Communications of the
Korean Mathematical Society CKMS

The object of the present paper is to introduce a type of non-flat Riemannian manifold, called a weakly cyclic generalized $B$-symmetric manifold $(WCGBS)_{n}$. We obtain a sufficient condition for a weakly cyclic generalized $B$-symmetric manifold to be a generalized quasi Einstein manifold. Next we consider conformally flat weakly cyclic generalized $B$-symmetric manifolds. Then we study Einstein $(WCGBS)_{n}$ $(n>2)$. Finally, it is shown that the semi-symmetry and Weyl semi-symmetry are equivalent in such a manifold.

Keywords: Generalized $B$-tensor, generalized quasi-Einstein manifold, Weyl semi-symmetry, conformally-flat manifold, generalized quasi-constant curvature, Einstein manifold

MSC numbers: Primary 53C15, 53C25