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

Abstract

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold $\left(M,g,f,\mu \right)$ 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 $\left(\lambda ,n+m\right)$-Einstein manifold $\left(M,g,w\right)$ having harmonic Weyl tensor, $\left({\nabla }^{j}w\right)\left({\nabla }^{m}w\right){\mathcal{C}}_{jklm}=0$ and ${\nabla }_{l}w{\nabla }^{l}w<0$ reduces to a perfect fluid generalized Robertson–Walker spacetime with Einstein fibers. Finally, $\left(M,g,w\right)$ reduces to a perfect fluid manifold if $\phi =-m\nabla \left(\ln w\right)$ is a $\phi \left(Ric\right)$-vector field on M and to an Einstein manifold if $\psi =\nabla w$ is a $\psi \left(Ric\right)$-vector field on M. Some consequences of these results are considered.