We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - \alpha g) \leqslant 1, for some \alpha \in R, or non-quasi-Einstein.
ARSLAN, KADRİ; DESZCZ, RYSZARD; EZENTAŞ, RIDVAN; HOTLOS, MARIAN; and MURATHAN, CENGİZHAN
"On generalized Robertson--Walker spacetimes satisfying some curvature condition,"
Turkish Journal of Mathematics: Vol. 38:
2, Article 16.
Available at: https://journals.tubitak.gov.tr/math/vol38/iss2/16