# Ricci tensor

From Encyclopedia of Mathematics

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].

#### 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, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) |

**How to Cite This Entry:**

Ricci tensor. L.A. Sidorov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Ricci_tensor&oldid=12398

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098