Some special tensors on the three-dimensional warped product manifolds

In this article, we classify the tensors ℜ, ρ, and ℜ[ρ] on particular class of the three-dimensional warped product manifolds. These tensors are used in the investigation of weakly-Einstein conditions on these manifolds. The concept of warped products is of particular importance in differential geometry and mathematical physics. This concept was first introduced by Bishop and O’Niell to construct examples of Riemannian manifolds with negative curvature. In the following, warped product spaces have been extensively studied and used to construct new manifolds with interesting curvature properties. Also, in Lorentzian geometry, some well-known solutions to Einstein field equations, such as Schwarzschild and Friedmann-Robertson-Walker metrics, can be expressed in terms of warped products. Thus, Lorentzian warped products have been used to obtain more solutions to Einstein field equations. he warped products are of particular importance from a curvature point of view, Since in many cases they are related to the structure of the Codazzi tensors and sometimes locally conformally flat manifolds.