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:

In a Riemannian space the Ricci tensor is symmetric: . The trace of the Ricci tensor with respect to the contravariant metric tensor of the space leads to a scalar, , called the curvature invariant or the scalar curvature of . The components of the Ricci tensor can be expressed in terms of the metric tensor of the space :

where and are the Christoffel symbols of the second kind (cf. Christoffel symbol) calculated with respect to the tensor .

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


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



[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
How to Cite This Entry:
Ricci tensor. L.A. Sidorov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098