On Warped Product Spaces With a Certain Ricci Condition

In this paper, we obtain the criteria that the Riemannian manifold B is Einstein or a gradient Ricci soliton from the information of the second derivative of f in the warped product space RfB with gradient Ricci solitons. Moreover, we construct new examples of non-Einstein gradient Ricci soliton spaces with an Einstein or non-Einstein gradient Ricci soliton leaf using our main theorems. Finally we also get analogous criteria for the Lorentzian version.