# Ricci tensor

A twice-covariant tensor obtained from the Riemann tensor $R^{l}_{jkl}$ by contracting the upper index with the first lower one: $$R_{ki} = R^{m}_{mki}.$$

In a Riemannian space $V_{n}$, the Ricci tensor is symmetric: $R_{ki} = R_{ik}$. The trace of the Ricci tensor with respect to the contravariant metric tensor $g^{ij}$ of the space $V_{n}$ leads to a scalar, $R = g^{ij} R_{ij}$, called the curvature invariant or the scalar curvature of $V_{n}$. The components of the Ricci tensor can be expressed in terms of the metric tensor $g_{ij}$ of the space $V_{n}$: $$R_{ij} = \frac{\partial^{2} \ln \sqrt{g}}{\partial x^{i} \partial x^{j}} - \frac{\partial}{\partial x^{k}} \Gamma^{k}_{ij} + \Gamma^{m}_{ik} \Gamma^{k}_{mj} - \Gamma^{m}_{ij} \frac{\partial \ln \sqrt{g}}{\partial x^{m}},$$ where $g = \det g_{ij}$ and $\Gamma^{k}_{ij}$ are the Christoffel symbols of the second kind calculated with respect to the tensor $g_{ij}$.

The tensor was introduced by G. Ricci in [1].

#### References

 [1] G. Ricci, “Atti R. Inst. Venelo”, 53: 2 (1903–1904), pp. 1233–1239. [2] L.P. Eisenhart, “Riemannian geometry”, Princeton Univ. Press (1949).