A Note on Generalized Quasi-Einstein and (lambda, n+m)-Einstein Manifolds with Harmonic Conformal Tensor

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold (M, g, f, mu) to be a generalized Robertson-Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that a (lambda, n + m)-Einstein manifold (M, g, w) having harmonic Weyl tensor, (del(j)w) (del(m)w)C-jklm = 0 and del(l)w del(l)w < 0 reduces to a perfect fluid generalized Robertson-Walker spacetime with Einstein fibers. Finally, (M, g, w) reduces to a perfect fluid manifold if phi = m del(ln w) is a phi(Ric)-vector field on M and to an Einstein manifold if psi = del w is a phi(Ric)-vector field on M. Some consequences of these results are considered.