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A twice-covariant tensor obtained from the [[Riemann tensor|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 symbol|Christoffel symbols]] of the second kind calculated with respect to the tensor $ g_{ij} $.

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

====References====

<table>
<TR><TD valign="top">[1]</TD><TD valign="top"> G. Ricci, “Atti R. Inst. Venelo”, '''53''': 2 (1903–1904), pp. 1233–1239.</TD></TR>
<TR><TD valign="top">[2]</TD><TD valign="top"> L.P. Eisenhart, “Riemannian geometry”, Princeton Univ. Press (1949).</TD></TR>
</table>

====Comments====

====References====
<table>
<TR><TD valign="top">[a1]</TD><TD valign="top"> S. Kobayashi and K. Nomizu, “Foundations of differential geometry”, '''1''', Interscience (1963).</TD></TR></table>