General Relativity: An Introduction for Physicists · Problem 14.1
The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby
Physics · curvature-tensor-properties
Problem
In an N-dimensional manifold, consider the tensor
Rijkl=K(gikgjl−gilgjk),
where K may be a function of position. Show that this tensor satisfies the symmetry properties and the cyclic identity of the curvature tensor. Show that, in order to satisfy the Bianchi identity, one requires K to be constant if N>2.