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 .
|||G. Ricci, Atti R. Inst. Venelo , 53 : 2 (1903–1904) pp. 1233–1239|
|||L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)|
|[a1]||S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)|
Ricci tensor. L.A. Sidorov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ricci_tensor&oldid=12398