ON THE GEOMETRIC STRUCTURES OF GENERALIZED $(k,\mu)$-SPACE FORMS

Abstract

In this paper, the geometric structures of generalized $(k,\mu)$-space forms and their quasi-umbilical hypersurface are analyzed. First $\xi$-$Q$ and conformally flat generalized $(k,\mu)$-space form are investigated and shown that a conformally flat generalized $(k,\mu)$-space form is Sasakian. Next, we prove that a generalized $(k,\mu)$-space form satisfying Ricci pseudosymmetry and $Q$-Ricci pseudosymmetry conditions is $\eta$-Einstein. We obtain the condition under which a quasi-umbilical hypersurface of a generalized $(k,\mu)$-space form is a generalized quasi Einstein hypersurface. Also $\xi$-sectional curvature of a quasi-umbilical hypersurface of generalized $(k,\mu)$-space form is obtained. Finally, the results obtained are verified by constructing an example of 3-dimensional generalized $(k,\mu)$-space form