Kenmotsu manifolds coupled with η-ρ-Einstein soliton admitting extended m-projective curvature tensor

Abstract

The object of the present paper is to study some curvature conditions on Kenmotsu manifolds. Initially, we analyze the condition $\xi$-$\mathcal{M}^{e}$ projective flat and $\varphi$-$\mathcal{M}^{e}$ semi-symmetric on Kenmotsu manifolds coupled with an $\eta$-$\rho$-Einstein soliton. Subsequently, we elaborate the conditions $\mathcal{M}^{e} \cdot\mathcal{R}$=$0$,\, $\mathcal{M}^{e} \cdot \mathcal{M}^{e}$=$0$ and $\mathcal{M}^{e} \cdot \mathcal{Q}$=$0$ on Kenmotsu manifolds in view of an $\eta$-$\rho$-Einstein soliton, where $\mathcal{M}^{e}$ is the extended $\mathcal{M}$-projective curvature tensor. In addition, we verify the results with a concrete example.