Covariant Derivative of the Curvature Tensor of Kenmotsu Manifolds

In this paper, we define a (1, 3)-tensor field T(X, Y)Z on Kenmotsu manifolds and give a necessary and sufficient condition for T to be a curvature-like tensor. Next, we present some properties related to the curvature-like tensor T and prove that M2m+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^{2m+1}$$\end{document} is an η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}-Einstein–Kenmotsu manifold if and only if ∑j=1mT(φ(ej),ej)X=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum ^{m}_{j=1}T( \varphi (e_j), e_j) X = 0$$\end{document}. Besides, we define a (1, 4)-tensor field t on the Kenmotsu manifold M which determines when M is a Chaki T-pseudo-symmetric manifold. Then, we obtain a formula for the covariant derivative of the curvature tensor of Kenmotsu manifold M. We also find some conditions under which an η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}-Einstein–Kenmotsu manifold is a Chaki T-pseudo-symmetric. Finally, we give an example to verify our results and prove that every three-dimensional Kenmotsu manifold is a generalized pseudo-symmetric manifold.