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:
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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
:
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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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_tensor&oldid=12398
Ricci tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_tensor&oldid=12398
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article