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Ricci tensor

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A twice-covariant tensor obtained from the Riemann tensor 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).

Comments

References

[a1] S. Kobayashi and K. Nomizu, “Foundations of differential geometry”, 1, Interscience (1963).
How to Cite This Entry:
Ricci tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_tensor&oldid=38665
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article