In this article, we define new kind of manifolds, namely pseudo W_2-symmetric manifolds. We first focus on studying the geometric properties of pseudo W_2-symmetric manifold. Pseudo W_2-symmetric spacetimes and Pseudo W_2-symmetric perfect fluid space-times in f(R) theory of gravity are invesitgated. We get in this case the form of the isotropic pressure p. Also, the form of energy density σ is obtained. As final point of study, some energy conditions are studied. The above mentioned studies and many others give us a great motivation to study the geometric properties of pseudo W_2-symmetric manifold and consider Pseudo W_2-symmetric spacetimes and Pseudo W_2-symmetric perfect fluid space-times in f(R) theory of gravity.
We organized this article as follows. I section 1, geometric properties of this manifold are considered. In section 2, general properties of pseudo W_2-symmetric manifold are considered. Next, pseudo W_2-symmetric spacetime and pseudo W_2-symmetric perfect fluid spacetime in f(R) gravity are investigated.