Warped product spaces with Ricci conditions

In this paper, we study the Ricci soliton in the Riemannian productsM=Rⁿ \ B and warped products M=R \ f B of theEuclidean space and Riemannian manifolds, and the gradient Riccisoliton in the warped products M=S¹ \ f B of 1-dimensionalcircle and Riemannian manifolds. Moreover, we introduce the concept ofthe generalized Ricci soliton and we suggest the method of constructionof the Riemannian manifold (M, g) with a Ricci soliton g. Finally, we also study the Lorentzian warped products with the Riccisoliton.