$$ {\varphi }$$ (Ric)-vector fields on warped product manifolds and applications

Sufficient and necessary conditions are provided on warped product manifolds their base fiber for a vector field $$\varphi _{j}$$ to be \left( \mathrm {Ric}\right) $$ -vector , that is, $$\nabla _{i}\varphi _{j}=\mu R_{ij}$$ where $$R_{ij}$$ is the Ricci tensor of M \mu scalar. Two space-times admitting fields considered. Lorentzian quasi-Einstein time-like {Ric} \right) shown either simple or perfect fluid GRW space-time. The generators generalized manifold eigenvectors with zero eigenvalue.