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

Abstract

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold M,g,f,μ 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 λ,n+m-Einstein manifold M,g,w having harmonic Weyl tensor, ∇jw∇mwCjklm=0 and ∇lw∇lw0 reduces to a perfect fluid generalized Robertson–Walker spacetime with Einstein fibers. Finally, M,g,w reduces to a perfect fluid manifold if φ=−m∇lnw is a φRic-vector field on M and to an Einstein manifold if ψ=∇w is a ψRic-vector field on M. Some consequences of these results are considered.

Keywords

• (<i>λ</i>, <i>n</i> + <i>m</i>)-Einstein manifolds
• generalized quasi-Einstein manifold
• perfect fluid
• torse-forming vector fields