Warped product spaces with Ricci conditions

In this paper, we study the Ricci soliton in the Riemannian products M = R-n x B and warped products M = R x (f) B of the Euclidean space and Riemannian manifolds, and the gradient Ricci soliton in the warped products M = S-1 x (f) B of 1-dimensional circle and Riemannian manifolds. Moreover, we introduce the concept of the generalized Ricci soliton and we suggest the method of construction of the Riemannian manifold (M, g) with a Ricci soliton Finally, we also study the Lorentzian warped products with the Ricci soliton.