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 M^2m+1 is an η-Einstein–Kenmotsu manifold if and only if ∑^m_j=1 T (ɕ(e j ), e j )X = 0. 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-Einstein–Kenmotsu manifold is a Chaki T -pseudosymmetric. Finally, we give an example to verify our results and prove that every three-dimensional Kenmotsu manifold is a generalized pseudo-symmetric manifold.