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).

References

[a1]	S. Kobayashi and K. Nomizu, “Foundations of differential geometry”, 1, Interscience (1963).