The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow g(t). The considered flow in covariant symmetric 2-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of g(t). Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most commonly addressed situation we express the Ricci flow equation in all four orthogonal separable coordinate systems of the plane.

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Abstract. The object of the present paper is to prove the existence of a generalized quasi Einstein spacetime, briefly G.QE/ 4 , by constructing a non-trivial Lorentzian metric and to study such spacetime. First, we prove that every W 2 -Ricci pseudosymmetric G.QE/ 4 is an N.k/-quasi Einstein spacetime which can be considered as a model of perfect fluid, in general relativity. Then, we consider Ricci symmetric G.QE/ 4 and we prove that in such spacetime satisfying Einstein's field equations, the energy density and the isotropic pressure are constants. As a consequence of this result, the expansion scalar and the acceleration vector vanish and also the possible local cosmological structures of this spacetime obeying Einstein's field equations are of Petrov I, D or O.

This paper deals with the study on (m,ρ)‐quasi Einstein manifolds. First, we give some characterizations of an (m,ρ)‐quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an (m,ρ)‐quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form I×eq/2M∗, where I is a real interval, (M∗,g∗) is an (n−1)‐dimensional Riemannian manifold and q is a smooth function on I. Finally, a non‐trivial example of an (m,ρ)‐quasi Einstein manifold verifying our results in terms of the potential function is presented.

We deal with a study of warped product manifold which is also a generalized quasi Einstein manifold. Then, we investigate the relationships between such warped products and certain manifolds that provide some Ricci-Hessian type equations, such as Ric = for some smooth function , where Ric denotes the-Bakery-Emery Ricci tensor. Finally, we obtain some rigidity conditions for such manifolds.

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The main aim of the present article is to study generalized quasi-Yamabe gradient solitons on warped product manifolds. First, we obtain some necessary and sufficient conditions for the existence of generalized quasi-Yamabe gradient solitons equipped on a warped product structure. Then we study some important applications in the Lorentzian and the neutral settings for the particular class, called as gradient Yamabe soliton. More explicitly, we prove the existence of the non-trivial gradient Yamabe soliton on generalized Robertson-Walker spacetimes, standard static spacetimes, Walker manifolds and pp-wave spacetimes.

In this paper, we study the sequential warped product manifolds, which are the natural generalizations of singly warped products. Many spacetime models that characterize the universe and the solutions of Einstein's field equations are known to have this new structure. For this reason, first, we investigate the geometry of sequential warped product manifold under some conditions of concircular curvature tensor. We also study the conformal and gradient almost Ricci solitons on the sequential warped product. These conditions allow us to obtain some interesting expressions for the Riemann curvature and the Ricci tensors of its base and fiber from the geometrical and the physical point of view. Then, we give two important applications of this concept in the Lorentzian settings, which are sequential generalized Robertson-Walker spacetimes and sequential standard static spacetimes and obtain the form of the warping functions. Also, by considering generalized quasi Einsteinian conditions on these spacetimes, we find some specific formulas for the Ricci tensors of the bases and fibers. Finally, we end this work with some examples for this structure.