φ (Ric)-vector fields on warped product manifolds and applications

Abstract

Sufficient and necessary conditions are provided on warped product manifolds and their base and fiber manifolds for a vector field φj to be a φ(Ric)-vector field , that is, ∇iφj=μRij where Rij is the Ricci tensor of M and μ is a scalar. Two warped product space-times admitting φ(Ric)-vector fields are considered. Lorentzian quasi-Einstein manifolds admitting a time-like φ(Ric)-vector field are shown to be either Ricci simple or a perfect fluid GRW space-time. The generators of a Lorentzian generalized quasi-Einstein manifold admitting a time-like φ(Ric)-vector field are eigenvectors of the Ricci tensor with zero eigenvalue.