Ricci tensor
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). |
Ricci tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_tensor&oldid=38665