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

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 $v\in T_xM$ , and let$$ S_x(v)=\{ u\in T_xM \colon g_x(u,u) = 1,\; g_x(u,v)=0 \}$$ 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 $T_xM$ , 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$$ \Ric_x(v,v)= (1-n)\int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x$$ 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
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:

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.