Communications of the
Korean Mathematical Society CKMS

The aim of the paper is to study some geometric properties of weakly $Z$-symmetric manifolds. Weakly $Z$-symmetric manifolds with Codazzi type and cyclic parallel $Z$ tensor are studied. We consider Einstein weakly $Z$-symmetric manifolds and conformally flat weakly $Z$-symmetric manifolds. Next, it is shown that a totally umbilical hypersurface of a conformally flat weakly $Z$-symmetric manifolds is of quasi constant curvature. Also, decomposable weakly $Z$-symmetric manifolds are studied and some examples are constructed to support the existence of such manifolds.

Keywords: Weakly $Z$-symmetric manifolds, Einstein weakly $Z$-symmetric manifolds, conformally flat weakly $Z$-symmetric manifolds, decomposable weakly $Z$-sym\-metric manifolds

MSC numbers: Primary 53B20, 53B21, 53C25