This article aimed to study and explore conformal vector fields on doubly warped product manifolds as well as on doubly warped space-times. Then we derive sufficient conditions for matter and Ricci collineations on doubly warped product manifolds. A special attention is paid to concurrent vector fields. Finally, Ricci solitons on doubly warped product space-times admitting conformal vector fields are considered.2000 Mathematics Subject Classification. Primary 53C21; Secondary 53C50, 53C80.

Abstract. The purpose of the present article is to study and characterize several types of symmetries of generalized Robertson-Walker space-times. Conformal vector fields, curvature and Ricci collineations are studied. Many implications for existence of these symmetries on generalied Robertson-Walker spacetimes are obtained. Finally, Ricci solitons on generalized RobertsonWalker space-times admitting conformal vector fields are investigated.

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A. Gray presented an interesting O(n) invariant decomposition of the covariant derivative of the Ricci tensor. Manifolds whose Ricci tensor satisfies the defining property of each orthogonal class are called Einstein-like manifolds. In the present paper, we answered the following question: Under what condition(s), does a factor manifold Mi,i = 1,2 of a doubly warped product manifold M =f2 M1 x f1 M2 lie in the same Einstein- like class of M? By imposing sufficient and necessary conditions on the warping functions, an inheritance property of each class is proved. As an application, Einstein-like doubly warped product space-times of type A,B or P are considered.

The purpose of this research is to investigate how a ρ-Einstein soliton structure on a warped product manifold affects its base and fiber factor manifolds. Firstly, the pertinent properties of ρ-Einstein solitons are provided. Secondly, numerous necessary and sufficient conditions of a ρ-Einstein soliton warped product manifold to make its factor ρ-Einstein soliton are examined. On a ρ-Einstein gradient soliton warped product manifold, necessary and sufficient conditions for making its factor ρ-Einstein gradient soliton are presented. ρ-Einstein solitons on warped product manifolds admitting a conformal vector field are also considered. Finally, the structure of ρ-Einstein solitons on some warped product space-times is investigated.