Pseudo B-symmetric manifolds

In this paper, we introduce a new tensor named B-tensor which generalizes the Z-tensor introduced by Mantica and Suh [Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-B-symmetric manifolds (PBS)(n) which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a (PBS)(n). Next, we prove that a pseudo-Riemannian manifold is B-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a (PBS)(n) to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a (PBS)(n) if the B-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-B-symmetric manifolds and prove that a (PBS)(n) ( n > 3) spacetime is a pp-wave under certain conditions.