{}^k{}_j = R^k{}_{jki} = R_{ji}.\] |
| It is also convenient to regard the Ricci tensor as a symmetric bilinear |
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 |
form. To that end for vector-fields $X,Y$ we will write |
| \[ \Ric(X,Y) = X^i Y^j R_{ij}.\] |
\[ \Ric(X,Y) = X^i Y^j R_{ij}.\] |
| \paragraph{Related objects.} |
\paragraph{Related objects.} |
| Contracting the Ricci tensor, we obtain an important scalar invariant |
Contracting the Ricci tensor, we obtain an important scalar invariant |
| \[R=R^i{}_i,\] called the scalar curvature, and sometimes also called |
\[R=R^i{}_i,\] called the scalar curvature, and sometimes also called |
| the Ricci scalar. Closely related to the Ricci tensor is the tensor |
the Ricci scalar. Closely related to the Ricci tensor is the tensor |
|
\[G_{ij} = R_{ij} - \frac{1}{2} R\, g_{ij},\] called the \emph{Einstein
|
\[G_{ij} = R_{ij} - \frac{1}{2} R\, g_{ij},\] called the Einstein
|
|
tensor}. The Einstein tensor is also known as the trace-reversed Ricci
|
tensor. The Einstein tensor is also known as the trace-reversed Ricci
|
| tensor owing to the fact that |
tensor owing to the fact that |
| \[ G^i{}_i = - R. \] |
\[ G^i{}_i = - R. \] |
| Another related tensor is |
Another related tensor is |
| \[S_{ij} = R_{ij} - \frac{1}{n} R\, g_{ij}.\] |
\[S_{ij} = R_{ij} - \frac{1}{n} R\, g_{ij}.\] |
| This is called |
This is called |
| the |
the |
| trace-free Ricci tensor, owing to the fact that the above definition |
trace-free Ricci tensor, owing to the fact that the above definition |
| implies that |
implies that |
| \[ S^i{}_i=0.\] |
\[ S^i{}_i=0.\] |
|
|
|
|
|
|
| \paragraph{Geometric interpretation.} |
\paragraph{Geometric interpretation.} |
| In Riemannian geometry, the Ricci tensor represents the average value |
In Riemannian geometry, the Ricci tensor represents the average value |
| of the sectional curvature along a particular direction. |
of the sectional curvature along a particular direction. |
| Let |
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} |
\[ K_x(u,v) = \frac{R_x(u,v,v,u)}{g_x(u,u) g_x(v,v) - g_x(u,v)^2} |
| \] |
\] |
| denote the sectional curvature of $M$ along the plane spanned by |
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 |
vectors $u,v\in T_x M$. Fix a point $x\in M$ and a tangent vector |
| $v\in T_xM$, and let |
$v\in T_xM$, and let |
| \[ |
\[ |
| S_x(v)=\{ u\in T_xM \colon g_x(u,u) = 1,\; g_x(u,v)=0 \} |
S_x(v)=\{ u\in T_xM \colon g_x(u,u) = 1,\; g_x(u,v)=0 \} |
| \] denote the $n-2$ dimensional |
\] denote the $n-2$ dimensional |
| sphere of those unit vectors at $x$ that are perpendicular to $v$. |
sphere of those unit vectors at $x$ that are perpendicular to $v$. |
| Let $\mu_x$ denote the natural |
Let $\mu_x$ denote the natural |
| $(n-2)$-dimensional volume measure on $T_xM$, normalized so that |
$(n-2)$-dimensional volume measure on $T_xM$, normalized so that |
| \[ \int_{S_x(v)} \mu_x = 1.\] |
\[ \int_{S_x(v)} \mu_x = 1.\] |
| In this way, the quantity |
In this way, the quantity |
| \[ \int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x, \] |
\[ \int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x, \] |
| describes the average value of the sectional curvature for all planes |
describes the average value of the sectional curvature for all planes |
| in $T_x M$ that contain $v$. It is possible to show that |
in $T_x M$ that contain $v$. It is possible to show that |
| \[ \Ric_x(v,v)= (1-n)\int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x,\] |
\[ \Ric_x(v,v)= (1-n)\int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x,\] |
| thereby giving us the desired geometric interpretation. |
thereby giving us the desired geometric interpretation. |
|
|
|
|
| \paragraph{Decomposition of the curvature tensor.} |
\paragraph{Decomposition of the curvature tensor.} |
| For $n\geq 3$, the Ricci tensor can be characterized in terms of the |
For $n\geq 3$, the Ricci tensor can be characterized in terms of the |
| decomposition of the full curvature tensor into three covariantly |
decomposition of the full curvature tensor into three covariantly |
| defined summands, namely |
defined summands, namely |
| \begin{align*} |
\begin{align*} |
| F_{ijkl} &= \tfrac{1}{n-2} \left( S_{jl}\, g_{ik}+S_{ik}\, |
F_{ijkl} &= \tfrac{1}{n-2} \left( S_{jl}\, g_{ik}+S_{ik}\, |
| g_{jl}-S_{il}\, g_{jk}-S_{jk}\, g_{il}\right),\\ |
g_{jl}-S_{il}\, g_{jk}-S_{jk}\, g_{il}\right),\\ |
| E_{ijkl} &= \tfrac{1}{n(n-1)}R \left( g_{jl}\,g_{ik} - |
E_{ijkl} &= \tfrac{1}{n(n-1)}R \left( g_{jl}\,g_{ik} - |
| g_{il}\,g_{jk}\right),\\ |
g_{il}\,g_{jk}\right),\\ |
| W_{ijkl} &= R_{ijkl}-F_{ijkl}-E_{ijkl}. |
W_{ijkl} &= R_{ijkl}-F_{ijkl}-E_{ijkl}. |
| \end{align*} |
\end{align*} |
| The $W_{ijkl}$ is called the \emph{Weyl curvature tensor}. It is |
The $W_{ijkl}$ is called the \emph{Weyl curvature tensor}. It is |
| the conformally invariant, trace-free part of the curvature tensor. |
the conformally invariant, trace-free part of the curvature tensor. |
| Indeed, with the above definitions, we have |
Indeed, with the above definitions, we have |
| \[ W^k{}_{ikj}=0.\] The $E_{ijkl}$ and $F_{ijkl}$ correspond to the |
\[ W^k{}_{ikj}=0.\] The $E_{ijkl}$ and $F_{ijkl}$ correspond to the |
| trace-free part of the Ricci curvature tensor, and to the Ricci |
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 |
scalar. Indeed, we can recover $S_{ij}$ and $R$ from $E_{ijkl}$ and |
| $F_{ijkl}$ as follows: |
$F_{ijkl}$ as follows: |
| \begin{align*} |
\begin{align*} |
| S_{ij} &= F^k{}_{ikj},\\ |
S_{ij} &= F^k{}_{ikj},\\ |
| E^{ij}{}_{ij} &= R. |
E^{ij}{}_{ij} &= R. |
| \end{align*} |
\end{align*} |
|
|
| \paragraph{Relativity.} The Ricci tensor also plays an important role |
\paragraph{Relativity.} The Ricci tensor also plays an important role |
| in the theory of general relativity. In this keystone application, |
in the theory of general relativity. In this keystone application, |
| $M$ is a 4-dimensional pseudo-Riemannian manifold with signature |
$M$ is a 4-dimensional pseudo-Riemannian manifold with signature |
| $(3,1)$. The Einstein field equations assert that the energy-momentum |
$(3,1)$. The Einstein field equations assert that the energy-momentum |
| tensor is proportional to the Einstein tensor. In particular, the |
tensor is proportional to the Einstein tensor. In particular, the |
| equation |
equation |
| \[ R_{ij}=0 \] |
\[ R_{ij}=0 \] |
| is the field equation for a vacuum space-time. In geometry, a |
is the field equation for a vacuum space-time. In geometry, a |
| pseudo-Riemannian manifold that satisfies this equation is called |
pseudo-Riemannian manifold that satisfies this equation is called |
| Ricci-flat. It is possible to prove that a manifold is Ricci flat if |
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. |
and only if locally, the manifold, is conformally equivalent to flat space. |
