Some Characterization Results on Generalized Weakly Symmetric Kenmotsu Manifolds

Abstract

In this paper, we prove that every non-Einstein generalized weakly symmetric Kenmotsu manifold is generalized pseudo-symmetric by showing that $A_i= -2\eta$ and $B_i = D_i = -\eta$ (when $i=1, 2$). Then, we give a necessary condition for Kenmotsu manifolds to be generalized weakly symmetric. We also demonstrate that weakly Ricci-symmetric Kenmotsu manifolds are Einstein. Thereafter, we illustrate that for every generalized weakly Ricci-symmetric Kenmotsu manifold with non-constant scalar curvature which satisfies $X(r) +2 (r + n(n-1))\eta (X) = 0$, the associated 1-forms satisfy $A_1=\frac{A_2}{(n-1)} = -2\eta$ and ${B_1}={D_1} =\frac{B_2}{(n-1)} =\frac{D_2}{(n-1)} =- \eta$. Finally, we give an example which verifies our results obtained in previous sections.