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.